LMFDB - Embedded newform 2320.2.a.u.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2320,2,Mod(1,2320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2320, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2320.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2320 = 2^{4} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2320.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:	$18.5252932689$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.580484.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 7x^{3} - 3x^{2} + 8x + 3 $ x^5 - 7x^3 - 3x^2 + 8*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1160)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.92335$ of defining polynomial
Character	$\chi$	$=$	2320.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.37998 q^{3} +1.00000 q^{5} -4.57754 q^{7} +2.66429 q^{9} -5.84671 q^{11} +2.86531 q^{13} +2.37998 q^{15} -6.90957 q^{17} +2.24183 q^{19} -10.8944 q^{21} +0.981401 q^{23} +1.00000 q^{25} -0.798983 q^{27} +1.00000 q^{29} -4.18242 q^{31} -13.9150 q^{33} -4.57754 q^{35} +2.31182 q^{37} +6.81936 q^{39} -7.15507 q^{41} +4.90957 q^{43} +2.66429 q^{45} -6.85016 q^{47} +13.9538 q^{49} -16.4446 q^{51} -6.06632 q^{53} -5.84671 q^{55} +5.33549 q^{57} -10.8281 q^{59} +10.9538 q^{61} -12.1959 q^{63} +2.86531 q^{65} -7.30837 q^{67} +2.33571 q^{69} -9.51991 q^{71} -12.7361 q^{73} +2.37998 q^{75} +26.7635 q^{77} -2.44306 q^{79} -9.89443 q^{81} +9.93524 q^{83} -6.90957 q^{85} +2.37998 q^{87} -9.41711 q^{89} -13.1160 q^{91} -9.95406 q^{93} +2.24183 q^{95} -4.95552 q^{97} -15.5773 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 3 q^{3} + 5 q^{5} - 7 q^{7} + 8 q^{9} - 10 q^{11} + q^{13} - 3 q^{15} - q^{17} - 10 q^{19} - 8 q^{21} - q^{23} + 5 q^{25} - 12 q^{27} + 5 q^{29} - 7 q^{31} - 8 q^{33} - 7 q^{35} - 8 q^{37} - 3 q^{39}+ \cdots - 32 q^{99}+O(q^{100}) $ 5 * q - 3 * q^3 + 5 * q^5 - 7 * q^7 + 8 * q^9 - 10 * q^11 + q^13 - 3 * q^15 - q^17 - 10 * q^19 - 8 * q^21 - q^23 + 5 * q^25 - 12 * q^27 + 5 * q^29 - 7 * q^31 - 8 * q^33 - 7 * q^35 - 8 * q^37 - 3 * q^39 - 4 * q^41 - 9 * q^43 + 8 * q^45 - 8 * q^47 + 16 * q^49 - 2 * q^51 - 9 * q^53 - 10 * q^55 + 2 * q^57 - 29 * q^59 + q^61 - 16 * q^63 + q^65 - 24 * q^67 + 17 * q^69 + 12 * q^71 - 9 * q^73 - 3 * q^75 + 2 * q^77 - 13 * q^79 - 3 * q^81 - 10 * q^83 - q^85 - 3 * q^87 + 4 * q^89 + 4 * q^91 - 26 * q^93 - 10 * q^95 - 15 * q^97 - 32 * q^99

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
$3$	2.37998	1.37408	0.687040	−	0.726619i	$-0.258910\pi$
$3$	2.37998	1.37408	0.687040	+	0.726619i	$0.258910\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−4.57754	−1.73015	−0.865073	−	0.501646i	$-0.832728\pi$
$7$	−4.57754	−1.73015	−0.865073	+	0.501646i	$0.832728\pi$
$8$	0	0
$9$	2.66429	0.888096
$10$	0	0
$11$	−5.84671	−1.76285	−0.881424	−	0.472326i	$-0.843415\pi$
$11$	−5.84671	−1.76285	−0.881424	+	0.472326i	$0.843415\pi$
$12$	0	0
$13$	2.86531	0.794693	0.397346	−	0.917669i	$-0.369931\pi$
$13$	2.86531	0.794693	0.397346	+	0.917669i	$0.369931\pi$
$14$	0	0
$15$	2.37998	0.614507
$16$	0	0
$17$	−6.90957	−1.67582	−0.837909	−	0.545810i	$-0.816222\pi$
$17$	−6.90957	−1.67582	−0.837909	+	0.545810i	$0.816222\pi$
$18$	0	0
$19$	2.24183	0.514310	0.257155	−	0.966370i	$-0.417215\pi$
$19$	2.24183	0.514310	0.257155	+	0.966370i	$0.417215\pi$
$20$	0	0
$21$	−10.8944	−2.37736
$22$	0	0
$23$	0.981401	0.204636	0.102318	−	0.994752i	$-0.467374\pi$
$23$	0.981401	0.204636	0.102318	+	0.994752i	$0.467374\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−0.798983	−0.153764
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−4.18242	−0.751184	−0.375592	−	0.926785i	$-0.622561\pi$
$31$	−4.18242	−0.751184	−0.375592	+	0.926785i	$0.622561\pi$
$32$	0	0
$33$	−13.9150	−2.42230
$34$	0	0
$35$	−4.57754	−0.773745
$36$	0	0
$37$	2.31182	0.380061	0.190031	−	0.981778i	$-0.439141\pi$
$37$	2.31182	0.380061	0.190031	+	0.981778i	$0.439141\pi$
$38$	0	0
$39$	6.81936	1.09197
$40$	0	0
$41$	−7.15507	−1.11743	−0.558717	−	0.829358i	$-0.688706\pi$
$41$	−7.15507	−1.11743	−0.558717	+	0.829358i	$0.688706\pi$
$42$	0	0
$43$	4.90957	0.748703	0.374351	−	0.927287i	$-0.377865\pi$
$43$	4.90957	0.748703	0.374351	+	0.927287i	$0.377865\pi$
$44$	0	0
$45$	2.66429	0.397169
$46$	0	0
$47$	−6.85016	−0.999199	−0.499600	−	0.866256i	$-0.666519\pi$
$47$	−6.85016	−0.999199	−0.499600	+	0.866256i	$0.666519\pi$
$48$	0	0
$49$	13.9538	1.99341
$50$	0	0
$51$	−16.4446	−2.30271
$52$	0	0
$53$	−6.06632	−0.833274	−0.416637	−	0.909073i	$-0.636791\pi$
$53$	−6.06632	−0.833274	−0.416637	+	0.909073i	$0.636791\pi$
$54$	0	0
$55$	−5.84671	−0.788370
$56$	0	0
$57$	5.33549	0.706703
$58$	0	0
$59$	−10.8281	−1.40970	−0.704850	−	0.709357i	$-0.748986\pi$
$59$	−10.8281	−1.40970	−0.704850	+	0.709357i	$0.748986\pi$
$60$	0	0
$61$	10.9538	1.40250	0.701248	−	0.712918i	$-0.252627\pi$
$61$	10.9538	1.40250	0.701248	+	0.712918i	$0.252627\pi$
$62$	0	0
$63$	−12.1959	−1.53654
$64$	0	0
$65$	2.86531	0.355397
$66$	0	0
$67$	−7.30837	−0.892859	−0.446430	−	0.894819i	$-0.647305\pi$
$67$	−7.30837	−0.892859	−0.446430	+	0.894819i	$0.647305\pi$
$68$	0	0
$69$	2.33571	0.281187
$70$	0	0
$71$	−9.51991	−1.12981	−0.564903	−	0.825158i	$-0.691086\pi$
$71$	−9.51991	−1.12981	−0.564903	+	0.825158i	$0.691086\pi$
$72$	0	0
$73$	−12.7361	−1.49064	−0.745322	−	0.666705i	$-0.767704\pi$
$73$	−12.7361	−1.49064	−0.745322	+	0.666705i	$0.767704\pi$
$74$	0	0
$75$	2.37998	0.274816
$76$	0	0
$77$	26.7635	3.04999
$78$	0	0
$79$	−2.44306	−0.274866	−0.137433	−	0.990511i	$-0.543885\pi$
$79$	−2.44306	−0.274866	−0.137433	+	0.990511i	$0.543885\pi$
$80$	0	0
$81$	−9.89443	−1.09938
$82$	0	0
$83$	9.93524	1.09053	0.545267	−	0.838262i	$-0.316428\pi$
$83$	9.93524	1.09053	0.545267	+	0.838262i	$0.316428\pi$
$84$	0	0
$85$	−6.90957	−0.749448
$86$	0	0
$87$	2.37998	0.255160
$88$	0	0
$89$	−9.41711	−0.998212	−0.499106	−	0.866541i	$-0.666338\pi$
$89$	−9.41711	−0.998212	−0.499106	+	0.866541i	$0.666338\pi$
$90$	0	0
$91$	−13.1160	−1.37493
$92$	0	0
$93$	−9.95406	−1.03219
$94$	0	0
$95$	2.24183	0.230006
$96$	0	0
$97$	−4.95552	−0.503156	−0.251578	−	0.967837i	$-0.580950\pi$
$97$	−4.95552	−0.503156	−0.251578	+	0.967837i	$0.580950\pi$
$98$	0	0
$99$	−15.5773	−1.56558
$100$	0	0
$101$	6.25948	0.622842	0.311421	−	0.950272i	$-0.399195\pi$
$101$	6.25948	0.622842	0.311421	+	0.950272i	$0.399195\pi$
$102$	0	0
$103$	7.00178	0.689906	0.344953	−	0.938620i	$-0.387895\pi$
$103$	7.00178	0.689906	0.344953	+	0.938620i	$0.387895\pi$
$104$	0	0
$105$	−10.8944	−1.06319
$106$	0	0
$107$	−0.415332	−0.0401517	−0.0200758	−	0.999798i	$-0.506391\pi$
$107$	−0.415332	−0.0401517	−0.0200758	+	0.999798i	$0.506391\pi$
$108$	0	0
$109$	10.1844	0.975490	0.487745	−	0.872986i	$-0.337820\pi$
$109$	10.1844	0.975490	0.487745	+	0.872986i	$0.337820\pi$
$110$	0	0
$111$	5.50209	0.522235
$112$	0	0
$113$	8.19221	0.770658	0.385329	−	0.922779i	$-0.374088\pi$
$113$	8.19221	0.770658	0.385329	+	0.922779i	$0.374088\pi$
$114$	0	0
$115$	0.981401	0.0915161
$116$	0	0
$117$	7.63400	0.705764
$118$	0	0
$119$	31.6288	2.89941
$120$	0	0
$121$	23.1840	2.10763
$122$	0	0
$123$	−17.0289	−1.53544
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−19.3338	−1.71560	−0.857799	−	0.513985i	$-0.828169\pi$
$127$	−19.3338	−1.71560	−0.857799	+	0.513985i	$0.828169\pi$
$128$	0	0
$129$	11.6847	1.02878
$130$	0	0
$131$	13.4038	1.17110	0.585548	−	0.810638i	$-0.300879\pi$
$131$	13.4038	1.17110	0.585548	+	0.810638i	$0.300879\pi$
$132$	0	0
$133$	−10.2620	−0.889832
$134$	0	0
$135$	−0.798983	−0.0687655
$136$	0	0
$137$	−20.4685	−1.74874	−0.874371	−	0.485257i	$-0.838726\pi$
$137$	−20.4685	−1.74874	−0.874371	+	0.485257i	$0.838726\pi$
$138$	0	0
$139$	−4.09661	−0.347470	−0.173735	−	0.984792i	$-0.555584\pi$
$139$	−4.09661	−0.347470	−0.173735	+	0.984792i	$0.555584\pi$
$140$	0	0
$141$	−16.3032	−1.37298
$142$	0	0
$143$	−16.7526	−1.40092
$144$	0	0
$145$	1.00000	0.0830455
$146$	0	0
$147$	33.2098	2.73910
$148$	0	0
$149$	18.0408	1.47796	0.738978	−	0.673729i	$-0.235309\pi$
$149$	18.0408	1.47796	0.738978	+	0.673729i	$0.235309\pi$
$150$	0	0
$151$	9.73061	0.791866	0.395933	−	0.918279i	$-0.370421\pi$
$151$	9.73061	0.791866	0.395933	+	0.918279i	$0.370421\pi$
$152$	0	0
$153$	−18.4091	−1.48829
$154$	0	0
$155$	−4.18242	−0.335940
$156$	0	0
$157$	8.54358	0.681852	0.340926	−	0.940090i	$-0.389259\pi$
$157$	8.54358	0.681852	0.340926	+	0.940090i	$0.389259\pi$
$158$	0	0
$159$	−14.4377	−1.14498
$160$	0	0
$161$	−4.49240	−0.354050
$162$	0	0
$163$	5.64732	0.442332	0.221166	−	0.975236i	$-0.429014\pi$
$163$	5.64732	0.442332	0.221166	+	0.975236i	$0.429014\pi$
$164$	0	0
$165$	−13.9150	−1.08328
$166$	0	0
$167$	16.7620	1.29708	0.648539	−	0.761181i	$-0.275380\pi$
$167$	16.7620	1.29708	0.648539	+	0.761181i	$0.275380\pi$
$168$	0	0
$169$	−4.79002	−0.368463
$170$	0	0
$171$	5.97287	0.456757
$172$	0	0
$173$	24.4260	1.85707	0.928536	−	0.371242i	$-0.121068\pi$
$173$	24.4260	1.85707	0.928536	+	0.371242i	$0.121068\pi$
$174$	0	0
$175$	−4.57754	−0.346029
$176$	0	0
$177$	−25.7706	−1.93704
$178$	0	0
$179$	−5.17189	−0.386565	−0.193283	−	0.981143i	$-0.561913\pi$
$179$	−5.17189	−0.386565	−0.193283	+	0.981143i	$0.561913\pi$
$180$	0	0
$181$	16.0513	1.19309	0.596543	−	0.802581i	$-0.296540\pi$
$181$	16.0513	1.19309	0.596543	+	0.802581i	$0.296540\pi$
$182$	0	0
$183$	26.0699	1.92714
$184$	0	0
$185$	2.31182	0.169969
$186$	0	0
$187$	40.3982	2.95421
$188$	0	0
$189$	3.65738	0.266035
$190$	0	0
$191$	12.1824	0.881489	0.440744	−	0.897633i	$-0.354715\pi$
$191$	12.1824	0.881489	0.440744	+	0.897633i	$0.354715\pi$
$192$	0	0
$193$	−1.84164	−0.132564	−0.0662819	−	0.997801i	$-0.521114\pi$
$193$	−1.84164	−0.132564	−0.0662819	+	0.997801i	$0.521114\pi$
$194$	0	0
$195$	6.81936	0.488345
$196$	0	0
$197$	15.9372	1.13548	0.567741	−	0.823208i	$-0.307818\pi$
$197$	15.9372	1.13548	0.567741	+	0.823208i	$0.307818\pi$
$198$	0	0
$199$	6.19133	0.438892	0.219446	−	0.975625i	$-0.429575\pi$
$199$	6.19133	0.438892	0.219446	+	0.975625i	$0.429575\pi$
$200$	0	0
$201$	−17.3937	−1.22686
$202$	0	0
$203$	−4.57754	−0.321280
$204$	0	0
$205$	−7.15507	−0.499732
$206$	0	0
$207$	2.61474	0.181737
$208$	0	0
$209$	−13.1073	−0.906651
$210$	0	0
$211$	−17.5401	−1.20751	−0.603756	−	0.797169i	$-0.706330\pi$
$211$	−17.5401	−1.20751	−0.603756	+	0.797169i	$0.706330\pi$
$212$	0	0
$213$	−22.6572	−1.55244
$214$	0	0
$215$	4.90957	0.334830
$216$	0	0
$217$	19.1452	1.29966
$218$	0	0
$219$	−30.3115	−2.04826
$220$	0	0
$221$	−19.7980	−1.33176
$222$	0	0
$223$	18.5013	1.23894	0.619470	−	0.785021i	$-0.287348\pi$
$223$	18.5013	1.23894	0.619470	+	0.785021i	$0.287348\pi$
$224$	0	0
$225$	2.66429	0.177619
$226$	0	0
$227$	−0.507552	−0.0336874	−0.0168437	−	0.999858i	$-0.505362\pi$
$227$	−0.507552	−0.0336874	−0.0168437	+	0.999858i	$0.505362\pi$
$228$	0	0
$229$	−7.17728	−0.474288	−0.237144	−	0.971474i	$-0.576211\pi$
$229$	−7.17728	−0.474288	−0.237144	+	0.971474i	$0.576211\pi$
$230$	0	0
$231$	63.6965	4.19092
$232$	0	0
$233$	−13.2098	−0.865400	−0.432700	−	0.901538i	$-0.642439\pi$
$233$	−13.2098	−0.865400	−0.432700	+	0.901538i	$0.642439\pi$
$234$	0	0
$235$	−6.85016	−0.446855
$236$	0	0
$237$	−5.81443	−0.377688
$238$	0	0
$239$	−20.6378	−1.33495	−0.667474	−	0.744634i	$-0.732624\pi$
$239$	−20.6378	−1.33495	−0.667474	+	0.744634i	$0.732624\pi$
$240$	0	0
$241$	16.8106	1.08287	0.541434	−	0.840744i	$-0.317882\pi$
$241$	16.8106	1.08287	0.541434	+	0.840744i	$0.317882\pi$
$242$	0	0
$243$	−21.1516	−1.35687
$244$	0	0
$245$	13.9538	0.891478
$246$	0	0
$247$	6.42352	0.408719
$248$	0	0
$249$	23.6456	1.49848
$250$	0	0
$251$	22.5993	1.42646	0.713228	−	0.700932i	$-0.247232\pi$
$251$	22.5993	1.42646	0.713228	+	0.700932i	$0.247232\pi$
$252$	0	0
$253$	−5.73796	−0.360743
$254$	0	0
$255$	−16.4446	−1.02980
$256$	0	0
$257$	−15.1551	−0.945347	−0.472674	−	0.881238i	$-0.656711\pi$
$257$	−15.1551	−0.945347	−0.472674	+	0.881238i	$0.656711\pi$
$258$	0	0
$259$	−10.5825	−0.657562
$260$	0	0
$261$	2.66429	0.164915
$262$	0	0
$263$	3.22235	0.198698	0.0993492	−	0.995053i	$-0.468324\pi$
$263$	3.22235	0.198698	0.0993492	+	0.995053i	$0.468324\pi$
$264$	0	0
$265$	−6.06632	−0.372651
$266$	0	0
$267$	−22.4125	−1.37162
$268$	0	0
$269$	−22.1549	−1.35081	−0.675403	−	0.737449i	$-0.736030\pi$
$269$	−22.1549	−1.35081	−0.675403	+	0.737449i	$0.736030\pi$
$270$	0	0
$271$	−22.5690	−1.37097	−0.685485	−	0.728086i	$-0.740410\pi$
$271$	−22.5690	−1.37097	−0.685485	+	0.728086i	$0.740410\pi$
$272$	0	0
$273$	−31.2159	−1.88927
$274$	0	0
$275$	−5.84671	−0.352570
$276$	0	0
$277$	−22.4941	−1.35154	−0.675770	−	0.737112i	$-0.736189\pi$
$277$	−22.4941	−1.35154	−0.675770	+	0.737112i	$0.736189\pi$
$278$	0	0
$279$	−11.1432	−0.667124
$280$	0	0
$281$	−7.31728	−0.436512	−0.218256	−	0.975892i	$-0.570037\pi$
$281$	−7.31728	−0.436512	−0.218256	+	0.975892i	$0.570037\pi$
$282$	0	0
$283$	5.38599	0.320164	0.160082	−	0.987104i	$-0.448824\pi$
$283$	5.38599	0.320164	0.160082	+	0.987104i	$0.448824\pi$
$284$	0	0
$285$	5.33549	0.316047
$286$	0	0
$287$	32.7526	1.93333
$288$	0	0
$289$	30.7422	1.80836
$290$	0	0
$291$	−11.7940	−0.691377
$292$	0	0
$293$	−33.2860	−1.94459	−0.972296	−	0.233755i	$-0.924899\pi$
$293$	−33.2860	−1.94459	−0.972296	+	0.233755i	$0.924899\pi$
$294$	0	0
$295$	−10.8281	−0.630437
$296$	0	0
$297$	4.67142	0.271063
$298$	0	0
$299$	2.81201	0.162623
$300$	0	0
$301$	−22.4737	−1.29537
$302$	0	0
$303$	14.8974	0.855835
$304$	0	0
$305$	10.9538	0.627215
$306$	0	0
$307$	6.43755	0.367411	0.183705	−	0.982981i	$-0.441191\pi$
$307$	6.43755	0.367411	0.183705	+	0.982981i	$0.441191\pi$
$308$	0	0
$309$	16.6641	0.947986
$310$	0	0
$311$	−2.66424	−0.151075	−0.0755375	−	0.997143i	$-0.524067\pi$
$311$	−2.66424	−0.151075	−0.0755375	+	0.997143i	$0.524067\pi$
$312$	0	0
$313$	19.2877	1.09021	0.545103	−	0.838369i	$-0.316490\pi$
$313$	19.2877	1.09021	0.545103	+	0.838369i	$0.316490\pi$
$314$	0	0
$315$	−12.1959	−0.687160
$316$	0	0
$317$	17.7161	0.995038	0.497519	−	0.867453i	$-0.334245\pi$
$317$	17.7161	0.995038	0.497519	+	0.867453i	$0.334245\pi$
$318$	0	0
$319$	−5.84671	−0.327353
$320$	0	0
$321$	−0.988481	−0.0551716
$322$	0	0
$323$	−15.4901	−0.861890
$324$	0	0
$325$	2.86531	0.158939
$326$	0	0
$327$	24.2387	1.34040
$328$	0	0
$329$	31.3569	1.72876
$330$	0	0
$331$	−4.50386	−0.247555	−0.123777	−	0.992310i	$-0.539501\pi$
$331$	−4.50386	−0.247555	−0.123777	+	0.992310i	$0.539501\pi$
$332$	0	0
$333$	6.15937	0.337531
$334$	0	0
$335$	−7.30837	−0.399299
$336$	0	0
$337$	−0.783841	−0.0426985	−0.0213493	−	0.999772i	$-0.506796\pi$
$337$	−0.783841	−0.0426985	−0.0213493	+	0.999772i	$0.506796\pi$
$338$	0	0
$339$	19.4973	1.05895
$340$	0	0
$341$	24.4534	1.32422
$342$	0	0
$343$	−31.8314	−1.71874
$344$	0	0
$345$	2.33571	0.125750
$346$	0	0
$347$	−0.289166	−0.0155232	−0.00776162	−	0.999970i	$-0.502471\pi$
$347$	−0.289166	−0.0155232	−0.00776162	+	0.999970i	$0.502471\pi$
$348$	0	0
$349$	−32.1577	−1.72136	−0.860681	−	0.509145i	$-0.829962\pi$
$349$	−32.1577	−1.72136	−0.860681	+	0.509145i	$0.829962\pi$
$350$	0	0
$351$	−2.28933	−0.122195
$352$	0	0
$353$	−16.2620	−0.865541	−0.432770	−	0.901504i	$-0.642464\pi$
$353$	−16.2620	−0.865541	−0.432770	+	0.901504i	$0.642464\pi$
$354$	0	0
$355$	−9.51991	−0.505264
$356$	0	0
$357$	75.2759	3.98402
$358$	0	0
$359$	22.6340	1.19457	0.597287	−	0.802027i	$-0.296245\pi$
$359$	22.6340	1.19457	0.597287	+	0.802027i	$0.296245\pi$
$360$	0	0
$361$	−13.9742	−0.735485
$362$	0	0
$363$	55.1773	2.89606
$364$	0	0
$365$	−12.7361	−0.666636
$366$	0	0
$367$	−24.6464	−1.28653	−0.643265	−	0.765644i	$-0.722421\pi$
$367$	−24.6464	−1.28653	−0.643265	+	0.765644i	$0.722421\pi$
$368$	0	0
$369$	−19.0632	−0.992390
$370$	0	0
$371$	27.7688	1.44168
$372$	0	0
$373$	−0.239829	−0.0124179	−0.00620894	−	0.999981i	$-0.501976\pi$
$373$	−0.239829	−0.0124179	−0.00620894	+	0.999981i	$0.501976\pi$
$374$	0	0
$375$	2.37998	0.122901
$376$	0	0
$377$	2.86531	0.147571
$378$	0	0
$379$	27.7241	1.42409	0.712046	−	0.702133i	$-0.247769\pi$
$379$	27.7241	1.42409	0.712046	+	0.702133i	$0.247769\pi$
$380$	0	0
$381$	−46.0140	−2.35737
$382$	0	0
$383$	−9.12678	−0.466357	−0.233178	−	0.972434i	$-0.574913\pi$
$383$	−9.12678	−0.466357	−0.233178	+	0.972434i	$0.574913\pi$
$384$	0	0
$385$	26.7635	1.36399
$386$	0	0
$387$	13.0805	0.664920
$388$	0	0
$389$	−39.3904	−1.99717	−0.998586	−	0.0531618i	$-0.983070\pi$
$389$	−39.3904	−1.99717	−0.998586	+	0.0531618i	$0.983070\pi$
$390$	0	0
$391$	−6.78106	−0.342933
$392$	0	0
$393$	31.9008	1.60918
$394$	0	0
$395$	−2.44306	−0.122924
$396$	0	0
$397$	−36.4751	−1.83063	−0.915317	−	0.402733i	$-0.868060\pi$
$397$	−36.4751	−1.83063	−0.915317	+	0.402733i	$0.868060\pi$
$398$	0	0
$399$	−24.4234	−1.22270
$400$	0	0
$401$	16.6845	0.833182	0.416591	−	0.909094i	$-0.363225\pi$
$401$	16.6845	0.833182	0.416591	+	0.909094i	$0.363225\pi$
$402$	0	0
$403$	−11.9839	−0.596961
$404$	0	0
$405$	−9.89443	−0.491658
$406$	0	0
$407$	−13.5166	−0.669991
$408$	0	0
$409$	2.74213	0.135590	0.0677948	−	0.997699i	$-0.478404\pi$
$409$	2.74213	0.135590	0.0677948	+	0.997699i	$0.478404\pi$
$410$	0	0
$411$	−48.7146	−2.40291
$412$	0	0
$413$	49.5661	2.43899
$414$	0	0
$415$	9.93524	0.487702
$416$	0	0
$417$	−9.74983	−0.477451
$418$	0	0
$419$	14.1691	0.692207	0.346103	−	0.938196i	$-0.387505\pi$
$419$	14.1691	0.692207	0.346103	+	0.938196i	$0.387505\pi$
$420$	0	0
$421$	−7.37991	−0.359675	−0.179837	−	0.983696i	$-0.557557\pi$
$421$	−7.37991	−0.359675	−0.179837	+	0.983696i	$0.557557\pi$
$422$	0	0
$423$	−18.2508	−0.887385
$424$	0	0
$425$	−6.90957	−0.335163
$426$	0	0
$427$	−50.1416	−2.42652
$428$	0	0
$429$	−39.8708	−1.92498
$430$	0	0
$431$	0.579533	0.0279151	0.0139576	−	0.999903i	$-0.495557\pi$
$431$	0.579533	0.0279151	0.0139576	+	0.999903i	$0.495557\pi$
$432$	0	0
$433$	5.82082	0.279731	0.139865	−	0.990171i	$-0.455333\pi$
$433$	5.82082	0.279731	0.139865	+	0.990171i	$0.455333\pi$
$434$	0	0
$435$	2.37998	0.114111
$436$	0	0
$437$	2.20013	0.105246
$438$	0	0
$439$	−10.0954	−0.481829	−0.240915	−	0.970546i	$-0.577447\pi$
$439$	−10.0954	−0.481829	−0.240915	+	0.970546i	$0.577447\pi$
$440$	0	0
$441$	37.1771	1.77034
$442$	0	0
$443$	−32.9361	−1.56484	−0.782422	−	0.622749i	$-0.786016\pi$
$443$	−32.9361	−1.56484	−0.782422	+	0.622749i	$0.786016\pi$
$444$	0	0
$445$	−9.41711	−0.446414
$446$	0	0
$447$	42.9366	2.03083
$448$	0	0
$449$	35.3426	1.66792	0.833960	−	0.551824i	$-0.186068\pi$
$449$	35.3426	1.66792	0.833960	+	0.551824i	$0.186068\pi$
$450$	0	0
$451$	41.8336	1.96987
$452$	0	0
$453$	23.1586	1.08809
$454$	0	0
$455$	−13.1160	−0.614890
$456$	0	0
$457$	5.58739	0.261367	0.130684	−	0.991424i	$-0.458283\pi$
$457$	5.58739	0.261367	0.130684	+	0.991424i	$0.458283\pi$
$458$	0	0
$459$	5.52063	0.257681
$460$	0	0
$461$	−8.95201	−0.416937	−0.208468	−	0.978029i	$-0.566848\pi$
$461$	−8.95201	−0.416937	−0.208468	+	0.978029i	$0.566848\pi$
$462$	0	0
$463$	1.43010	0.0664626	0.0332313	−	0.999448i	$-0.489420\pi$
$463$	1.43010	0.0664626	0.0332313	+	0.999448i	$0.489420\pi$
$464$	0	0
$465$	−9.95406	−0.461608
$466$	0	0
$467$	−31.5846	−1.46156	−0.730779	−	0.682614i	$-0.760843\pi$
$467$	−31.5846	−1.46156	−0.730779	+	0.682614i	$0.760843\pi$
$468$	0	0
$469$	33.4543	1.54478
$470$	0	0
$471$	20.3335	0.936919
$472$	0	0
$473$	−28.7048	−1.31985
$474$	0	0
$475$	2.24183	0.102862
$476$	0	0
$477$	−16.1624	−0.740027
$478$	0	0
$479$	19.6550	0.898059	0.449030	−	0.893517i	$-0.351770\pi$
$479$	19.6550	0.898059	0.449030	+	0.893517i	$0.351770\pi$
$480$	0	0
$481$	6.62408	0.302032
$482$	0	0
$483$	−10.6918	−0.486494
$484$	0	0
$485$	−4.95552	−0.225018
$486$	0	0
$487$	3.50225	0.158702	0.0793511	−	0.996847i	$-0.474715\pi$
$487$	3.50225	0.158702	0.0793511	+	0.996847i	$0.474715\pi$
$488$	0	0
$489$	13.4405	0.607800
$490$	0	0
$491$	30.3886	1.37142	0.685709	−	0.727876i	$-0.259492\pi$
$491$	30.3886	1.37142	0.685709	+	0.727876i	$0.259492\pi$
$492$	0	0
$493$	−6.90957	−0.311191
$494$	0	0
$495$	−15.5773	−0.700148
$496$	0	0
$497$	43.5777	1.95473
$498$	0	0
$499$	−39.6591	−1.77539	−0.887693	−	0.460437i	$-0.847693\pi$
$499$	−39.6591	−1.77539	−0.887693	+	0.460437i	$0.847693\pi$
$500$	0	0
$501$	39.8931	1.78229
$502$	0	0
$503$	−27.6775	−1.23408	−0.617039	−	0.786933i	$-0.711668\pi$
$503$	−27.6775	−1.23408	−0.617039	+	0.786933i	$0.711668\pi$
$504$	0	0
$505$	6.25948	0.278543
$506$	0	0
$507$	−11.4001	−0.506298
$508$	0	0
$509$	−22.3009	−0.988470	−0.494235	−	0.869328i	$-0.664552\pi$
$509$	−22.3009	−0.988470	−0.494235	+	0.869328i	$0.664552\pi$
$510$	0	0
$511$	58.2998	2.57903
$512$	0	0
$513$	−1.79118	−0.0790826
$514$	0	0
$515$	7.00178	0.308535
$516$	0	0
$517$	40.0509	1.76144
$518$	0	0
$519$	58.1332	2.55177
$520$	0	0
$521$	−24.6454	−1.07974	−0.539868	−	0.841750i	$-0.681526\pi$
$521$	−24.6454	−1.07974	−0.539868	+	0.841750i	$0.681526\pi$
$522$	0	0
$523$	−2.29608	−0.100401	−0.0502003	−	0.998739i	$-0.515986\pi$
$523$	−2.29608	−0.100401	−0.0502003	+	0.998739i	$0.515986\pi$
$524$	0	0
$525$	−10.8944	−0.475472
$526$	0	0
$527$	28.8987	1.25885
$528$	0	0
$529$	−22.0369	−0.958124
$530$	0	0
$531$	−28.8492	−1.25195
$532$	0	0
$533$	−20.5015	−0.888017
$534$	0	0
$535$	−0.415332	−0.0179564
$536$	0	0
$537$	−12.3090	−0.531172
$538$	0	0
$539$	−81.5840	−3.51407
$540$	0	0
$541$	33.2626	1.43007	0.715035	−	0.699088i	$-0.246411\pi$
$541$	33.2626	1.43007	0.715035	+	0.699088i	$0.246411\pi$
$542$	0	0
$543$	38.2018	1.63940
$544$	0	0
$545$	10.1844	0.436252
$546$	0	0
$547$	−17.9352	−0.766855	−0.383428	−	0.923571i	$-0.625256\pi$
$547$	−17.9352	−0.766855	−0.383428	+	0.923571i	$0.625256\pi$
$548$	0	0
$549$	29.1842	1.24555
$550$	0	0
$551$	2.24183	0.0955050
$552$	0	0
$553$	11.1832	0.475558
$554$	0	0
$555$	5.50209	0.233551
$556$	0	0
$557$	−33.3929	−1.41490	−0.707452	−	0.706761i	$-0.750156\pi$
$557$	−33.3929	−1.41490	−0.707452	+	0.706761i	$0.750156\pi$
$558$	0	0
$559$	14.0674	0.594989
$560$	0	0
$561$	96.1469	4.05932
$562$	0	0
$563$	−2.71685	−0.114501	−0.0572507	−	0.998360i	$-0.518233\pi$
$563$	−2.71685	−0.114501	−0.0572507	+	0.998360i	$0.518233\pi$
$564$	0	0
$565$	8.19221	0.344649
$566$	0	0
$567$	45.2921	1.90209
$568$	0	0
$569$	0.740245	0.0310327	0.0155163	−	0.999880i	$-0.495061\pi$
$569$	0.740245	0.0310327	0.0155163	+	0.999880i	$0.495061\pi$
$570$	0	0
$571$	27.7348	1.16067	0.580333	−	0.814379i	$-0.302922\pi$
$571$	27.7348	1.16067	0.580333	+	0.814379i	$0.302922\pi$
$572$	0	0
$573$	28.9939	1.21124
$574$	0	0
$575$	0.981401	0.0409272
$576$	0	0
$577$	−8.87606	−0.369515	−0.184758	−	0.982784i	$-0.559150\pi$
$577$	−8.87606	−0.369515	−0.184758	+	0.982784i	$0.559150\pi$
$578$	0	0
$579$	−4.38305	−0.182153
$580$	0	0
$581$	−45.4789	−1.88678
$582$	0	0
$583$	35.4680	1.46893
$584$	0	0
$585$	7.63400	0.315627
$586$	0	0
$587$	42.7976	1.76644	0.883222	−	0.468955i	$-0.155369\pi$
$587$	42.7976	1.76644	0.883222	+	0.468955i	$0.155369\pi$
$588$	0	0
$589$	−9.37625	−0.386342
$590$	0	0
$591$	37.9303	1.56024
$592$	0	0
$593$	−32.7635	−1.34544	−0.672718	−	0.739899i	$-0.734873\pi$
$593$	−32.7635	−1.34544	−0.672718	+	0.739899i	$0.734873\pi$
$594$	0	0
$595$	31.6288	1.29666
$596$	0	0
$597$	14.7352	0.603072
$598$	0	0
$599$	−16.6005	−0.678277	−0.339138	−	0.940736i	$-0.610136\pi$
$599$	−16.6005	−0.678277	−0.339138	+	0.940736i	$0.610136\pi$
$600$	0	0
$601$	36.7383	1.49859	0.749294	−	0.662237i	$-0.230393\pi$
$601$	36.7383	1.49859	0.749294	+	0.662237i	$0.230393\pi$
$602$	0	0
$603$	−19.4716	−0.792945
$604$	0	0
$605$	23.1840	0.942563
$606$	0	0
$607$	−37.8761	−1.53735	−0.768673	−	0.639642i	$-0.779083\pi$
$607$	−37.8761	−1.53735	−0.768673	+	0.639642i	$0.779083\pi$
$608$	0	0
$609$	−10.8944	−0.441465
$610$	0	0
$611$	−19.6278	−0.794056
$612$	0	0
$613$	13.3293	0.538366	0.269183	−	0.963089i	$-0.413246\pi$
$613$	13.3293	0.538366	0.269183	+	0.963089i	$0.413246\pi$
$614$	0	0
$615$	−17.0289	−0.686672
$616$	0	0
$617$	−29.8255	−1.20073	−0.600366	−	0.799726i	$-0.704978\pi$
$617$	−29.8255	−1.20073	−0.600366	+	0.799726i	$0.704978\pi$
$618$	0	0
$619$	17.0316	0.684559	0.342279	−	0.939598i	$-0.388801\pi$
$619$	17.0316	0.684559	0.342279	+	0.939598i	$0.388801\pi$
$620$	0	0
$621$	−0.784123	−0.0314658
$622$	0	0
$623$	43.1072	1.72705
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−31.1951	−1.24581
$628$	0	0
$629$	−15.9737	−0.636913
$630$	0	0
$631$	32.6571	1.30006	0.650028	−	0.759910i	$-0.274757\pi$
$631$	32.6571	1.30006	0.650028	+	0.759910i	$0.274757\pi$
$632$	0	0
$633$	−41.7451	−1.65922
$634$	0	0
$635$	−19.3338	−0.767239
$636$	0	0
$637$	39.9820	1.58415
$638$	0	0
$639$	−25.3638	−1.00338
$640$	0	0
$641$	−43.1622	−1.70480	−0.852402	−	0.522887i	$-0.824855\pi$
$641$	−43.1622	−1.70480	−0.852402	+	0.522887i	$0.824855\pi$
$642$	0	0
$643$	−24.8243	−0.978974	−0.489487	−	0.872011i	$-0.662816\pi$
$643$	−24.8243	−0.978974	−0.489487	+	0.872011i	$0.662816\pi$
$644$	0	0
$645$	11.6847	0.460083
$646$	0	0
$647$	31.2444	1.22835	0.614173	−	0.789172i	$-0.289490\pi$
$647$	31.2444	1.22835	0.614173	+	0.789172i	$0.289490\pi$
$648$	0	0
$649$	63.3088	2.48509
$650$	0	0
$651$	45.5651	1.78584
$652$	0	0
$653$	17.5550	0.686980	0.343490	−	0.939156i	$-0.388391\pi$
$653$	17.5550	0.686980	0.343490	+	0.939156i	$0.388391\pi$
$654$	0	0
$655$	13.4038	0.523730
$656$	0	0
$657$	−33.9326	−1.32384
$658$	0	0
$659$	28.9383	1.12728	0.563638	−	0.826022i	$-0.309401\pi$
$659$	28.9383	1.12728	0.563638	+	0.826022i	$0.309401\pi$
$660$	0	0
$661$	−17.9008	−0.696259	−0.348129	−	0.937446i	$-0.613183\pi$
$661$	−17.9008	−0.696259	−0.348129	+	0.937446i	$0.613183\pi$
$662$	0	0
$663$	−47.1189	−1.82995
$664$	0	0
$665$	−10.2620	−0.397945
$666$	0	0
$667$	0.981401	0.0380000
$668$	0	0
$669$	44.0327	1.70240
$670$	0	0
$671$	−64.0439	−2.47239
$672$	0	0
$673$	10.8518	0.418308	0.209154	−	0.977883i	$-0.432929\pi$
$673$	10.8518	0.418308	0.209154	+	0.977883i	$0.432929\pi$
$674$	0	0
$675$	−0.798983	−0.0307529
$676$	0	0
$677$	−24.2851	−0.933352	−0.466676	−	0.884428i	$-0.654549\pi$
$677$	−24.2851	−0.933352	−0.466676	+	0.884428i	$0.654549\pi$
$678$	0	0
$679$	22.6841	0.870534
$680$	0	0
$681$	−1.20796	−0.0462892
$682$	0	0
$683$	−47.5295	−1.81867	−0.909334	−	0.416068i	$-0.863408\pi$
$683$	−47.5295	−1.81867	−0.909334	+	0.416068i	$0.863408\pi$
$684$	0	0
$685$	−20.4685	−0.782062
$686$	0	0
$687$	−17.0818	−0.651710
$688$	0	0
$689$	−17.3819	−0.662197
$690$	0	0
$691$	45.9262	1.74712	0.873558	−	0.486720i	$-0.161807\pi$
$691$	45.9262	1.74712	0.873558	+	0.486720i	$0.161807\pi$
$692$	0	0
$693$	71.3057	2.70868
$694$	0	0
$695$	−4.09661	−0.155393
$696$	0	0
$697$	49.4385	1.87262
$698$	0	0
$699$	−31.4389	−1.18913
$700$	0	0
$701$	11.5300	0.435484	0.217742	−	0.976006i	$-0.430131\pi$
$701$	11.5300	0.435484	0.217742	+	0.976006i	$0.430131\pi$
$702$	0	0
$703$	5.18270	0.195469
$704$	0	0
$705$	−16.3032	−0.614015
$706$	0	0
$707$	−28.6530	−1.07761
$708$	0	0
$709$	13.7237	0.515404	0.257702	−	0.966224i	$-0.417035\pi$
$709$	13.7237	0.515404	0.257702	+	0.966224i	$0.417035\pi$
$710$	0	0
$711$	−6.50902	−0.244107
$712$	0	0
$713$	−4.10463	−0.153719
$714$	0	0
$715$	−16.7526	−0.626512
$716$	0	0
$717$	−49.1174	−1.83432
$718$	0	0
$719$	7.42047	0.276737	0.138368	−	0.990381i	$-0.455814\pi$
$719$	7.42047	0.276737	0.138368	+	0.990381i	$0.455814\pi$
$720$	0	0
$721$	−32.0509	−1.19364
$722$	0	0
$723$	40.0089	1.48795
$724$	0	0
$725$	1.00000	0.0371391
$726$	0	0
$727$	1.52514	0.0565644	0.0282822	−	0.999600i	$-0.490996\pi$
$727$	1.52514	0.0565644	0.0282822	+	0.999600i	$0.490996\pi$
$728$	0	0
$729$	−20.6569	−0.765072
$730$	0	0
$731$	−33.9230	−1.25469
$732$	0	0
$733$	−1.38528	−0.0511664	−0.0255832	−	0.999673i	$-0.508144\pi$
$733$	−1.38528	−0.0511664	−0.0255832	+	0.999673i	$0.508144\pi$
$734$	0	0
$735$	33.2098	1.22496
$736$	0	0
$737$	42.7299	1.57398
$738$	0	0
$739$	50.6505	1.86321	0.931605	−	0.363472i	$-0.118409\pi$
$739$	50.6505	1.86321	0.931605	+	0.363472i	$0.118409\pi$
$740$	0	0
$741$	15.2878	0.561612
$742$	0	0
$743$	8.47107	0.310773	0.155387	−	0.987854i	$-0.450338\pi$
$743$	8.47107	0.310773	0.155387	+	0.987854i	$0.450338\pi$
$744$	0	0
$745$	18.0408	0.660962
$746$	0	0
$747$	26.4704	0.968500
$748$	0	0
$749$	1.90120	0.0694682
$750$	0	0
$751$	19.2566	0.702683	0.351342	−	0.936247i	$-0.385726\pi$
$751$	19.2566	0.702683	0.351342	+	0.936247i	$0.385726\pi$
$752$	0	0
$753$	53.7858	1.96006
$754$	0	0
$755$	9.73061	0.354133
$756$	0	0
$757$	−29.1406	−1.05913	−0.529567	−	0.848268i	$-0.677645\pi$
$757$	−29.1406	−1.05913	−0.529567	+	0.848268i	$0.677645\pi$
$758$	0	0
$759$	−13.6562	−0.495689
$760$	0	0
$761$	13.3242	0.483002	0.241501	−	0.970401i	$-0.422360\pi$
$761$	13.3242	0.483002	0.241501	+	0.970401i	$0.422360\pi$
$762$	0	0
$763$	−46.6195	−1.68774
$764$	0	0
$765$	−18.4091	−0.665582
$766$	0	0
$767$	−31.0258	−1.12028
$768$	0	0
$769$	3.75578	0.135437	0.0677185	−	0.997704i	$-0.478428\pi$
$769$	3.75578	0.135437	0.0677185	+	0.997704i	$0.478428\pi$
$770$	0	0
$771$	−36.0687	−1.29898
$772$	0	0
$773$	−17.9841	−0.646842	−0.323421	−	0.946255i	$-0.604833\pi$
$773$	−17.9841	−0.646842	−0.323421	+	0.946255i	$0.604833\pi$
$774$	0	0
$775$	−4.18242	−0.150237
$776$	0	0
$777$	−25.1860	−0.903543
$778$	0	0
$779$	−16.0404	−0.574708
$780$	0	0
$781$	55.6601	1.99168
$782$	0	0
$783$	−0.798983	−0.0285533
$784$	0	0
$785$	8.54358	0.304933
$786$	0	0
$787$	−39.9640	−1.42456	−0.712281	−	0.701895i	$-0.752338\pi$
$787$	−39.9640	−1.42456	−0.712281	+	0.701895i	$0.752338\pi$
$788$	0	0
$789$	7.66911	0.273028
$790$	0	0
$791$	−37.5001	−1.33335
$792$	0	0
$793$	31.3861	1.11455
$794$	0	0
$795$	−14.4377	−0.512053
$796$	0	0
$797$	−20.6216	−0.730456	−0.365228	−	0.930918i	$-0.619009\pi$
$797$	−20.6216	−0.730456	−0.365228	+	0.930918i	$0.619009\pi$
$798$	0	0
$799$	47.3317	1.67448
$800$	0	0
$801$	−25.0899	−0.886508
$802$	0	0
$803$	74.4640	2.62778
$804$	0	0
$805$	−4.49240	−0.158336
$806$	0	0
$807$	−52.7280	−1.85611
$808$	0	0
$809$	−14.0885	−0.495326	−0.247663	−	0.968846i	$-0.579663\pi$
$809$	−14.0885	−0.495326	−0.247663	+	0.968846i	$0.579663\pi$
$810$	0	0
$811$	2.25084	0.0790377	0.0395188	−	0.999219i	$-0.487417\pi$
$811$	2.25084	0.0790377	0.0395188	+	0.999219i	$0.487417\pi$
$812$	0	0
$813$	−53.7138	−1.88382
$814$	0	0
$815$	5.64732	0.197817
$816$	0	0
$817$	11.0064	0.385065
$818$	0	0
$819$	−34.9449	−1.22107
$820$	0	0
$821$	37.9664	1.32504	0.662518	−	0.749046i	$-0.269488\pi$
$821$	37.9664	1.32504	0.662518	+	0.749046i	$0.269488\pi$
$822$	0	0
$823$	−14.9282	−0.520365	−0.260183	−	0.965559i	$-0.583783\pi$
$823$	−14.9282	−0.520365	−0.260183	+	0.965559i	$0.583783\pi$
$824$	0	0
$825$	−13.9150	−0.484459
$826$	0	0
$827$	27.8054	0.966888	0.483444	−	0.875375i	$-0.339386\pi$
$827$	27.8054	0.966888	0.483444	+	0.875375i	$0.339386\pi$
$828$	0	0
$829$	14.5463	0.505213	0.252607	−	0.967569i	$-0.418712\pi$
$829$	14.5463	0.505213	0.252607	+	0.967569i	$0.418712\pi$
$830$	0	0
$831$	−53.5355	−1.85713
$832$	0	0
$833$	−96.4151	−3.34058
$834$	0	0
$835$	16.7620	0.580071
$836$	0	0
$837$	3.34168	0.115505
$838$	0	0
$839$	32.8687	1.13475	0.567377	−	0.823458i	$-0.307958\pi$
$839$	32.8687	1.13475	0.567377	+	0.823458i	$0.307958\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−17.4149	−0.599803
$844$	0	0
$845$	−4.79002	−0.164782
$846$	0	0
$847$	−106.126	−3.64652
$848$	0	0
$849$	12.8185	0.439931
$850$	0	0
$851$	2.26882	0.0777743
$852$	0	0
$853$	−16.3843	−0.560989	−0.280494	−	0.959856i	$-0.590498\pi$
$853$	−16.3843	−0.560989	−0.280494	+	0.959856i	$0.590498\pi$
$854$	0	0
$855$	5.97287	0.204268
$856$	0	0
$857$	26.1985	0.894925	0.447463	−	0.894303i	$-0.352328\pi$
$857$	26.1985	0.894925	0.447463	+	0.894303i	$0.352328\pi$
$858$	0	0
$859$	−40.1498	−1.36989	−0.684947	−	0.728593i	$-0.740175\pi$
$859$	−40.1498	−1.36989	−0.684947	+	0.728593i	$0.740175\pi$
$860$	0	0
$861$	77.9504	2.65654
$862$	0	0
$863$	−2.19808	−0.0748234	−0.0374117	−	0.999300i	$-0.511911\pi$
$863$	−2.19808	−0.0748234	−0.0374117	+	0.999300i	$0.511911\pi$
$864$	0	0
$865$	24.4260	0.830508
$866$	0	0
$867$	73.1657	2.48484
$868$	0	0
$869$	14.2839	0.484547
$870$	0	0
$871$	−20.9407	−0.709549
$872$	0	0
$873$	−13.2029	−0.446851
$874$	0	0
$875$	−4.57754	−0.154749
$876$	0	0
$877$	−18.3093	−0.618263	−0.309131	−	0.951019i	$-0.600038\pi$
$877$	−18.3093	−0.618263	−0.309131	+	0.951019i	$0.600038\pi$
$878$	0	0
$879$	−79.2200	−2.67202
$880$	0	0
$881$	−30.9107	−1.04141	−0.520704	−	0.853737i	$-0.674331\pi$
$881$	−30.9107	−1.04141	−0.520704	+	0.853737i	$0.674331\pi$
$882$	0	0
$883$	−49.2370	−1.65696	−0.828478	−	0.560021i	$-0.810793\pi$
$883$	−49.2370	−1.65696	−0.828478	+	0.560021i	$0.810793\pi$
$884$	0	0
$885$	−25.7706	−0.866271
$886$	0	0
$887$	13.3812	0.449295	0.224648	−	0.974440i	$-0.427877\pi$
$887$	13.3812	0.449295	0.224648	+	0.974440i	$0.427877\pi$
$888$	0	0
$889$	88.5012	2.96824
$890$	0	0
$891$	57.8498	1.93804
$892$	0	0
$893$	−15.3569	−0.513898
$894$	0	0
$895$	−5.17189	−0.172877
$896$	0	0
$897$	6.69253	0.223457
$898$	0	0
$899$	−4.18242	−0.139491
$900$	0	0
$901$	41.9157	1.39641
$902$	0	0
$903$	−53.4870	−1.77994
$904$	0	0
$905$	16.0513	0.533564
$906$	0	0
$907$	38.1302	1.26609	0.633047	−	0.774113i	$-0.281804\pi$
$907$	38.1302	1.26609	0.633047	+	0.774113i	$0.281804\pi$
$908$	0	0
$909$	16.6771	0.553144
$910$	0	0
$911$	−32.4907	−1.07647	−0.538233	−	0.842796i	$-0.680908\pi$
$911$	−32.4907	−1.07647	−0.538233	+	0.842796i	$0.680908\pi$
$912$	0	0
$913$	−58.0884	−1.92245
$914$	0	0
$915$	26.0699	0.861844
$916$	0	0
$917$	−61.3564	−2.02617
$918$	0	0
$919$	25.7117	0.848150	0.424075	−	0.905627i	$-0.360599\pi$
$919$	25.7117	0.848150	0.424075	+	0.905627i	$0.360599\pi$
$920$	0	0
$921$	15.3212	0.504852
$922$	0	0
$923$	−27.2774	−0.897848
$924$	0	0
$925$	2.31182	0.0760123
$926$	0	0
$927$	18.6548	0.612703
$928$	0	0
$929$	−50.4591	−1.65551	−0.827755	−	0.561090i	$-0.810382\pi$
$929$	−50.4591	−1.65551	−0.827755	+	0.561090i	$0.810382\pi$
$930$	0	0
$931$	31.2821	1.02523
$932$	0	0
$933$	−6.34082	−0.207589
$934$	0	0
$935$	40.3982	1.32116
$936$	0	0
$937$	−9.03333	−0.295106	−0.147553	−	0.989054i	$-0.547140\pi$
$937$	−9.03333	−0.295106	−0.147553	+	0.989054i	$0.547140\pi$
$938$	0	0
$939$	45.9043	1.49803
$940$	0	0
$941$	−9.74488	−0.317674	−0.158837	−	0.987305i	$-0.550774\pi$
$941$	−9.74488	−0.317674	−0.158837	+	0.987305i	$0.550774\pi$
$942$	0	0
$943$	−7.02199	−0.228668
$944$	0	0
$945$	3.65738	0.118974
$946$	0	0
$947$	21.8008	0.708430	0.354215	−	0.935164i	$-0.384748\pi$
$947$	21.8008	0.708430	0.354215	+	0.935164i	$0.384748\pi$
$948$	0	0
$949$	−36.4927	−1.18460
$950$	0	0
$951$	42.1640	1.36726
$952$	0	0
$953$	−48.6037	−1.57443	−0.787214	−	0.616680i	$-0.788477\pi$
$953$	−48.6037	−1.57443	−0.787214	+	0.616680i	$0.788477\pi$
$954$	0	0
$955$	12.1824	0.394214
$956$	0	0
$957$	−13.9150	−0.449809
$958$	0	0
$959$	93.6953	3.02558
$960$	0	0
$961$	−13.5074	−0.435722
$962$	0	0
$963$	−1.10656	−0.0356586
$964$	0	0
$965$	−1.84164	−0.0592844
$966$	0	0
$967$	−24.4922	−0.787617	−0.393809	−	0.919192i	$-0.628843\pi$
$967$	−24.4922	−0.787617	−0.393809	+	0.919192i	$0.628843\pi$
$968$	0	0
$969$	−36.8660	−1.18431
$970$	0	0
$971$	−43.6840	−1.40189	−0.700943	−	0.713217i	$-0.747237\pi$
$971$	−43.6840	−1.40189	−0.700943	+	0.713217i	$0.747237\pi$
$972$	0	0
$973$	18.7524	0.601173
$974$	0	0
$975$	6.81936	0.218394
$976$	0	0
$977$	33.4898	1.07143	0.535717	−	0.844397i	$-0.320041\pi$
$977$	33.4898	1.07143	0.535717	+	0.844397i	$0.320041\pi$
$978$	0	0
$979$	55.0591	1.75970
$980$	0	0
$981$	27.1342	0.866329
$982$	0	0
$983$	−27.0615	−0.863128	−0.431564	−	0.902082i	$-0.642038\pi$
$983$	−27.0615	−0.863128	−0.431564	+	0.902082i	$0.642038\pi$
$984$	0	0
$985$	15.9372	0.507803
$986$	0	0
$987$	74.6286	2.37546
$988$	0	0
$989$	4.81826	0.153212
$990$	0	0
$991$	50.1984	1.59461	0.797303	−	0.603579i	$-0.206259\pi$
$991$	50.1984	1.59461	0.797303	+	0.603579i	$0.206259\pi$
$992$	0	0
$993$	−10.7191	−0.340160
$994$	0	0
$995$	6.19133	0.196278
$996$	0	0
$997$	8.13064	0.257500	0.128750	−	0.991677i	$-0.458904\pi$
$997$	8.13064	0.257500	0.128750	+	0.991677i	$0.458904\pi$
$998$	0	0
$999$	−1.84711	−0.0584399

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2320.2.a.u.1.5		5
4.3	odd	2		1160.2.a.j.1.1	✓	5
8.3	odd	2		9280.2.a.cg.1.5		5
8.5	even	2		9280.2.a.cl.1.1		5
20.19	odd	2		5800.2.a.t.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1160.2.a.j.1.1	✓	5	4.3	odd	2
2320.2.a.u.1.5		5	1.1	even	1	trivial
5800.2.a.t.1.5		5	20.19	odd	2
9280.2.a.cg.1.5		5	8.3	odd	2
9280.2.a.cl.1.1		5	8.5	even	2

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:	\( N \)	\(=\)	\( 2320 = 2^{4} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2320.a (trivial)

\(f(q)\)	\(=\)	\(q+2.37998 q^{3} +1.00000 q^{5} -4.57754 q^{7} +2.66429 q^{9} -5.84671 q^{11} +2.86531 q^{13} +2.37998 q^{15} -6.90957 q^{17} +2.24183 q^{19} -10.8944 q^{21} +0.981401 q^{23} +1.00000 q^{25} -0.798983 q^{27} +1.00000 q^{29} -4.18242 q^{31} -13.9150 q^{33} -4.57754 q^{35} +2.31182 q^{37} +6.81936 q^{39} -7.15507 q^{41} +4.90957 q^{43} +2.66429 q^{45} -6.85016 q^{47} +13.9538 q^{49} -16.4446 q^{51} -6.06632 q^{53} -5.84671 q^{55} +5.33549 q^{57} -10.8281 q^{59} +10.9538 q^{61} -12.1959 q^{63} +2.86531 q^{65} -7.30837 q^{67} +2.33571 q^{69} -9.51991 q^{71} -12.7361 q^{73} +2.37998 q^{75} +26.7635 q^{77} -2.44306 q^{79} -9.89443 q^{81} +9.93524 q^{83} -6.90957 q^{85} +2.37998 q^{87} -9.41711 q^{89} -13.1160 q^{91} -9.95406 q^{93} +2.24183 q^{95} -4.95552 q^{97} -15.5773 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 3 q^{3} + 5 q^{5} - 7 q^{7} + 8 q^{9} - 10 q^{11} + q^{13} - 3 q^{15} - q^{17} - 10 q^{19} - 8 q^{21} - q^{23} + 5 q^{25} - 12 q^{27} + 5 q^{29} - 7 q^{31} - 8 q^{33} - 7 q^{35} - 8 q^{37} - 3 q^{39}+ \cdots - 32 q^{99}+O(q^{100}) \) 5 * q - 3 * q^3 + 5 * q^5 - 7 * q^7 + 8 * q^9 - 10 * q^11 + q^13 - 3 * q^15 - q^17 - 10 * q^19 - 8 * q^21 - q^23 + 5 * q^25 - 12 * q^27 + 5 * q^29 - 7 * q^31 - 8 * q^33 - 7 * q^35 - 8 * q^37 - 3 * q^39 - 4 * q^41 - 9 * q^43 + 8 * q^45 - 8 * q^47 + 16 * q^49 - 2 * q^51 - 9 * q^53 - 10 * q^55 + 2 * q^57 - 29 * q^59 + q^61 - 16 * q^63 + q^65 - 24 * q^67 + 17 * q^69 + 12 * q^71 - 9 * q^73 - 3 * q^75 + 2 * q^77 - 13 * q^79 - 3 * q^81 - 10 * q^83 - q^85 - 3 * q^87 + 4 * q^89 + 4 * q^91 - 26 * q^93 - 10 * q^95 - 15 * q^97 - 32 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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