LMFDB - Embedded newform 5520.2.a.bs.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5520,2,Mod(1,5520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5520, 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("5520.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5520.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:	$44.0774219157$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 690)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	5520.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} +5.12311 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} +5.12311 q^{7} +1.00000 q^{9} -5.12311 q^{11} +2.00000 q^{13} +1.00000 q^{15} +7.12311 q^{17} -4.00000 q^{19} +5.12311 q^{21} -1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +2.00000 q^{29} -5.12311 q^{33} +5.12311 q^{35} -7.12311 q^{37} +2.00000 q^{39} +2.00000 q^{41} +1.00000 q^{45} +8.00000 q^{47} +19.2462 q^{49} +7.12311 q^{51} -4.24621 q^{53} -5.12311 q^{55} -4.00000 q^{57} +14.2462 q^{59} +0.876894 q^{61} +5.12311 q^{63} +2.00000 q^{65} +8.00000 q^{67} -1.00000 q^{69} -6.24621 q^{71} +12.2462 q^{73} +1.00000 q^{75} -26.2462 q^{77} +5.12311 q^{79} +1.00000 q^{81} -11.3693 q^{83} +7.12311 q^{85} +2.00000 q^{87} +3.12311 q^{89} +10.2462 q^{91} -4.00000 q^{95} +0.246211 q^{97} -5.12311 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} - 2 q^{11} + 4 q^{13} + 2 q^{15} + 6 q^{17} - 8 q^{19} + 2 q^{21} - 2 q^{23} + 2 q^{25} + 2 q^{27} + 4 q^{29} - 2 q^{33} + 2 q^{35} - 6 q^{37} + 4 q^{39} + 4 q^{41} + 2 q^{45} + 16 q^{47} + 22 q^{49} + 6 q^{51} + 8 q^{53} - 2 q^{55} - 8 q^{57} + 12 q^{59} + 10 q^{61} + 2 q^{63} + 4 q^{65} + 16 q^{67} - 2 q^{69} + 4 q^{71} + 8 q^{73} + 2 q^{75} - 36 q^{77} + 2 q^{79} + 2 q^{81} + 2 q^{83} + 6 q^{85} + 4 q^{87} - 2 q^{89} + 4 q^{91} - 8 q^{95} - 16 q^{97} - 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9 - 2 * q^11 + 4 * q^13 + 2 * q^15 + 6 * q^17 - 8 * q^19 + 2 * q^21 - 2 * q^23 + 2 * q^25 + 2 * q^27 + 4 * q^29 - 2 * q^33 + 2 * q^35 - 6 * q^37 + 4 * q^39 + 4 * q^41 + 2 * q^45 + 16 * q^47 + 22 * q^49 + 6 * q^51 + 8 * q^53 - 2 * q^55 - 8 * q^57 + 12 * q^59 + 10 * q^61 + 2 * q^63 + 4 * q^65 + 16 * q^67 - 2 * q^69 + 4 * q^71 + 8 * q^73 + 2 * q^75 - 36 * q^77 + 2 * q^79 + 2 * q^81 + 2 * q^83 + 6 * q^85 + 4 * q^87 - 2 * q^89 + 4 * q^91 - 8 * q^95 - 16 * q^97 - 2 * 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
$7$	5.12311	1.93635	0.968176	−	0.250270i	$-0.0805195\pi$
$7$	5.12311	1.93635	0.968176	+	0.250270i	$0.0805195\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.12311	−1.54467	−0.772337	−	0.635213i	$-0.780912\pi$
$11$	−5.12311	−1.54467	−0.772337	+	0.635213i	$0.780912\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	7.12311	1.72761	0.863803	−	0.503829i	$-0.168076\pi$
$17$	7.12311	1.72761	0.863803	+	0.503829i	$0.168076\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	5.12311	1.11795
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	−5.12311	−0.891818
$34$	0	0
$35$	5.12311	0.865963
$36$	0	0
$37$	−7.12311	−1.17103	−0.585516	−	0.810661i	$-0.699108\pi$
$37$	−7.12311	−1.17103	−0.585516	+	0.810661i	$0.699108\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	19.2462	2.74946
$50$	0	0
$51$	7.12311	0.997434
$52$	0	0
$53$	−4.24621	−0.583262	−0.291631	−	0.956531i	$-0.594198\pi$
$53$	−4.24621	−0.583262	−0.291631	+	0.956531i	$0.594198\pi$
$54$	0	0
$55$	−5.12311	−0.690799
$56$	0	0
$57$	−4.00000	−0.529813
$58$	0	0
$59$	14.2462	1.85470	0.927349	−	0.374197i	$-0.122082\pi$
$59$	14.2462	1.85470	0.927349	+	0.374197i	$0.122082\pi$
$60$	0	0
$61$	0.876894	0.112275	0.0561374	−	0.998423i	$-0.482122\pi$
$61$	0.876894	0.112275	0.0561374	+	0.998423i	$0.482122\pi$
$62$	0	0
$63$	5.12311	0.645451
$64$	0	0
$65$	2.00000	0.248069
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−6.24621	−0.741289	−0.370644	−	0.928775i	$-0.620863\pi$
$71$	−6.24621	−0.741289	−0.370644	+	0.928775i	$0.620863\pi$
$72$	0	0
$73$	12.2462	1.43331	0.716655	−	0.697428i	$-0.245672\pi$
$73$	12.2462	1.43331	0.716655	+	0.697428i	$0.245672\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−26.2462	−2.99103
$78$	0	0
$79$	5.12311	0.576394	0.288197	−	0.957571i	$-0.406944\pi$
$79$	5.12311	0.576394	0.288197	+	0.957571i	$0.406944\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−11.3693	−1.24794	−0.623972	−	0.781446i	$-0.714482\pi$
$83$	−11.3693	−1.24794	−0.623972	+	0.781446i	$0.714482\pi$
$84$	0	0
$85$	7.12311	0.772609
$86$	0	0
$87$	2.00000	0.214423
$88$	0	0
$89$	3.12311	0.331049	0.165524	−	0.986206i	$-0.447068\pi$
$89$	3.12311	0.331049	0.165524	+	0.986206i	$0.447068\pi$
$90$	0	0
$91$	10.2462	1.07409
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−4.00000	−0.410391
$96$	0	0
$97$	0.246211	0.0249990	0.0124995	−	0.999922i	$-0.496021\pi$
$97$	0.246211	0.0249990	0.0124995	+	0.999922i	$0.496021\pi$
$98$	0	0
$99$	−5.12311	−0.514891
$100$	0	0
$101$	−16.2462	−1.61656	−0.808279	−	0.588799i	$-0.799601\pi$
$101$	−16.2462	−1.61656	−0.808279	+	0.588799i	$0.799601\pi$
$102$	0	0
$103$	−15.3693	−1.51438	−0.757192	−	0.653192i	$-0.773429\pi$
$103$	−15.3693	−1.51438	−0.757192	+	0.653192i	$0.773429\pi$
$104$	0	0
$105$	5.12311	0.499964
$106$	0	0
$107$	−11.3693	−1.09911	−0.549557	−	0.835456i	$-0.685203\pi$
$107$	−11.3693	−1.09911	−0.549557	+	0.835456i	$0.685203\pi$
$108$	0	0
$109$	11.1231	1.06540	0.532700	−	0.846304i	$-0.321177\pi$
$109$	11.1231	1.06540	0.532700	+	0.846304i	$0.321177\pi$
$110$	0	0
$111$	−7.12311	−0.676095
$112$	0	0
$113$	−0.876894	−0.0824913	−0.0412456	−	0.999149i	$-0.513133\pi$
$113$	−0.876894	−0.0824913	−0.0412456	+	0.999149i	$0.513133\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	36.4924	3.34525
$120$	0	0
$121$	15.2462	1.38602
$122$	0	0
$123$	2.00000	0.180334
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	22.2462	1.97403	0.987016	−	0.160622i	$-0.0513500\pi$
$127$	22.2462	1.97403	0.987016	+	0.160622i	$0.0513500\pi$
$128$	0	0
$129$	0	0
$130$	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
$133$	−20.4924	−1.77692
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−19.1231	−1.63380	−0.816899	−	0.576781i	$-0.804308\pi$
$137$	−19.1231	−1.63380	−0.816899	+	0.576781i	$0.804308\pi$
$138$	0	0
$139$	16.4924	1.39887	0.699435	−	0.714697i	$-0.253435\pi$
$139$	16.4924	1.39887	0.699435	+	0.714697i	$0.253435\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	0	0
$143$	−10.2462	−0.856831
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	19.2462	1.58740
$148$	0	0
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	10.2462	0.833825	0.416912	−	0.908947i	$-0.363112\pi$
$151$	10.2462	0.833825	0.416912	+	0.908947i	$0.363112\pi$
$152$	0	0
$153$	7.12311	0.575869
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0.876894	0.0699838	0.0349919	−	0.999388i	$-0.488859\pi$
$157$	0.876894	0.0699838	0.0349919	+	0.999388i	$0.488859\pi$
$158$	0	0
$159$	−4.24621	−0.336746
$160$	0	0
$161$	−5.12311	−0.403757
$162$	0	0
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	0	0
$165$	−5.12311	−0.398833
$166$	0	0
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−4.00000	−0.305888
$172$	0	0
$173$	3.75379	0.285395	0.142698	−	0.989766i	$-0.454422\pi$
$173$	3.75379	0.285395	0.142698	+	0.989766i	$0.454422\pi$
$174$	0	0
$175$	5.12311	0.387270
$176$	0	0
$177$	14.2462	1.07081
$178$	0	0
$179$	−24.4924	−1.83065	−0.915325	−	0.402716i	$-0.868066\pi$
$179$	−24.4924	−1.83065	−0.915325	+	0.402716i	$0.868066\pi$
$180$	0	0
$181$	19.1231	1.42141	0.710705	−	0.703491i	$-0.248376\pi$
$181$	19.1231	1.42141	0.710705	+	0.703491i	$0.248376\pi$
$182$	0	0
$183$	0.876894	0.0648219
$184$	0	0
$185$	−7.12311	−0.523701
$186$	0	0
$187$	−36.4924	−2.66859
$188$	0	0
$189$	5.12311	0.372651
$190$	0	0
$191$	20.4924	1.48278	0.741390	−	0.671075i	$-0.234167\pi$
$191$	20.4924	1.48278	0.741390	+	0.671075i	$0.234167\pi$
$192$	0	0
$193$	−8.24621	−0.593575	−0.296788	−	0.954944i	$-0.595915\pi$
$193$	−8.24621	−0.593575	−0.296788	+	0.954944i	$0.595915\pi$
$194$	0	0
$195$	2.00000	0.143223
$196$	0	0
$197$	−10.0000	−0.712470	−0.356235	−	0.934396i	$-0.615940\pi$
$197$	−10.0000	−0.712470	−0.356235	+	0.934396i	$0.615940\pi$
$198$	0	0
$199$	−10.8769	−0.771043	−0.385521	−	0.922699i	$-0.625978\pi$
$199$	−10.8769	−0.771043	−0.385521	+	0.922699i	$0.625978\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	0	0
$203$	10.2462	0.719143
$204$	0	0
$205$	2.00000	0.139686
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	20.4924	1.41749
$210$	0	0
$211$	−16.4924	−1.13539	−0.567693	−	0.823241i	$-0.692164\pi$
$211$	−16.4924	−1.13539	−0.567693	+	0.823241i	$0.692164\pi$
$212$	0	0
$213$	−6.24621	−0.427983
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	12.2462	0.827522
$220$	0	0
$221$	14.2462	0.958304
$222$	0	0
$223$	−6.24621	−0.418277	−0.209139	−	0.977886i	$-0.567066\pi$
$223$	−6.24621	−0.418277	−0.209139	+	0.977886i	$0.567066\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−19.3693	−1.28559	−0.642793	−	0.766040i	$-0.722225\pi$
$227$	−19.3693	−1.28559	−0.642793	+	0.766040i	$0.722225\pi$
$228$	0	0
$229$	−1.36932	−0.0904870	−0.0452435	−	0.998976i	$-0.514406\pi$
$229$	−1.36932	−0.0904870	−0.0452435	+	0.998976i	$0.514406\pi$
$230$	0	0
$231$	−26.2462	−1.72687
$232$	0	0
$233$	−20.7386	−1.35863	−0.679317	−	0.733845i	$-0.737724\pi$
$233$	−20.7386	−1.35863	−0.679317	+	0.733845i	$0.737724\pi$
$234$	0	0
$235$	8.00000	0.521862
$236$	0	0
$237$	5.12311	0.332781
$238$	0	0
$239$	1.75379	0.113443	0.0567216	−	0.998390i	$-0.481935\pi$
$239$	1.75379	0.113443	0.0567216	+	0.998390i	$0.481935\pi$
$240$	0	0
$241$	−2.49242	−0.160551	−0.0802755	−	0.996773i	$-0.525580\pi$
$241$	−2.49242	−0.160551	−0.0802755	+	0.996773i	$0.525580\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	19.2462	1.22960
$246$	0	0
$247$	−8.00000	−0.509028
$248$	0	0
$249$	−11.3693	−0.720501
$250$	0	0
$251$	7.36932	0.465147	0.232574	−	0.972579i	$-0.425285\pi$
$251$	7.36932	0.465147	0.232574	+	0.972579i	$0.425285\pi$
$252$	0	0
$253$	5.12311	0.322087
$254$	0	0
$255$	7.12311	0.446066
$256$	0	0
$257$	−8.24621	−0.514385	−0.257192	−	0.966360i	$-0.582797\pi$
$257$	−8.24621	−0.514385	−0.257192	+	0.966360i	$0.582797\pi$
$258$	0	0
$259$	−36.4924	−2.26753
$260$	0	0
$261$	2.00000	0.123797
$262$	0	0
$263$	−10.2462	−0.631808	−0.315904	−	0.948791i	$-0.602308\pi$
$263$	−10.2462	−0.631808	−0.315904	+	0.948791i	$0.602308\pi$
$264$	0	0
$265$	−4.24621	−0.260843
$266$	0	0
$267$	3.12311	0.191131
$268$	0	0
$269$	−16.2462	−0.990549	−0.495274	−	0.868737i	$-0.664933\pi$
$269$	−16.2462	−0.990549	−0.495274	+	0.868737i	$0.664933\pi$
$270$	0	0
$271$	−10.2462	−0.622413	−0.311207	−	0.950342i	$-0.600733\pi$
$271$	−10.2462	−0.622413	−0.311207	+	0.950342i	$0.600733\pi$
$272$	0	0
$273$	10.2462	0.620129
$274$	0	0
$275$	−5.12311	−0.308935
$276$	0	0
$277$	20.2462	1.21648	0.608238	−	0.793754i	$-0.291876\pi$
$277$	20.2462	1.21648	0.608238	+	0.793754i	$0.291876\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−12.8769	−0.768171	−0.384086	−	0.923298i	$-0.625483\pi$
$281$	−12.8769	−0.768171	−0.384086	+	0.923298i	$0.625483\pi$
$282$	0	0
$283$	2.24621	0.133523	0.0667617	−	0.997769i	$-0.478733\pi$
$283$	2.24621	0.133523	0.0667617	+	0.997769i	$0.478733\pi$
$284$	0	0
$285$	−4.00000	−0.236940
$286$	0	0
$287$	10.2462	0.604815
$288$	0	0
$289$	33.7386	1.98463
$290$	0	0
$291$	0.246211	0.0144332
$292$	0	0
$293$	11.7538	0.686664	0.343332	−	0.939214i	$-0.388444\pi$
$293$	11.7538	0.686664	0.343332	+	0.939214i	$0.388444\pi$
$294$	0	0
$295$	14.2462	0.829446
$296$	0	0
$297$	−5.12311	−0.297273
$298$	0	0
$299$	−2.00000	−0.115663
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−16.2462	−0.933320
$304$	0	0
$305$	0.876894	0.0502108
$306$	0	0
$307$	−0.492423	−0.0281040	−0.0140520	−	0.999901i	$-0.504473\pi$
$307$	−0.492423	−0.0281040	−0.0140520	+	0.999901i	$0.504473\pi$
$308$	0	0
$309$	−15.3693	−0.874330
$310$	0	0
$311$	−26.7386	−1.51621	−0.758104	−	0.652133i	$-0.773874\pi$
$311$	−26.7386	−1.51621	−0.758104	+	0.652133i	$0.773874\pi$
$312$	0	0
$313$	10.4924	0.593067	0.296533	−	0.955022i	$-0.404169\pi$
$313$	10.4924	0.593067	0.296533	+	0.955022i	$0.404169\pi$
$314$	0	0
$315$	5.12311	0.288654
$316$	0	0
$317$	20.7386	1.16480	0.582399	−	0.812903i	$-0.302114\pi$
$317$	20.7386	1.16480	0.582399	+	0.812903i	$0.302114\pi$
$318$	0	0
$319$	−10.2462	−0.573678
$320$	0	0
$321$	−11.3693	−0.634573
$322$	0	0
$323$	−28.4924	−1.58536
$324$	0	0
$325$	2.00000	0.110940
$326$	0	0
$327$	11.1231	0.615109
$328$	0	0
$329$	40.9848	2.25957
$330$	0	0
$331$	32.4924	1.78595	0.892973	−	0.450111i	$-0.148616\pi$
$331$	32.4924	1.78595	0.892973	+	0.450111i	$0.148616\pi$
$332$	0	0
$333$	−7.12311	−0.390344
$334$	0	0
$335$	8.00000	0.437087
$336$	0	0
$337$	−6.49242	−0.353665	−0.176832	−	0.984241i	$-0.556585\pi$
$337$	−6.49242	−0.353665	−0.176832	+	0.984241i	$0.556585\pi$
$338$	0	0
$339$	−0.876894	−0.0476264
$340$	0	0
$341$	0	0
$342$	0	0
$343$	62.7386	3.38757
$344$	0	0
$345$	−1.00000	−0.0538382
$346$	0	0
$347$	34.7386	1.86487	0.932434	−	0.361341i	$-0.117681\pi$
$347$	34.7386	1.86487	0.932434	+	0.361341i	$0.117681\pi$
$348$	0	0
$349$	−24.7386	−1.32423	−0.662114	−	0.749403i	$-0.730341\pi$
$349$	−24.7386	−1.32423	−0.662114	+	0.749403i	$0.730341\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	0	0
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	0	0
$355$	−6.24621	−0.331514
$356$	0	0
$357$	36.4924	1.93138
$358$	0	0
$359$	12.4924	0.659325	0.329662	−	0.944099i	$-0.393065\pi$
$359$	12.4924	0.659325	0.329662	+	0.944099i	$0.393065\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	15.2462	0.800219
$364$	0	0
$365$	12.2462	0.640996
$366$	0	0
$367$	−13.1231	−0.685021	−0.342510	−	0.939514i	$-0.611277\pi$
$367$	−13.1231	−0.685021	−0.342510	+	0.939514i	$0.611277\pi$
$368$	0	0
$369$	2.00000	0.104116
$370$	0	0
$371$	−21.7538	−1.12940
$372$	0	0
$373$	13.3693	0.692237	0.346118	−	0.938191i	$-0.387500\pi$
$373$	13.3693	0.692237	0.346118	+	0.938191i	$0.387500\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	4.00000	0.206010
$378$	0	0
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	0	0
$381$	22.2462	1.13971
$382$	0	0
$383$	−13.7538	−0.702786	−0.351393	−	0.936228i	$-0.614292\pi$
$383$	−13.7538	−0.702786	−0.351393	+	0.936228i	$0.614292\pi$
$384$	0	0
$385$	−26.2462	−1.33763
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	0	0
$391$	−7.12311	−0.360231
$392$	0	0
$393$	4.00000	0.201773
$394$	0	0
$395$	5.12311	0.257771
$396$	0	0
$397$	−34.4924	−1.73113	−0.865563	−	0.500801i	$-0.833039\pi$
$397$	−34.4924	−1.73113	−0.865563	+	0.500801i	$0.833039\pi$
$398$	0	0
$399$	−20.4924	−1.02590
$400$	0	0
$401$	−17.3693	−0.867382	−0.433691	−	0.901062i	$-0.642789\pi$
$401$	−17.3693	−0.867382	−0.433691	+	0.901062i	$0.642789\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	36.4924	1.80886
$408$	0	0
$409$	−16.2462	−0.803323	−0.401662	−	0.915788i	$-0.631567\pi$
$409$	−16.2462	−0.803323	−0.401662	+	0.915788i	$0.631567\pi$
$410$	0	0
$411$	−19.1231	−0.943273
$412$	0	0
$413$	72.9848	3.59135
$414$	0	0
$415$	−11.3693	−0.558098
$416$	0	0
$417$	16.4924	0.807637
$418$	0	0
$419$	1.61553	0.0789237	0.0394619	−	0.999221i	$-0.487436\pi$
$419$	1.61553	0.0789237	0.0394619	+	0.999221i	$0.487436\pi$
$420$	0	0
$421$	13.3693	0.651581	0.325790	−	0.945442i	$-0.394370\pi$
$421$	13.3693	0.651581	0.325790	+	0.945442i	$0.394370\pi$
$422$	0	0
$423$	8.00000	0.388973
$424$	0	0
$425$	7.12311	0.345521
$426$	0	0
$427$	4.49242	0.217404
$428$	0	0
$429$	−10.2462	−0.494692
$430$	0	0
$431$	−26.2462	−1.26424	−0.632118	−	0.774872i	$-0.717814\pi$
$431$	−26.2462	−1.26424	−0.632118	+	0.774872i	$0.717814\pi$
$432$	0	0
$433$	10.4924	0.504234	0.252117	−	0.967697i	$-0.418873\pi$
$433$	10.4924	0.504234	0.252117	+	0.967697i	$0.418873\pi$
$434$	0	0
$435$	2.00000	0.0958927
$436$	0	0
$437$	4.00000	0.191346
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	19.2462	0.916486
$442$	0	0
$443$	24.4924	1.16367	0.581835	−	0.813307i	$-0.302335\pi$
$443$	24.4924	1.16367	0.581835	+	0.813307i	$0.302335\pi$
$444$	0	0
$445$	3.12311	0.148049
$446$	0	0
$447$	−10.0000	−0.472984
$448$	0	0
$449$	−10.4924	−0.495168	−0.247584	−	0.968866i	$-0.579637\pi$
$449$	−10.4924	−0.495168	−0.247584	+	0.968866i	$0.579637\pi$
$450$	0	0
$451$	−10.2462	−0.482475
$452$	0	0
$453$	10.2462	0.481409
$454$	0	0
$455$	10.2462	0.480350
$456$	0	0
$457$	−32.7386	−1.53145	−0.765724	−	0.643169i	$-0.777619\pi$
$457$	−32.7386	−1.53145	−0.765724	+	0.643169i	$0.777619\pi$
$458$	0	0
$459$	7.12311	0.332478
$460$	0	0
$461$	32.7386	1.52479	0.762395	−	0.647112i	$-0.224023\pi$
$461$	32.7386	1.52479	0.762395	+	0.647112i	$0.224023\pi$
$462$	0	0
$463$	36.0000	1.67306	0.836531	−	0.547920i	$-0.184580\pi$
$463$	36.0000	1.67306	0.836531	+	0.547920i	$0.184580\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	3.36932	0.155913	0.0779567	−	0.996957i	$-0.475160\pi$
$467$	3.36932	0.155913	0.0779567	+	0.996957i	$0.475160\pi$
$468$	0	0
$469$	40.9848	1.89250
$470$	0	0
$471$	0.876894	0.0404052
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−4.00000	−0.183533
$476$	0	0
$477$	−4.24621	−0.194421
$478$	0	0
$479$	20.4924	0.936323	0.468161	−	0.883643i	$-0.344917\pi$
$479$	20.4924	0.936323	0.468161	+	0.883643i	$0.344917\pi$
$480$	0	0
$481$	−14.2462	−0.649571
$482$	0	0
$483$	−5.12311	−0.233109
$484$	0	0
$485$	0.246211	0.0111799
$486$	0	0
$487$	−40.4924	−1.83489	−0.917443	−	0.397866i	$-0.869751\pi$
$487$	−40.4924	−1.83489	−0.917443	+	0.397866i	$0.869751\pi$
$488$	0	0
$489$	12.0000	0.542659
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	14.2462	0.641617
$494$	0	0
$495$	−5.12311	−0.230266
$496$	0	0
$497$	−32.0000	−1.43540
$498$	0	0
$499$	28.9848	1.29754	0.648770	−	0.760985i	$-0.275284\pi$
$499$	28.9848	1.29754	0.648770	+	0.760985i	$0.275284\pi$
$500$	0	0
$501$	8.00000	0.357414
$502$	0	0
$503$	6.73863	0.300461	0.150230	−	0.988651i	$-0.451998\pi$
$503$	6.73863	0.300461	0.150230	+	0.988651i	$0.451998\pi$
$504$	0	0
$505$	−16.2462	−0.722947
$506$	0	0
$507$	−9.00000	−0.399704
$508$	0	0
$509$	34.0000	1.50702	0.753512	−	0.657434i	$-0.228358\pi$
$509$	34.0000	1.50702	0.753512	+	0.657434i	$0.228358\pi$
$510$	0	0
$511$	62.7386	2.77539
$512$	0	0
$513$	−4.00000	−0.176604
$514$	0	0
$515$	−15.3693	−0.677253
$516$	0	0
$517$	−40.9848	−1.80251
$518$	0	0
$519$	3.75379	0.164773
$520$	0	0
$521$	8.87689	0.388904	0.194452	−	0.980912i	$-0.437707\pi$
$521$	8.87689	0.388904	0.194452	+	0.980912i	$0.437707\pi$
$522$	0	0
$523$	28.4924	1.24589	0.622943	−	0.782267i	$-0.285937\pi$
$523$	28.4924	1.24589	0.622943	+	0.782267i	$0.285937\pi$
$524$	0	0
$525$	5.12311	0.223591
$526$	0	0
$527$	0	0
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	14.2462	0.618233
$532$	0	0
$533$	4.00000	0.173259
$534$	0	0
$535$	−11.3693	−0.491538
$536$	0	0
$537$	−24.4924	−1.05693
$538$	0	0
$539$	−98.6004	−4.24702
$540$	0	0
$541$	−15.7538	−0.677308	−0.338654	−	0.940911i	$-0.609972\pi$
$541$	−15.7538	−0.677308	−0.338654	+	0.940911i	$0.609972\pi$
$542$	0	0
$543$	19.1231	0.820651
$544$	0	0
$545$	11.1231	0.476461
$546$	0	0
$547$	−0.492423	−0.0210545	−0.0105272	−	0.999945i	$-0.503351\pi$
$547$	−0.492423	−0.0210545	−0.0105272	+	0.999945i	$0.503351\pi$
$548$	0	0
$549$	0.876894	0.0374249
$550$	0	0
$551$	−8.00000	−0.340811
$552$	0	0
$553$	26.2462	1.11610
$554$	0	0
$555$	−7.12311	−0.302359
$556$	0	0
$557$	−2.00000	−0.0847427	−0.0423714	−	0.999102i	$-0.513491\pi$
$557$	−2.00000	−0.0847427	−0.0423714	+	0.999102i	$0.513491\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−36.4924	−1.54071
$562$	0	0
$563$	−11.3693	−0.479160	−0.239580	−	0.970877i	$-0.577010\pi$
$563$	−11.3693	−0.479160	−0.239580	+	0.970877i	$0.577010\pi$
$564$	0	0
$565$	−0.876894	−0.0368912
$566$	0	0
$567$	5.12311	0.215150
$568$	0	0
$569$	27.1231	1.13706	0.568530	−	0.822663i	$-0.307512\pi$
$569$	27.1231	1.13706	0.568530	+	0.822663i	$0.307512\pi$
$570$	0	0
$571$	−34.7386	−1.45377	−0.726883	−	0.686761i	$-0.759032\pi$
$571$	−34.7386	−1.45377	−0.726883	+	0.686761i	$0.759032\pi$
$572$	0	0
$573$	20.4924	0.856083
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	14.4924	0.603327	0.301664	−	0.953414i	$-0.402458\pi$
$577$	14.4924	0.603327	0.301664	+	0.953414i	$0.402458\pi$
$578$	0	0
$579$	−8.24621	−0.342701
$580$	0	0
$581$	−58.2462	−2.41646
$582$	0	0
$583$	21.7538	0.900950
$584$	0	0
$585$	2.00000	0.0826898
$586$	0	0
$587$	9.75379	0.402582	0.201291	−	0.979531i	$-0.435486\pi$
$587$	9.75379	0.402582	0.201291	+	0.979531i	$0.435486\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−10.0000	−0.411345
$592$	0	0
$593$	7.75379	0.318410	0.159205	−	0.987246i	$-0.449107\pi$
$593$	7.75379	0.318410	0.159205	+	0.987246i	$0.449107\pi$
$594$	0	0
$595$	36.4924	1.49604
$596$	0	0
$597$	−10.8769	−0.445162
$598$	0	0
$599$	6.24621	0.255213	0.127607	−	0.991825i	$-0.459270\pi$
$599$	6.24621	0.255213	0.127607	+	0.991825i	$0.459270\pi$
$600$	0	0
$601$	−6.00000	−0.244745	−0.122373	−	0.992484i	$-0.539050\pi$
$601$	−6.00000	−0.244745	−0.122373	+	0.992484i	$0.539050\pi$
$602$	0	0
$603$	8.00000	0.325785
$604$	0	0
$605$	15.2462	0.619847
$606$	0	0
$607$	−26.7386	−1.08529	−0.542644	−	0.839963i	$-0.682577\pi$
$607$	−26.7386	−1.08529	−0.542644	+	0.839963i	$0.682577\pi$
$608$	0	0
$609$	10.2462	0.415197
$610$	0	0
$611$	16.0000	0.647291
$612$	0	0
$613$	43.1231	1.74173	0.870863	−	0.491526i	$-0.163561\pi$
$613$	43.1231	1.74173	0.870863	+	0.491526i	$0.163561\pi$
$614$	0	0
$615$	2.00000	0.0806478
$616$	0	0
$617$	40.1080	1.61469	0.807343	−	0.590083i	$-0.200905\pi$
$617$	40.1080	1.61469	0.807343	+	0.590083i	$0.200905\pi$
$618$	0	0
$619$	36.0000	1.44696	0.723481	−	0.690344i	$-0.242541\pi$
$619$	36.0000	1.44696	0.723481	+	0.690344i	$0.242541\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	16.0000	0.641026
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	20.4924	0.818389
$628$	0	0
$629$	−50.7386	−2.02308
$630$	0	0
$631$	39.3693	1.56727	0.783634	−	0.621223i	$-0.213364\pi$
$631$	39.3693	1.56727	0.783634	+	0.621223i	$0.213364\pi$
$632$	0	0
$633$	−16.4924	−0.655515
$634$	0	0
$635$	22.2462	0.882814
$636$	0	0
$637$	38.4924	1.52513
$638$	0	0
$639$	−6.24621	−0.247096
$640$	0	0
$641$	−9.36932	−0.370066	−0.185033	−	0.982732i	$-0.559239\pi$
$641$	−9.36932	−0.370066	−0.185033	+	0.982732i	$0.559239\pi$
$642$	0	0
$643$	−36.4924	−1.43912	−0.719560	−	0.694430i	$-0.755657\pi$
$643$	−36.4924	−1.43912	−0.719560	+	0.694430i	$0.755657\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−11.5076	−0.452410	−0.226205	−	0.974080i	$-0.572632\pi$
$647$	−11.5076	−0.452410	−0.226205	+	0.974080i	$0.572632\pi$
$648$	0	0
$649$	−72.9848	−2.86491
$650$	0	0
$651$	0	0
$652$	0	0
$653$	10.4924	0.410600	0.205300	−	0.978699i	$-0.434183\pi$
$653$	10.4924	0.410600	0.205300	+	0.978699i	$0.434183\pi$
$654$	0	0
$655$	4.00000	0.156293
$656$	0	0
$657$	12.2462	0.477770
$658$	0	0
$659$	7.36932	0.287068	0.143534	−	0.989645i	$-0.454153\pi$
$659$	7.36932	0.287068	0.143534	+	0.989645i	$0.454153\pi$
$660$	0	0
$661$	33.8617	1.31707	0.658535	−	0.752551i	$-0.271177\pi$
$661$	33.8617	1.31707	0.658535	+	0.752551i	$0.271177\pi$
$662$	0	0
$663$	14.2462	0.553277
$664$	0	0
$665$	−20.4924	−0.794662
$666$	0	0
$667$	−2.00000	−0.0774403
$668$	0	0
$669$	−6.24621	−0.241492
$670$	0	0
$671$	−4.49242	−0.173428
$672$	0	0
$673$	−46.9848	−1.81113	−0.905566	−	0.424205i	$-0.860554\pi$
$673$	−46.9848	−1.81113	−0.905566	+	0.424205i	$0.860554\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−46.4924	−1.78685	−0.893424	−	0.449213i	$-0.851704\pi$
$677$	−46.4924	−1.78685	−0.893424	+	0.449213i	$0.851704\pi$
$678$	0	0
$679$	1.26137	0.0484068
$680$	0	0
$681$	−19.3693	−0.742234
$682$	0	0
$683$	−17.7538	−0.679330	−0.339665	−	0.940547i	$-0.610314\pi$
$683$	−17.7538	−0.679330	−0.339665	+	0.940547i	$0.610314\pi$
$684$	0	0
$685$	−19.1231	−0.730656
$686$	0	0
$687$	−1.36932	−0.0522427
$688$	0	0
$689$	−8.49242	−0.323536
$690$	0	0
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	0	0
$693$	−26.2462	−0.997011
$694$	0	0
$695$	16.4924	0.625593
$696$	0	0
$697$	14.2462	0.539614
$698$	0	0
$699$	−20.7386	−0.784407
$700$	0	0
$701$	−36.2462	−1.36900	−0.684500	−	0.729013i	$-0.739980\pi$
$701$	−36.2462	−1.36900	−0.684500	+	0.729013i	$0.739980\pi$
$702$	0	0
$703$	28.4924	1.07461
$704$	0	0
$705$	8.00000	0.301297
$706$	0	0
$707$	−83.2311	−3.13023
$708$	0	0
$709$	13.3693	0.502095	0.251048	−	0.967975i	$-0.419225\pi$
$709$	13.3693	0.502095	0.251048	+	0.967975i	$0.419225\pi$
$710$	0	0
$711$	5.12311	0.192131
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−10.2462	−0.383187
$716$	0	0
$717$	1.75379	0.0654964
$718$	0	0
$719$	9.75379	0.363755	0.181877	−	0.983321i	$-0.441783\pi$
$719$	9.75379	0.363755	0.181877	+	0.983321i	$0.441783\pi$
$720$	0	0
$721$	−78.7386	−2.93238
$722$	0	0
$723$	−2.49242	−0.0926942
$724$	0	0
$725$	2.00000	0.0742781
$726$	0	0
$727$	−27.8617	−1.03333	−0.516667	−	0.856186i	$-0.672828\pi$
$727$	−27.8617	−1.03333	−0.516667	+	0.856186i	$0.672828\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−13.8617	−0.511995	−0.255998	−	0.966677i	$-0.582404\pi$
$733$	−13.8617	−0.511995	−0.255998	+	0.966677i	$0.582404\pi$
$734$	0	0
$735$	19.2462	0.709907
$736$	0	0
$737$	−40.9848	−1.50970
$738$	0	0
$739$	−40.4924	−1.48954	−0.744769	−	0.667322i	$-0.767440\pi$
$739$	−40.4924	−1.48954	−0.744769	+	0.667322i	$0.767440\pi$
$740$	0	0
$741$	−8.00000	−0.293887
$742$	0	0
$743$	14.7386	0.540708	0.270354	−	0.962761i	$-0.412859\pi$
$743$	14.7386	0.540708	0.270354	+	0.962761i	$0.412859\pi$
$744$	0	0
$745$	−10.0000	−0.366372
$746$	0	0
$747$	−11.3693	−0.415982
$748$	0	0
$749$	−58.2462	−2.12827
$750$	0	0
$751$	−43.8617	−1.60054	−0.800269	−	0.599641i	$-0.795310\pi$
$751$	−43.8617	−1.60054	−0.800269	+	0.599641i	$0.795310\pi$
$752$	0	0
$753$	7.36932	0.268553
$754$	0	0
$755$	10.2462	0.372898
$756$	0	0
$757$	9.86174	0.358431	0.179216	−	0.983810i	$-0.442644\pi$
$757$	9.86174	0.358431	0.179216	+	0.983810i	$0.442644\pi$
$758$	0	0
$759$	5.12311	0.185957
$760$	0	0
$761$	−48.2462	−1.74892	−0.874462	−	0.485094i	$-0.838785\pi$
$761$	−48.2462	−1.74892	−0.874462	+	0.485094i	$0.838785\pi$
$762$	0	0
$763$	56.9848	2.06299
$764$	0	0
$765$	7.12311	0.257536
$766$	0	0
$767$	28.4924	1.02880
$768$	0	0
$769$	−20.7386	−0.747854	−0.373927	−	0.927458i	$-0.621989\pi$
$769$	−20.7386	−0.747854	−0.373927	+	0.927458i	$0.621989\pi$
$770$	0	0
$771$	−8.24621	−0.296980
$772$	0	0
$773$	−36.2462	−1.30369	−0.651843	−	0.758354i	$-0.726004\pi$
$773$	−36.2462	−1.30369	−0.651843	+	0.758354i	$0.726004\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−36.4924	−1.30916
$778$	0	0
$779$	−8.00000	−0.286630
$780$	0	0
$781$	32.0000	1.14505
$782$	0	0
$783$	2.00000	0.0714742
$784$	0	0
$785$	0.876894	0.0312977
$786$	0	0
$787$	−48.0000	−1.71102	−0.855508	−	0.517790i	$-0.826755\pi$
$787$	−48.0000	−1.71102	−0.855508	+	0.517790i	$0.826755\pi$
$788$	0	0
$789$	−10.2462	−0.364775
$790$	0	0
$791$	−4.49242	−0.159732
$792$	0	0
$793$	1.75379	0.0622789
$794$	0	0
$795$	−4.24621	−0.150598
$796$	0	0
$797$	−22.4924	−0.796722	−0.398361	−	0.917229i	$-0.630421\pi$
$797$	−22.4924	−0.796722	−0.398361	+	0.917229i	$0.630421\pi$
$798$	0	0
$799$	56.9848	2.01598
$800$	0	0
$801$	3.12311	0.110350
$802$	0	0
$803$	−62.7386	−2.21400
$804$	0	0
$805$	−5.12311	−0.180566
$806$	0	0
$807$	−16.2462	−0.571894
$808$	0	0
$809$	4.24621	0.149289	0.0746444	−	0.997210i	$-0.476218\pi$
$809$	4.24621	0.149289	0.0746444	+	0.997210i	$0.476218\pi$
$810$	0	0
$811$	36.0000	1.26413	0.632065	−	0.774915i	$-0.282207\pi$
$811$	36.0000	1.26413	0.632065	+	0.774915i	$0.282207\pi$
$812$	0	0
$813$	−10.2462	−0.359350
$814$	0	0
$815$	12.0000	0.420342
$816$	0	0
$817$	0	0
$818$	0	0
$819$	10.2462	0.358032
$820$	0	0
$821$	−19.7538	−0.689412	−0.344706	−	0.938711i	$-0.612021\pi$
$821$	−19.7538	−0.689412	−0.344706	+	0.938711i	$0.612021\pi$
$822$	0	0
$823$	4.00000	0.139431	0.0697156	−	0.997567i	$-0.477791\pi$
$823$	4.00000	0.139431	0.0697156	+	0.997567i	$0.477791\pi$
$824$	0	0
$825$	−5.12311	−0.178364
$826$	0	0
$827$	−17.1231	−0.595429	−0.297714	−	0.954655i	$-0.596224\pi$
$827$	−17.1231	−0.595429	−0.297714	+	0.954655i	$0.596224\pi$
$828$	0	0
$829$	32.2462	1.11996	0.559979	−	0.828507i	$-0.310809\pi$
$829$	32.2462	1.11996	0.559979	+	0.828507i	$0.310809\pi$
$830$	0	0
$831$	20.2462	0.702333
$832$	0	0
$833$	137.093	4.74998
$834$	0	0
$835$	8.00000	0.276851
$836$	0	0
$837$	0	0
$838$	0	0
$839$	26.2462	0.906120	0.453060	−	0.891480i	$-0.350332\pi$
$839$	26.2462	0.906120	0.453060	+	0.891480i	$0.350332\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	−12.8769	−0.443504
$844$	0	0
$845$	−9.00000	−0.309609
$846$	0	0
$847$	78.1080	2.68382
$848$	0	0
$849$	2.24621	0.0770898
$850$	0	0
$851$	7.12311	0.244177
$852$	0	0
$853$	−6.00000	−0.205436	−0.102718	−	0.994711i	$-0.532754\pi$
$853$	−6.00000	−0.205436	−0.102718	+	0.994711i	$0.532754\pi$
$854$	0	0
$855$	−4.00000	−0.136797
$856$	0	0
$857$	24.7386	0.845056	0.422528	−	0.906350i	$-0.361143\pi$
$857$	24.7386	0.845056	0.422528	+	0.906350i	$0.361143\pi$
$858$	0	0
$859$	−48.4924	−1.65454	−0.827270	−	0.561804i	$-0.810107\pi$
$859$	−48.4924	−1.65454	−0.827270	+	0.561804i	$0.810107\pi$
$860$	0	0
$861$	10.2462	0.349190
$862$	0	0
$863$	−48.9848	−1.66746	−0.833732	−	0.552170i	$-0.813800\pi$
$863$	−48.9848	−1.66746	−0.833732	+	0.552170i	$0.813800\pi$
$864$	0	0
$865$	3.75379	0.127633
$866$	0	0
$867$	33.7386	1.14582
$868$	0	0
$869$	−26.2462	−0.890342
$870$	0	0
$871$	16.0000	0.542139
$872$	0	0
$873$	0.246211	0.00833299
$874$	0	0
$875$	5.12311	0.173193
$876$	0	0
$877$	3.26137	0.110129	0.0550643	−	0.998483i	$-0.482464\pi$
$877$	3.26137	0.110129	0.0550643	+	0.998483i	$0.482464\pi$
$878$	0	0
$879$	11.7538	0.396445
$880$	0	0
$881$	31.6155	1.06515	0.532577	−	0.846381i	$-0.321224\pi$
$881$	31.6155	1.06515	0.532577	+	0.846381i	$0.321224\pi$
$882$	0	0
$883$	20.9848	0.706196	0.353098	−	0.935586i	$-0.385128\pi$
$883$	20.9848	0.706196	0.353098	+	0.935586i	$0.385128\pi$
$884$	0	0
$885$	14.2462	0.478881
$886$	0	0
$887$	−52.4924	−1.76252	−0.881262	−	0.472629i	$-0.843305\pi$
$887$	−52.4924	−1.76252	−0.881262	+	0.472629i	$0.843305\pi$
$888$	0	0
$889$	113.970	3.82242
$890$	0	0
$891$	−5.12311	−0.171630
$892$	0	0
$893$	−32.0000	−1.07084
$894$	0	0
$895$	−24.4924	−0.818691
$896$	0	0
$897$	−2.00000	−0.0667781
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−30.2462	−1.00765
$902$	0	0
$903$	0	0
$904$	0	0
$905$	19.1231	0.635674
$906$	0	0
$907$	−37.7538	−1.25359	−0.626797	−	0.779183i	$-0.715634\pi$
$907$	−37.7538	−1.25359	−0.626797	+	0.779183i	$0.715634\pi$
$908$	0	0
$909$	−16.2462	−0.538853
$910$	0	0
$911$	13.7538	0.455683	0.227842	−	0.973698i	$-0.426833\pi$
$911$	13.7538	0.455683	0.227842	+	0.973698i	$0.426833\pi$
$912$	0	0
$913$	58.2462	1.92767
$914$	0	0
$915$	0.876894	0.0289892
$916$	0	0
$917$	20.4924	0.676719
$918$	0	0
$919$	−10.8769	−0.358796	−0.179398	−	0.983777i	$-0.557415\pi$
$919$	−10.8769	−0.358796	−0.179398	+	0.983777i	$0.557415\pi$
$920$	0	0
$921$	−0.492423	−0.0162259
$922$	0	0
$923$	−12.4924	−0.411193
$924$	0	0
$925$	−7.12311	−0.234206
$926$	0	0
$927$	−15.3693	−0.504795
$928$	0	0
$929$	−30.9848	−1.01658	−0.508290	−	0.861186i	$-0.669722\pi$
$929$	−30.9848	−1.01658	−0.508290	+	0.861186i	$0.669722\pi$
$930$	0	0
$931$	−76.9848	−2.52308
$932$	0	0
$933$	−26.7386	−0.875384
$934$	0	0
$935$	−36.4924	−1.19343
$936$	0	0
$937$	−16.7386	−0.546827	−0.273414	−	0.961897i	$-0.588153\pi$
$937$	−16.7386	−0.546827	−0.273414	+	0.961897i	$0.588153\pi$
$938$	0	0
$939$	10.4924	0.342407
$940$	0	0
$941$	−6.49242	−0.211647	−0.105823	−	0.994385i	$-0.533748\pi$
$941$	−6.49242	−0.211647	−0.105823	+	0.994385i	$0.533748\pi$
$942$	0	0
$943$	−2.00000	−0.0651290
$944$	0	0
$945$	5.12311	0.166655
$946$	0	0
$947$	−22.2462	−0.722905	−0.361452	−	0.932391i	$-0.617719\pi$
$947$	−22.2462	−0.722905	−0.361452	+	0.932391i	$0.617719\pi$
$948$	0	0
$949$	24.4924	0.795058
$950$	0	0
$951$	20.7386	0.672496
$952$	0	0
$953$	−33.8617	−1.09689	−0.548445	−	0.836187i	$-0.684780\pi$
$953$	−33.8617	−1.09689	−0.548445	+	0.836187i	$0.684780\pi$
$954$	0	0
$955$	20.4924	0.663119
$956$	0	0
$957$	−10.2462	−0.331213
$958$	0	0
$959$	−97.9697	−3.16361
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	−11.3693	−0.366371
$964$	0	0
$965$	−8.24621	−0.265455
$966$	0	0
$967$	4.98485	0.160302	0.0801509	−	0.996783i	$-0.474460\pi$
$967$	4.98485	0.160302	0.0801509	+	0.996783i	$0.474460\pi$
$968$	0	0
$969$	−28.4924	−0.915308
$970$	0	0
$971$	−34.8769	−1.11925	−0.559626	−	0.828745i	$-0.689055\pi$
$971$	−34.8769	−1.11925	−0.559626	+	0.828745i	$0.689055\pi$
$972$	0	0
$973$	84.4924	2.70870
$974$	0	0
$975$	2.00000	0.0640513
$976$	0	0
$977$	21.8617	0.699419	0.349710	−	0.936858i	$-0.386280\pi$
$977$	21.8617	0.699419	0.349710	+	0.936858i	$0.386280\pi$
$978$	0	0
$979$	−16.0000	−0.511362
$980$	0	0
$981$	11.1231	0.355133
$982$	0	0
$983$	26.2462	0.837124	0.418562	−	0.908188i	$-0.362534\pi$
$983$	26.2462	0.837124	0.418562	+	0.908188i	$0.362534\pi$
$984$	0	0
$985$	−10.0000	−0.318626
$986$	0	0
$987$	40.9848	1.30456
$988$	0	0
$989$	0	0
$990$	0	0
$991$	37.4773	1.19050	0.595252	−	0.803539i	$-0.297052\pi$
$991$	37.4773	1.19050	0.595252	+	0.803539i	$0.297052\pi$
$992$	0	0
$993$	32.4924	1.03112
$994$	0	0
$995$	−10.8769	−0.344821
$996$	0	0
$997$	20.2462	0.641204	0.320602	−	0.947214i	$-0.396115\pi$
$997$	20.2462	0.641204	0.320602	+	0.947214i	$0.396115\pi$
$998$	0	0
$999$	−7.12311	−0.225365

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5520.2.a.bs.1.2		2
4.3	odd	2		690.2.a.l.1.1	✓	2
12.11	even	2		2070.2.a.t.1.1		2
20.3	even	4		3450.2.d.v.2899.2		4
20.7	even	4		3450.2.d.v.2899.3		4
20.19	odd	2		3450.2.a.bi.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.2.a.l.1.1	✓	2	4.3	odd	2
2070.2.a.t.1.1		2	12.11	even	2
3450.2.a.bi.1.2		2	20.19	odd	2
3450.2.d.v.2899.2		4	20.3	even	4
3450.2.d.v.2899.3		4	20.7	even	4
5520.2.a.bs.1.2		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:	\( N \)	\(=\)	\( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5520.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} +5.12311 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} +5.12311 q^{7} +1.00000 q^{9} -5.12311 q^{11} +2.00000 q^{13} +1.00000 q^{15} +7.12311 q^{17} -4.00000 q^{19} +5.12311 q^{21} -1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +2.00000 q^{29} -5.12311 q^{33} +5.12311 q^{35} -7.12311 q^{37} +2.00000 q^{39} +2.00000 q^{41} +1.00000 q^{45} +8.00000 q^{47} +19.2462 q^{49} +7.12311 q^{51} -4.24621 q^{53} -5.12311 q^{55} -4.00000 q^{57} +14.2462 q^{59} +0.876894 q^{61} +5.12311 q^{63} +2.00000 q^{65} +8.00000 q^{67} -1.00000 q^{69} -6.24621 q^{71} +12.2462 q^{73} +1.00000 q^{75} -26.2462 q^{77} +5.12311 q^{79} +1.00000 q^{81} -11.3693 q^{83} +7.12311 q^{85} +2.00000 q^{87} +3.12311 q^{89} +10.2462 q^{91} -4.00000 q^{95} +0.246211 q^{97} -5.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} - 2 q^{11} + 4 q^{13} + 2 q^{15} + 6 q^{17} - 8 q^{19} + 2 q^{21} - 2 q^{23} + 2 q^{25} + 2 q^{27} + 4 q^{29} - 2 q^{33} + 2 q^{35} - 6 q^{37} + 4 q^{39} + 4 q^{41} + 2 q^{45} + 16 q^{47} + 22 q^{49} + 6 q^{51} + 8 q^{53} - 2 q^{55} - 8 q^{57} + 12 q^{59} + 10 q^{61} + 2 q^{63} + 4 q^{65} + 16 q^{67} - 2 q^{69} + 4 q^{71} + 8 q^{73} + 2 q^{75} - 36 q^{77} + 2 q^{79} + 2 q^{81} + 2 q^{83} + 6 q^{85} + 4 q^{87} - 2 q^{89} + 4 q^{91} - 8 q^{95} - 16 q^{97} - 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9 - 2 * q^11 + 4 * q^13 + 2 * q^15 + 6 * q^17 - 8 * q^19 + 2 * q^21 - 2 * q^23 + 2 * q^25 + 2 * q^27 + 4 * q^29 - 2 * q^33 + 2 * q^35 - 6 * q^37 + 4 * q^39 + 4 * q^41 + 2 * q^45 + 16 * q^47 + 22 * q^49 + 6 * q^51 + 8 * q^53 - 2 * q^55 - 8 * q^57 + 12 * q^59 + 10 * q^61 + 2 * q^63 + 4 * q^65 + 16 * q^67 - 2 * q^69 + 4 * q^71 + 8 * q^73 + 2 * q^75 - 36 * q^77 + 2 * q^79 + 2 * q^81 + 2 * q^83 + 6 * q^85 + 4 * q^87 - 2 * q^89 + 4 * q^91 - 8 * q^95 - 16 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more