LMFDB - Embedded newform 5520.2.be.c.1471.32

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5520,2,Mod(1471,5520)]

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([1, 0, 0, 0, 1]))

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

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

chi := DirichletCharacter("5520.1471");

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.be (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$44.0774219157$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1471.32
Character	$\chi$	$=$	5520.1471
Dual form			5520.2.be.c.1471.31

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +1.00000i q^{5} +2.30439 q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +1.00000i q^{5} +2.30439 q^{7} -1.00000 q^{9} +0.945051 q^{11} -5.38485 q^{13} -1.00000 q^{15} -6.56693i q^{17} +2.72915 q^{19} +2.30439i q^{21} +(0.880622 - 4.71429i) q^{23} -1.00000 q^{25} -1.00000i q^{27} -6.26801 q^{29} +5.84526i q^{31} +0.945051i q^{33} +2.30439i q^{35} +6.23066i q^{37} -5.38485i q^{39} +11.7635 q^{41} -8.79376 q^{43} -1.00000i q^{45} +9.81739i q^{47} -1.68979 q^{49} +6.56693 q^{51} +12.2599i q^{53} +0.945051i q^{55} +2.72915i q^{57} +5.84785i q^{59} +10.7499i q^{61} -2.30439 q^{63} -5.38485i q^{65} -6.05298 q^{67} +(4.71429 + 0.880622i) q^{69} +9.51338i q^{71} +5.20690 q^{73} -1.00000i q^{75} +2.17776 q^{77} -2.04834 q^{79} +1.00000 q^{81} -3.94714 q^{83} +6.56693 q^{85} -6.26801i q^{87} +2.77495i q^{89} -12.4088 q^{91} -5.84526 q^{93} +2.72915i q^{95} +9.24518i q^{97} -0.945051 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 8 q^{7} - 32 q^{9}+O(q^{10}) $ 32 * q - 8 * q^7 - 32 * q^9	$ 32 q - 8 q^{7} - 32 q^{9} + 8 q^{11} - 8 q^{13} - 32 q^{15} - 32 q^{25} + 4 q^{29} + 20 q^{41} + 52 q^{49} - 4 q^{51} + 8 q^{63} + 32 q^{67} - 40 q^{73} - 24 q^{77} + 32 q^{79} + 32 q^{81} - 4 q^{85} - 48 q^{91} - 8 q^{99}+O(q^{100}) $ 32 * q - 8 * q^7 - 32 * q^9 + 8 * q^11 - 8 * q^13 - 32 * q^15 - 32 * q^25 + 4 * q^29 + 20 * q^41 + 52 * q^49 - 4 * q^51 + 8 * q^63 + 32 * q^67 - 40 * q^73 - 24 * q^77 + 32 * q^79 + 32 * q^81 - 4 * q^85 - 48 * q^91 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5520\mathbb{Z}\right)^\times$.

$n$	$1201$	$1381$	$1841$	$4417$	$4831$
$\chi(n)$	$-1$	$1$	$1$	$1$	$-1$

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
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$	2.30439			0.870977			0.435488	−	0.900194i	$-0.356576\pi$
$7$	2.30439			0.870977			0.435488	+	0.900194i	$0.356576\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	0.945051			0.284944			0.142472	−	0.989799i	$-0.454495\pi$
$11$	0.945051			0.284944			0.142472	+	0.989799i	$0.454495\pi$
$12$		0			0
$13$	−5.38485			−1.49349			−0.746744	−	0.665112i	$-0.768384\pi$
$13$	−5.38485			−1.49349			−0.746744	+	0.665112i	$0.768384\pi$
$14$		0			0
$15$	−1.00000			−0.258199
$16$		0			0
$17$		−	6.56693i		−	1.59272i	−0.604826	−	0.796358i	$-0.706757\pi$
$17$		−	6.56693i		−	1.59272i	0.604826	−	0.796358i	$-0.293243\pi$
$18$		0			0
$19$	2.72915			0.626110			0.313055	−	0.949735i	$-0.398648\pi$
$19$	2.72915			0.626110			0.313055	+	0.949735i	$0.398648\pi$
$20$		0			0
$21$			2.30439i			0.502859i
$22$		0			0
$23$	0.880622	−	4.71429i	0.183622	−	0.982997i
$23$	0.880622	−	4.71429i	0.183622	−	0.982997i
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	−6.26801			−1.16394			−0.581970	−	0.813210i	$-0.697718\pi$
$29$	−6.26801			−1.16394			−0.581970	+	0.813210i	$0.697718\pi$
$30$		0			0
$31$			5.84526i			1.04984i	0.851152	+	0.524919i	$0.175905\pi$
$31$			5.84526i			1.04984i	−0.851152	+	0.524919i	$0.824095\pi$
$32$		0			0
$33$			0.945051i			0.164512i
$34$		0			0
$35$			2.30439i			0.389513i
$36$		0			0
$37$			6.23066i			1.02431i	0.858892	+	0.512157i	$0.171153\pi$
$37$			6.23066i			1.02431i	−0.858892	+	0.512157i	$0.828847\pi$
$38$		0			0
$39$		−	5.38485i		−	0.862266i
$40$		0			0
$41$	11.7635			1.83715			0.918577	−	0.395243i	$-0.129340\pi$
$41$	11.7635			1.83715			0.918577	+	0.395243i	$0.129340\pi$
$42$		0			0
$43$	−8.79376			−1.34104			−0.670518	−	0.741893i	$-0.733928\pi$
$43$	−8.79376			−1.34104			−0.670518	+	0.741893i	$0.733928\pi$
$44$		0			0
$45$		−	1.00000i		−	0.149071i
$46$		0			0
$47$			9.81739i			1.43201i	0.698094	+	0.716007i	$0.254032\pi$
$47$			9.81739i			1.43201i	−0.698094	+	0.716007i	$0.745968\pi$
$48$		0			0
$49$	−1.68979			−0.241399
$50$		0			0
$51$	6.56693			0.919555
$52$		0			0
$53$			12.2599i			1.68403i	0.539456	+	0.842014i	$0.318630\pi$
$53$			12.2599i			1.68403i	−0.539456	+	0.842014i	$0.681370\pi$
$54$		0			0
$55$			0.945051i			0.127431i
$56$		0			0
$57$			2.72915i			0.361485i
$58$		0			0
$59$			5.84785i			0.761326i	0.924714	+	0.380663i	$0.124304\pi$
$59$			5.84785i			0.761326i	−0.924714	+	0.380663i	$0.875696\pi$
$60$		0			0
$61$			10.7499i			1.37638i	0.725529	+	0.688191i	$0.241595\pi$
$61$			10.7499i			1.37638i	−0.725529	+	0.688191i	$0.758405\pi$
$62$		0			0
$63$	−2.30439			−0.290326
$64$		0			0
$65$		−	5.38485i		−	0.667908i
$66$		0			0
$67$	−6.05298			−0.739489			−0.369744	−	0.929134i	$-0.620555\pi$
$67$	−6.05298			−0.739489			−0.369744	+	0.929134i	$0.620555\pi$
$68$		0			0
$69$	4.71429	+	0.880622i	0.567533	+	0.106014i
$70$		0			0
$71$			9.51338i			1.12903i	0.825423	+	0.564515i	$0.190937\pi$
$71$			9.51338i			1.12903i	−0.825423	+	0.564515i	$0.809063\pi$
$72$		0			0
$73$	5.20690			0.609421			0.304711	−	0.952445i	$-0.401440\pi$
$73$	5.20690			0.609421			0.304711	+	0.952445i	$0.401440\pi$
$74$		0			0
$75$		−	1.00000i		−	0.115470i
$76$		0			0
$77$	2.17776			0.248179
$78$		0			0
$79$	−2.04834			−0.230456			−0.115228	−	0.993339i	$-0.536760\pi$
$79$	−2.04834			−0.230456			−0.115228	+	0.993339i	$0.536760\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	−3.94714			−0.433255			−0.216628	−	0.976254i	$-0.569506\pi$
$83$	−3.94714			−0.433255			−0.216628	+	0.976254i	$0.569506\pi$
$84$		0			0
$85$	6.56693			0.712284
$86$		0			0
$87$		−	6.26801i		−	0.672002i
$88$		0			0
$89$			2.77495i			0.294144i	0.989126	+	0.147072i	$0.0469850\pi$
$89$			2.77495i			0.294144i	−0.989126	+	0.147072i	$0.953015\pi$
$90$		0			0
$91$	−12.4088			−1.30079
$92$		0			0
$93$	−5.84526			−0.606125
$94$		0			0
$95$			2.72915i			0.280005i
$96$		0			0
$97$			9.24518i			0.938706i	0.883011	+	0.469353i	$0.155513\pi$
$97$			9.24518i			0.938706i	−0.883011	+	0.469353i	$0.844487\pi$
$98$		0			0
$99$	−0.945051			−0.0949812
$100$		0			0
$101$	5.52273			0.549532			0.274766	−	0.961511i	$-0.411400\pi$
$101$	5.52273			0.549532			0.274766	+	0.961511i	$0.411400\pi$
$102$		0			0
$103$	4.36891			0.430482			0.215241	−	0.976561i	$-0.430946\pi$
$103$	4.36891			0.430482			0.215241	+	0.976561i	$0.430946\pi$
$104$		0			0
$105$	−2.30439			−0.224885
$106$		0			0
$107$	−12.8331			−1.24062			−0.620311	−	0.784356i	$-0.712993\pi$
$107$	−12.8331			−1.24062			−0.620311	+	0.784356i	$0.712993\pi$
$108$		0			0
$109$		−	10.6709i		−	1.02209i	−0.859554	−	0.511045i	$-0.829258\pi$
$109$		−	10.6709i		−	1.02209i	0.859554	−	0.511045i	$-0.170742\pi$
$110$		0			0
$111$	−6.23066			−0.591388
$112$		0			0
$113$			17.8706i			1.68112i	0.541715	+	0.840562i	$0.317775\pi$
$113$			17.8706i			1.68112i	−0.541715	+	0.840562i	$0.682225\pi$
$114$		0			0
$115$	4.71429	+	0.880622i	0.439610	+	0.0821184i
$116$		0			0
$117$	5.38485			0.497829
$118$		0			0
$119$		−	15.1328i		−	1.38722i
$120$		0			0
$121$	−10.1069			−0.918807
$122$		0			0
$123$			11.7635i			1.06068i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$		−	7.62437i		−	0.676554i	−0.941047	−	0.338277i	$-0.890156\pi$
$127$		−	7.62437i		−	0.676554i	0.941047	−	0.338277i	$-0.109844\pi$
$128$		0			0
$129$		−	8.79376i		−	0.774248i
$130$		0			0
$131$			9.92569i			0.867211i	0.901103	+	0.433606i	$0.142759\pi$
$131$			9.92569i			0.867211i	−0.901103	+	0.433606i	$0.857241\pi$
$132$		0			0
$133$	6.28902			0.545327
$134$		0			0
$135$	1.00000			0.0860663
$136$		0			0
$137$			7.12966i			0.609128i	0.952492	+	0.304564i	$0.0985107\pi$
$137$			7.12966i			0.609128i	−0.952492	+	0.304564i	$0.901489\pi$
$138$		0			0
$139$		−	4.03149i		−	0.341946i	−0.985276	−	0.170973i	$-0.945309\pi$
$139$		−	4.03149i		−	0.341946i	0.985276	−	0.170973i	$-0.0546911\pi$
$140$		0			0
$141$	−9.81739			−0.826773
$142$		0			0
$143$	−5.08896			−0.425560
$144$		0			0
$145$		−	6.26801i		−	0.520530i
$146$		0			0
$147$		−	1.68979i		−	0.139372i
$148$		0			0
$149$			21.4949i			1.76093i	0.474112	+	0.880465i	$0.342769\pi$
$149$			21.4949i			1.76093i	−0.474112	+	0.880465i	$0.657231\pi$
$150$		0			0
$151$			10.3126i			0.839226i	0.907703	+	0.419613i	$0.137834\pi$
$151$			10.3126i			0.839226i	−0.907703	+	0.419613i	$0.862166\pi$
$152$		0			0
$153$			6.56693i			0.530905i
$154$		0			0
$155$	−5.84526			−0.469502
$156$		0			0
$157$		−	3.65626i		−	0.291801i	−0.989299	−	0.145901i	$-0.953392\pi$
$157$		−	3.65626i		−	0.291801i	0.989299	−	0.145901i	$-0.0466080\pi$
$158$		0			0
$159$	−12.2599			−0.972274
$160$		0			0
$161$	2.02930	−	10.8635i	0.159931	−	0.856168i
$162$		0			0
$163$		−	17.6263i		−	1.38060i	−0.723525	−	0.690299i	$-0.757479\pi$
$163$		−	17.6263i		−	1.38060i	0.723525	−	0.690299i	$-0.242521\pi$
$164$		0			0
$165$	−0.945051			−0.0735721
$166$		0			0
$167$		−	7.18702i		−	0.556148i	−0.960560	−	0.278074i	$-0.910304\pi$
$167$		−	7.18702i		−	0.556148i	0.960560	−	0.278074i	$-0.0896961\pi$
$168$		0			0
$169$	15.9966			1.23051
$170$		0			0
$171$	−2.72915			−0.208703
$172$		0			0
$173$	−3.18259			−0.241968			−0.120984	−	0.992654i	$-0.538605\pi$
$173$	−3.18259			−0.241968			−0.120984	+	0.992654i	$0.538605\pi$
$174$		0			0
$175$	−2.30439			−0.174195
$176$		0			0
$177$	−5.84785			−0.439552
$178$		0			0
$179$			1.33852i			0.100046i	0.998748	+	0.0500229i	$0.0159294\pi$
$179$			1.33852i			0.100046i	−0.998748	+	0.0500229i	$0.984071\pi$
$180$		0			0
$181$			5.39261i			0.400829i	0.979711	+	0.200415i	$0.0642290\pi$
$181$			5.39261i			0.400829i	−0.979711	+	0.200415i	$0.935771\pi$
$182$		0			0
$183$	−10.7499			−0.794655
$184$		0			0
$185$	−6.23066			−0.458087
$186$		0			0
$187$		−	6.20609i		−	0.453834i
$188$		0			0
$189$		−	2.30439i		−	0.167620i
$190$		0			0
$191$	1.10757			0.0801409			0.0400705	−	0.999197i	$-0.487242\pi$
$191$	1.10757			0.0801409			0.0400705	+	0.999197i	$0.487242\pi$
$192$		0			0
$193$	23.3586			1.68139			0.840693	−	0.541512i	$-0.182148\pi$
$193$	23.3586			1.68139			0.840693	+	0.541512i	$0.182148\pi$
$194$		0			0
$195$	5.38485			0.385617
$196$		0			0
$197$	−4.45281			−0.317250			−0.158625	−	0.987339i	$-0.550706\pi$
$197$	−4.45281			−0.317250			−0.158625	+	0.987339i	$0.550706\pi$
$198$		0			0
$199$	−11.9066			−0.844037			−0.422018	−	0.906587i	$-0.638678\pi$
$199$	−11.9066			−0.844037			−0.422018	+	0.906587i	$0.638678\pi$
$200$		0			0
$201$		−	6.05298i		−	0.426944i
$202$		0			0
$203$	−14.4439			−1.01377
$204$		0			0
$205$			11.7635i			0.821600i
$206$		0			0
$207$	−0.880622	+	4.71429i	−0.0612075	+	0.327666i
$208$		0			0
$209$	2.57918			0.178406
$210$		0			0
$211$		−	9.70730i		−	0.668278i	−0.942524	−	0.334139i	$-0.891555\pi$
$211$		−	9.70730i		−	0.668278i	0.942524	−	0.334139i	$-0.108445\pi$
$212$		0			0
$213$	−9.51338			−0.651846
$214$		0			0
$215$		−	8.79376i		−	0.599730i
$216$		0			0
$217$			13.4697i			0.914385i
$218$		0			0
$219$			5.20690i			0.351850i
$220$		0			0
$221$			35.3619i			2.37870i
$222$		0			0
$223$		−	26.1472i		−	1.75095i	−0.483265	−	0.875474i	$-0.660549\pi$
$223$		−	26.1472i		−	1.75095i	0.483265	−	0.875474i	$-0.339451\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$	−5.66996			−0.376329			−0.188164	−	0.982138i	$-0.560254\pi$
$227$	−5.66996			−0.376329			−0.188164	+	0.982138i	$0.560254\pi$
$228$		0			0
$229$		−	27.9937i		−	1.84987i	−0.380122	−	0.924936i	$-0.624118\pi$
$229$		−	27.9937i		−	1.84987i	0.380122	−	0.924936i	$-0.375882\pi$
$230$		0			0
$231$			2.17776i			0.143286i
$232$		0			0
$233$	−4.84614			−0.317481			−0.158741	−	0.987320i	$-0.550743\pi$
$233$	−4.84614			−0.317481			−0.158741	+	0.987320i	$0.550743\pi$
$234$		0			0
$235$	−9.81739			−0.640416
$236$		0			0
$237$		−	2.04834i		−	0.133054i
$238$		0			0
$239$		−	6.00897i		−	0.388688i	−0.980933	−	0.194344i	$-0.937742\pi$
$239$		−	6.00897i		−	0.388688i	0.980933	−	0.194344i	$-0.0622578\pi$
$240$		0			0
$241$		−	2.02440i		−	0.130403i	−0.997872	−	0.0652017i	$-0.979231\pi$
$241$		−	2.02440i		−	0.130403i	0.997872	−	0.0652017i	$-0.0207691\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$		−	1.68979i		−	0.107957i
$246$		0			0
$247$	−14.6960			−0.935087
$248$		0			0
$249$		−	3.94714i		−	0.250140i
$250$		0			0
$251$	17.5448			1.10742			0.553710	−	0.832709i	$-0.313212\pi$
$251$	17.5448			1.10742			0.553710	+	0.832709i	$0.313212\pi$
$252$		0			0
$253$	0.832233	−	4.45524i	0.0523220	−	0.280099i
$254$		0			0
$255$			6.56693i			0.411237i
$256$		0			0
$257$	8.58849			0.535735			0.267868	−	0.963456i	$-0.413681\pi$
$257$	8.58849			0.535735			0.267868	+	0.963456i	$0.413681\pi$
$258$		0			0
$259$			14.3579i			0.892154i
$260$		0			0
$261$	6.26801			0.387980
$262$		0			0
$263$	31.3534			1.93333			0.966666	−	0.256041i	$-0.0824183\pi$
$263$	31.3534			1.93333			0.966666	+	0.256041i	$0.0824183\pi$
$264$		0			0
$265$	−12.2599			−0.753120
$266$		0			0
$267$	−2.77495			−0.169824
$268$		0			0
$269$	16.8328			1.02632			0.513158	−	0.858294i	$-0.328475\pi$
$269$	16.8328			1.02632			0.513158	+	0.858294i	$0.328475\pi$
$270$		0			0
$271$			21.3986i			1.29987i	0.759988	+	0.649937i	$0.225205\pi$
$271$			21.3986i			1.29987i	−0.759988	+	0.649937i	$0.774795\pi$
$272$		0			0
$273$		−	12.4088i		−	0.751014i
$274$		0			0
$275$	−0.945051			−0.0569887
$276$		0			0
$277$	28.4043			1.70665			0.853323	−	0.521382i	$-0.174583\pi$
$277$	28.4043			1.70665			0.853323	+	0.521382i	$0.174583\pi$
$278$		0			0
$279$		−	5.84526i		−	0.349946i
$280$		0			0
$281$			16.8544i			1.00545i	0.864447	+	0.502724i	$0.167669\pi$
$281$			16.8544i			1.00545i	−0.864447	+	0.502724i	$0.832331\pi$
$282$		0			0
$283$	−0.578653			−0.0343973			−0.0171987	−	0.999852i	$-0.505475\pi$
$283$	−0.578653			−0.0343973			−0.0171987	+	0.999852i	$0.505475\pi$
$284$		0			0
$285$	−2.72915			−0.161661
$286$		0			0
$287$	27.1077			1.60012
$288$		0			0
$289$	−26.1246			−1.53674
$290$		0			0
$291$	−9.24518			−0.541962
$292$		0			0
$293$			4.43678i			0.259200i	0.991566	+	0.129600i	$0.0413693\pi$
$293$			4.43678i			0.259200i	−0.991566	+	0.129600i	$0.958631\pi$
$294$		0			0
$295$	−5.84785			−0.340475
$296$		0			0
$297$		−	0.945051i		−	0.0548374i
$298$		0			0
$299$	−4.74202	+	25.3857i	−0.274238	+	1.46809i
$300$		0			0
$301$	−20.2642			−1.16801
$302$		0			0
$303$			5.52273i			0.317273i
$304$		0			0
$305$	−10.7499			−0.615537
$306$		0			0
$307$			13.3426i			0.761504i	0.924677	+	0.380752i	$0.124335\pi$
$307$			13.3426i			0.761504i	−0.924677	+	0.380752i	$0.875665\pi$
$308$		0			0
$309$			4.36891i			0.248539i
$310$		0			0
$311$		−	11.4921i		−	0.651659i	−0.945428	−	0.325830i	$-0.894356\pi$
$311$		−	11.4921i		−	0.651659i	0.945428	−	0.325830i	$-0.105644\pi$
$312$		0			0
$313$		−	15.8304i		−	0.894786i	−0.894338	−	0.447393i	$-0.852353\pi$
$313$		−	15.8304i		−	0.894786i	0.894338	−	0.447393i	$-0.147647\pi$
$314$		0			0
$315$		−	2.30439i		−	0.129838i
$316$		0			0
$317$	1.17611			0.0660567			0.0330284	−	0.999454i	$-0.489485\pi$
$317$	1.17611			0.0660567			0.0330284	+	0.999454i	$0.489485\pi$
$318$		0			0
$319$	−5.92359			−0.331658
$320$		0			0
$321$		−	12.8331i		−	0.716273i
$322$		0			0
$323$		−	17.9221i		−	0.997214i
$324$		0			0
$325$	5.38485			0.298698
$326$		0			0
$327$	10.6709			0.590104
$328$		0			0
$329$			22.6231i			1.24725i
$330$		0			0
$331$			23.1828i			1.27424i	0.770764	+	0.637121i	$0.219875\pi$
$331$			23.1828i			1.27424i	−0.770764	+	0.637121i	$0.780125\pi$
$332$		0			0
$333$		−	6.23066i		−	0.341438i
$334$		0			0
$335$		−	6.05298i		−	0.330709i
$336$		0			0
$337$		−	30.9158i		−	1.68409i	−0.539406	−	0.842046i	$-0.681351\pi$
$337$		−	30.9158i		−	1.68409i	0.539406	−	0.842046i	$-0.318649\pi$
$338$		0			0
$339$	−17.8706			−0.970598
$340$		0			0
$341$			5.52406i			0.299145i
$342$		0			0
$343$	−20.0247			−1.08123
$344$		0			0
$345$	−0.880622	+	4.71429i	−0.0474111	+	0.253809i
$346$		0			0
$347$			7.70586i			0.413672i	0.978376	+	0.206836i	$0.0663167\pi$
$347$			7.70586i			0.413672i	−0.978376	+	0.206836i	$0.933683\pi$
$348$		0			0
$349$	−2.80376			−0.150082			−0.0750410	−	0.997180i	$-0.523909\pi$
$349$	−2.80376			−0.150082			−0.0750410	+	0.997180i	$0.523909\pi$
$350$		0			0
$351$			5.38485i			0.287422i
$352$		0			0
$353$	−30.8217			−1.64047			−0.820237	−	0.572024i	$-0.806158\pi$
$353$	−30.8217			−1.64047			−0.820237	+	0.572024i	$0.806158\pi$
$354$		0			0
$355$	−9.51338			−0.504918
$356$		0			0
$357$	15.1328			0.800911
$358$		0			0
$359$	−28.6936			−1.51439			−0.757195	−	0.653189i	$-0.773431\pi$
$359$	−28.6936			−1.51439			−0.757195	+	0.653189i	$0.773431\pi$
$360$		0			0
$361$	−11.5517			−0.607987
$362$		0			0
$363$		−	10.1069i		−	0.530474i
$364$		0			0
$365$			5.20690i			0.272542i
$366$		0			0
$367$	−25.2579			−1.31845			−0.659227	−	0.751944i	$-0.729116\pi$
$367$	−25.2579			−1.31845			−0.659227	+	0.751944i	$0.729116\pi$
$368$		0			0
$369$	−11.7635			−0.612384
$370$		0			0
$371$			28.2516i			1.46675i
$372$		0			0
$373$			11.9213i			0.617259i	0.951182	+	0.308630i	$0.0998703\pi$
$373$			11.9213i			0.617259i	−0.951182	+	0.308630i	$0.900130\pi$
$374$		0			0
$375$	1.00000			0.0516398
$376$		0			0
$377$	33.7523			1.73833
$378$		0			0
$379$	−24.2908			−1.24773			−0.623866	−	0.781531i	$-0.714439\pi$
$379$	−24.2908			−1.24773			−0.623866	+	0.781531i	$0.714439\pi$
$380$		0			0
$381$	7.62437			0.390608
$382$		0			0
$383$	2.30689			0.117876			0.0589382	−	0.998262i	$-0.481229\pi$
$383$	2.30689			0.117876			0.0589382	+	0.998262i	$0.481229\pi$
$384$		0			0
$385$			2.17776i			0.110989i
$386$		0			0
$387$	8.79376			0.447012
$388$		0			0
$389$			11.0868i			0.562124i	0.959690	+	0.281062i	$0.0906866\pi$
$389$			11.0868i			0.562124i	−0.959690	+	0.281062i	$0.909313\pi$
$390$		0			0
$391$	−30.9584	−	5.78299i	−1.56563	−	0.292458i
$392$		0			0
$393$	−9.92569			−0.500685
$394$		0			0
$395$		−	2.04834i		−	0.103063i
$396$		0			0
$397$	−9.92306			−0.498024			−0.249012	−	0.968500i	$-0.580106\pi$
$397$	−9.92306			−0.498024			−0.249012	+	0.968500i	$0.580106\pi$
$398$		0			0
$399$			6.28902i			0.314845i
$400$		0			0
$401$			4.42003i			0.220726i	0.993891	+	0.110363i	$0.0352013\pi$
$401$			4.42003i			0.220726i	−0.993891	+	0.110363i	$0.964799\pi$
$402$		0			0
$403$		−	31.4758i		−	1.56792i
$404$		0			0
$405$			1.00000i			0.0496904i
$406$		0			0
$407$			5.88829i			0.291872i
$408$		0			0
$409$	−5.31943			−0.263029			−0.131514	−	0.991314i	$-0.541984\pi$
$409$	−5.31943			−0.263029			−0.131514	+	0.991314i	$0.541984\pi$
$410$		0			0
$411$	−7.12966			−0.351680
$412$		0			0
$413$			13.4757i			0.663097i
$414$		0			0
$415$		−	3.94714i		−	0.193758i
$416$		0			0
$417$	4.03149			0.197423
$418$		0			0
$419$	−26.0876			−1.27446			−0.637232	−	0.770672i	$-0.719920\pi$
$419$	−26.0876			−1.27446			−0.637232	+	0.770672i	$0.719920\pi$
$420$		0			0
$421$		−	1.93135i		−	0.0941280i	−0.998892	−	0.0470640i	$-0.985014\pi$
$421$		−	1.93135i		−	0.0941280i	0.998892	−	0.0470640i	$-0.0149865\pi$
$422$		0			0
$423$		−	9.81739i		−	0.477338i
$424$		0			0
$425$			6.56693i			0.318543i
$426$		0			0
$427$			24.7719i			1.19880i
$428$		0			0
$429$		−	5.08896i		−	0.245697i
$430$		0			0
$431$	−3.04962			−0.146895			−0.0734475	−	0.997299i	$-0.523400\pi$
$431$	−3.04962			−0.146895			−0.0734475	+	0.997299i	$0.523400\pi$
$432$		0			0
$433$		−	19.9714i		−	0.959766i	−0.877332	−	0.479883i	$-0.840679\pi$
$433$		−	19.9714i		−	0.959766i	0.877332	−	0.479883i	$-0.159321\pi$
$434$		0			0
$435$	6.26801			0.300528
$436$		0			0
$437$	2.40335	−	12.8660i	0.114968	−	0.615464i
$438$		0			0
$439$			35.0414i			1.67244i	0.548398	+	0.836218i	$0.315238\pi$
$439$			35.0414i			1.67244i	−0.548398	+	0.836218i	$0.684762\pi$
$440$		0			0
$441$	1.68979			0.0804664
$442$		0			0
$443$		−	3.03636i		−	0.144262i	−0.997395	−	0.0721308i	$-0.977020\pi$
$443$		−	3.03636i		−	0.144262i	0.997395	−	0.0721308i	$-0.0229799\pi$
$444$		0			0
$445$	−2.77495			−0.131545
$446$		0			0
$447$	−21.4949			−1.01667
$448$		0			0
$449$	26.7881			1.26421			0.632103	−	0.774884i	$-0.282192\pi$
$449$	26.7881			1.26421			0.632103	+	0.774884i	$0.282192\pi$
$450$		0			0
$451$	11.1171			0.523485
$452$		0			0
$453$	−10.3126			−0.484527
$454$		0			0
$455$		−	12.4088i		−	0.581733i
$456$		0			0
$457$			26.3251i			1.23144i	0.787966	+	0.615719i	$0.211134\pi$
$457$			26.3251i			1.23144i	−0.787966	+	0.615719i	$0.788866\pi$
$458$		0			0
$459$	−6.56693			−0.306518
$460$		0			0
$461$	−27.8829			−1.29864			−0.649319	−	0.760516i	$-0.724946\pi$
$461$	−27.8829			−1.29864			−0.649319	+	0.760516i	$0.724946\pi$
$462$		0			0
$463$			10.9751i			0.510057i	0.966934	+	0.255028i	$0.0820848\pi$
$463$			10.9751i			0.510057i	−0.966934	+	0.255028i	$0.917915\pi$
$464$		0			0
$465$		−	5.84526i		−	0.271067i
$466$		0			0
$467$	−39.4892			−1.82734			−0.913671	−	0.406455i	$-0.866765\pi$
$467$	−39.4892			−1.82734			−0.913671	+	0.406455i	$0.866765\pi$
$468$		0			0
$469$	−13.9484			−0.644078
$470$		0			0
$471$	3.65626			0.168472
$472$		0			0
$473$	−8.31056			−0.382120
$474$		0			0
$475$	−2.72915			−0.125222
$476$		0			0
$477$		−	12.2599i		−	0.561343i
$478$		0			0
$479$	21.9979			1.00511			0.502554	−	0.864546i	$-0.332394\pi$
$479$	21.9979			1.00511			0.502554	+	0.864546i	$0.332394\pi$
$480$		0			0
$481$		−	33.5512i		−	1.52980i
$482$		0			0
$483$	10.8635	+	2.02930i	0.494309	+	0.0923361i
$484$		0			0
$485$	−9.24518			−0.419802
$486$		0			0
$487$			33.1409i			1.50176i	0.660440	+	0.750879i	$0.270370\pi$
$487$			33.1409i			1.50176i	−0.660440	+	0.750879i	$0.729630\pi$
$488$		0			0
$489$	17.6263			0.797088
$490$		0			0
$491$		−	1.98700i		−	0.0896722i	−0.998994	−	0.0448361i	$-0.985723\pi$
$491$		−	1.98700i		−	0.0896722i	0.998994	−	0.0448361i	$-0.0142766\pi$
$492$		0			0
$493$			41.1616i			1.85383i
$494$		0			0
$495$		−	0.945051i		−	0.0424769i
$496$		0			0
$497$			21.9225i			0.983360i
$498$		0			0
$499$		−	40.4328i		−	1.81002i	−0.425387	−	0.905011i	$-0.639862\pi$
$499$		−	40.4328i		−	1.81002i	0.425387	−	0.905011i	$-0.360138\pi$
$500$		0			0
$501$	7.18702			0.321092
$502$		0			0
$503$	29.1638			1.30035			0.650175	−	0.759785i	$-0.274696\pi$
$503$	29.1638			1.30035			0.650175	+	0.759785i	$0.274696\pi$
$504$		0			0
$505$			5.52273i			0.245758i
$506$		0			0
$507$			15.9966i			0.710433i
$508$		0			0
$509$	33.7684			1.49676			0.748378	−	0.663272i	$-0.230833\pi$
$509$	33.7684			1.49676			0.748378	+	0.663272i	$0.230833\pi$
$510$		0			0
$511$	11.9987			0.530792
$512$		0			0
$513$		−	2.72915i		−	0.120495i
$514$		0			0
$515$			4.36891i			0.192517i
$516$		0			0
$517$			9.27793i			0.408043i
$518$		0			0
$519$		−	3.18259i		−	0.139700i
$520$		0			0
$521$		−	20.5854i		−	0.901861i	−0.892559	−	0.450931i	$-0.851092\pi$
$521$		−	20.5854i		−	0.901861i	0.892559	−	0.450931i	$-0.148908\pi$
$522$		0			0
$523$	9.02662			0.394707			0.197353	−	0.980332i	$-0.436765\pi$
$523$	9.02662			0.394707			0.197353	+	0.980332i	$0.436765\pi$
$524$		0			0
$525$		−	2.30439i		−	0.100572i
$526$		0			0
$527$	38.3854			1.67209
$528$		0			0
$529$	−21.4490	−	8.30301i	−0.932566	−	0.361000i
$530$		0			0
$531$		−	5.84785i		−	0.253775i
$532$		0			0
$533$	−63.3448			−2.74377
$534$		0			0
$535$		−	12.8331i		−	0.554823i
$536$		0			0
$537$	−1.33852			−0.0577615
$538$		0			0
$539$	−1.59694			−0.0687851
$540$		0			0
$541$	37.8304			1.62645			0.813227	−	0.581946i	$-0.197708\pi$
$541$	37.8304			1.62645			0.813227	+	0.581946i	$0.197708\pi$
$542$		0			0
$543$	−5.39261			−0.231419
$544$		0			0
$545$	10.6709			0.457093
$546$		0			0
$547$		−	6.91343i		−	0.295597i	−0.989017	−	0.147799i	$-0.952781\pi$
$547$		−	6.91343i		−	0.295597i	0.989017	−	0.147799i	$-0.0472187\pi$
$548$		0			0
$549$		−	10.7499i		−	0.458794i
$550$		0			0
$551$	−17.1063			−0.728755
$552$		0			0
$553$	−4.72017			−0.200722
$554$		0			0
$555$		−	6.23066i		−	0.264477i
$556$		0			0
$557$			20.9538i			0.887842i	0.896066	+	0.443921i	$0.146413\pi$
$557$			20.9538i			0.887842i	−0.896066	+	0.443921i	$0.853587\pi$
$558$		0			0
$559$	47.3531			2.00282
$560$		0			0
$561$	6.20609			0.262021
$562$		0			0
$563$	−32.2350			−1.35854			−0.679271	−	0.733887i	$-0.737704\pi$
$563$	−32.2350			−1.35854			−0.679271	+	0.733887i	$0.737704\pi$
$564$		0			0
$565$	−17.8706			−0.751822
$566$		0			0
$567$	2.30439			0.0967752
$568$		0			0
$569$			32.7177i			1.37160i	0.727791	+	0.685799i	$0.240547\pi$
$569$			32.7177i			1.37160i	−0.727791	+	0.685799i	$0.759453\pi$
$570$		0			0
$571$	28.2525			1.18233			0.591165	−	0.806551i	$-0.298668\pi$
$571$	28.2525			1.18233			0.591165	+	0.806551i	$0.298668\pi$
$572$		0			0
$573$			1.10757i			0.0462694i
$574$		0			0
$575$	−0.880622	+	4.71429i	−0.0367245	+	0.196599i
$576$		0			0
$577$	11.4996			0.478735			0.239367	−	0.970929i	$-0.423060\pi$
$577$	11.4996			0.478735			0.239367	+	0.970929i	$0.423060\pi$
$578$		0			0
$579$			23.3586i			0.970748i
$580$		0			0
$581$	−9.09576			−0.377356
$582$		0			0
$583$			11.5862i			0.479853i
$584$		0			0
$585$			5.38485i			0.222636i
$586$		0			0
$587$		−	45.9280i		−	1.89565i	−0.318791	−	0.947825i	$-0.603277\pi$
$587$		−	45.9280i		−	1.89565i	0.318791	−	0.947825i	$-0.396723\pi$
$588$		0			0
$589$			15.9526i			0.657314i
$590$		0			0
$591$		−	4.45281i		−	0.183164i
$592$		0			0
$593$	31.1128			1.27765			0.638826	−	0.769351i	$-0.279420\pi$
$593$	31.1128			1.27765			0.638826	+	0.769351i	$0.279420\pi$
$594$		0			0
$595$	15.1328			0.620383
$596$		0			0
$597$		−	11.9066i		−	0.487305i
$598$		0			0
$599$		−	32.7015i		−	1.33615i	−0.744096	−	0.668073i	$-0.767119\pi$
$599$		−	32.7015i		−	1.33615i	0.744096	−	0.668073i	$-0.232881\pi$
$600$		0			0
$601$	−13.0680			−0.533056			−0.266528	−	0.963827i	$-0.585877\pi$
$601$	−13.0680			−0.533056			−0.266528	+	0.963827i	$0.585877\pi$
$602$		0			0
$603$	6.05298			0.246496
$604$		0			0
$605$		−	10.1069i		−	0.410903i
$606$		0			0
$607$		−	7.50391i		−	0.304574i	−0.988336	−	0.152287i	$-0.951336\pi$
$607$		−	7.50391i		−	0.304574i	0.988336	−	0.152287i	$-0.0486639\pi$
$608$		0			0
$609$		−	14.4439i		−	0.585298i
$610$		0			0
$611$		−	52.8651i		−	2.13869i
$612$		0			0
$613$		−	22.0113i		−	0.889026i	−0.895772	−	0.444513i	$-0.853377\pi$
$613$		−	22.0113i		−	0.889026i	0.895772	−	0.444513i	$-0.146623\pi$
$614$		0			0
$615$	−11.7635			−0.474351
$616$		0			0
$617$		−	37.6212i		−	1.51457i	−0.653083	−	0.757286i	$-0.726525\pi$
$617$		−	37.6212i		−	1.51457i	0.653083	−	0.757286i	$-0.273475\pi$
$618$		0			0
$619$	27.1512			1.09130			0.545650	−	0.838013i	$-0.316283\pi$
$619$	27.1512			1.09130			0.545650	+	0.838013i	$0.316283\pi$
$620$		0			0
$621$	−4.71429	−	0.880622i	−0.189178	−	0.0353381i
$622$		0			0
$623$			6.39457i			0.256193i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$			2.57918i			0.103003i
$628$		0			0
$629$	40.9163			1.63144
$630$		0			0
$631$	9.02826			0.359409			0.179705	−	0.983721i	$-0.442486\pi$
$631$	9.02826			0.359409			0.179705	+	0.983721i	$0.442486\pi$
$632$		0			0
$633$	9.70730			0.385830
$634$		0			0
$635$	7.62437			0.302564
$636$		0			0
$637$	9.09928			0.360527
$638$		0			0
$639$		−	9.51338i		−	0.376344i
$640$		0			0
$641$		−	5.80482i		−	0.229277i	−0.993407	−	0.114638i	$-0.963429\pi$
$641$		−	5.80482i		−	0.229277i	0.993407	−	0.114638i	$-0.0365709\pi$
$642$		0			0
$643$	−38.0823			−1.50182			−0.750909	−	0.660405i	$-0.770385\pi$
$643$	−38.0823			−1.50182			−0.750909	+	0.660405i	$0.770385\pi$
$644$		0			0
$645$	8.79376			0.346254
$646$		0			0
$647$			36.6890i			1.44239i	0.692731	+	0.721196i	$0.256407\pi$
$647$			36.6890i			1.44239i	−0.692731	+	0.721196i	$0.743593\pi$
$648$		0			0
$649$			5.52652i			0.216935i
$650$		0			0
$651$	−13.4697			−0.527921
$652$		0			0
$653$	9.39454			0.367637			0.183818	−	0.982960i	$-0.441154\pi$
$653$	9.39454			0.367637			0.183818	+	0.982960i	$0.441154\pi$
$654$		0			0
$655$	−9.92569			−0.387829
$656$		0			0
$657$	−5.20690			−0.203140
$658$		0			0
$659$	21.2469			0.827663			0.413832	−	0.910353i	$-0.364190\pi$
$659$	21.2469			0.827663			0.413832	+	0.910353i	$0.364190\pi$
$660$		0			0
$661$		−	23.8980i		−	0.929525i	−0.885435	−	0.464763i	$-0.846140\pi$
$661$		−	23.8980i		−	0.929525i	0.885435	−	0.464763i	$-0.153860\pi$
$662$		0			0
$663$	−35.3619			−1.37334
$664$		0			0
$665$			6.28902i			0.243878i
$666$		0			0
$667$	−5.51975	+	29.5492i	−0.213726	+	1.14415i
$668$		0			0
$669$	26.1472			1.01091
$670$		0			0
$671$			10.1592i			0.392191i
$672$		0			0
$673$	12.2997			0.474117			0.237059	−	0.971495i	$-0.423817\pi$
$673$	12.2997			0.474117			0.237059	+	0.971495i	$0.423817\pi$
$674$		0			0
$675$			1.00000i			0.0384900i
$676$		0			0
$677$			21.5734i			0.829131i	0.910019	+	0.414566i	$0.136066\pi$
$677$			21.5734i			0.829131i	−0.910019	+	0.414566i	$0.863934\pi$
$678$		0			0
$679$			21.3045i			0.817591i
$680$		0			0
$681$		−	5.66996i		−	0.217273i
$682$		0			0
$683$		−	25.0898i		−	0.960035i	−0.877259	−	0.480017i	$-0.840630\pi$
$683$		−	25.0898i		−	0.960035i	0.877259	−	0.480017i	$-0.159370\pi$
$684$		0			0
$685$	−7.12966			−0.272410
$686$		0			0
$687$	27.9937			1.06802
$688$		0			0
$689$		−	66.0177i		−	2.51508i
$690$		0			0
$691$			20.3102i			0.772635i	0.922366	+	0.386317i	$0.126253\pi$
$691$			20.3102i			0.772635i	−0.922366	+	0.386317i	$0.873747\pi$
$692$		0			0
$693$	−2.17776			−0.0827264
$694$		0			0
$695$	4.03149			0.152923
$696$		0			0
$697$		−	77.2502i		−	2.92606i
$698$		0			0
$699$		−	4.84614i		−	0.183298i
$700$		0			0
$701$			19.2492i			0.727031i	0.931588	+	0.363516i	$0.118424\pi$
$701$			19.2492i			0.727031i	−0.931588	+	0.363516i	$0.881576\pi$
$702$		0			0
$703$			17.0044i			0.641333i
$704$		0			0
$705$		−	9.81739i		−	0.369744i
$706$		0			0
$707$	12.7265			0.478630
$708$		0			0
$709$			29.4070i			1.10440i	0.833711	+	0.552201i	$0.186212\pi$
$709$			29.4070i			1.10440i	−0.833711	+	0.552201i	$0.813788\pi$
$710$		0			0
$711$	2.04834			0.0768187
$712$		0			0
$713$	27.5562	+	5.14746i	1.03199	+	0.192774i
$714$		0			0
$715$		−	5.08896i		−	0.190316i
$716$		0			0
$717$	6.00897			0.224409
$718$		0			0
$719$			7.22290i			0.269369i	0.990889	+	0.134684i	$0.0430020\pi$
$719$			7.22290i			0.269369i	−0.990889	+	0.134684i	$0.956998\pi$
$720$		0			0
$721$	10.0677			0.374940
$722$		0			0
$723$	2.02440			0.0752884
$724$		0			0
$725$	6.26801			0.232788
$726$		0			0
$727$	22.0079			0.816227			0.408113	−	0.912931i	$-0.366187\pi$
$727$	22.0079			0.816227			0.408113	+	0.912931i	$0.366187\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$			57.7481i			2.13589i
$732$		0			0
$733$			8.31712i			0.307200i	0.988133	+	0.153600i	$0.0490867\pi$
$733$			8.31712i			0.307200i	−0.988133	+	0.153600i	$0.950913\pi$
$734$		0			0
$735$	1.68979			0.0623290
$736$		0			0
$737$	−5.72037			−0.210713
$738$		0			0
$739$			20.3628i			0.749059i	0.927215	+	0.374529i	$0.122196\pi$
$739$			20.3628i			0.749059i	−0.927215	+	0.374529i	$0.877804\pi$
$740$		0			0
$741$		−	14.6960i		−	0.539873i
$742$		0			0
$743$	−20.7820			−0.762416			−0.381208	−	0.924489i	$-0.624492\pi$
$743$	−20.7820			−0.762416			−0.381208	+	0.924489i	$0.624492\pi$
$744$		0			0
$745$	−21.4949			−0.787511
$746$		0			0
$747$	3.94714			0.144418
$748$		0			0
$749$	−29.5724			−1.08055
$750$		0			0
$751$	−38.5957			−1.40838			−0.704188	−	0.710014i	$-0.748689\pi$
$751$	−38.5957			−1.40838			−0.704188	+	0.710014i	$0.748689\pi$
$752$		0			0
$753$			17.5448i			0.639369i
$754$		0			0
$755$	−10.3126			−0.375313
$756$		0			0
$757$			6.49702i			0.236138i	0.993005	+	0.118069i	$0.0376704\pi$
$757$			6.49702i			0.236138i	−0.993005	+	0.118069i	$0.962330\pi$
$758$		0			0
$759$	4.45524	+	0.832233i	0.161715	+	0.0302081i
$760$		0			0
$761$	9.72045			0.352366			0.176183	−	0.984357i	$-0.443625\pi$
$761$	9.72045			0.352366			0.176183	+	0.984357i	$0.443625\pi$
$762$		0			0
$763$		−	24.5900i		−	0.890218i
$764$		0			0
$765$	−6.56693			−0.237428
$766$		0			0
$767$		−	31.4898i		−	1.13703i
$768$		0			0
$769$			23.4553i			0.845819i	0.906172	+	0.422910i	$0.138991\pi$
$769$			23.4553i			0.845819i	−0.906172	+	0.422910i	$0.861009\pi$
$770$		0			0
$771$			8.58849i			0.309307i
$772$		0			0
$773$			1.29495i			0.0465760i	0.999729	+	0.0232880i	$0.00741347\pi$
$773$			1.29495i			0.0465760i	−0.999729	+	0.0232880i	$0.992587\pi$
$774$		0			0
$775$		−	5.84526i		−	0.209968i
$776$		0			0
$777$	−14.3579			−0.515085
$778$		0			0
$779$	32.1044			1.15026
$780$		0			0
$781$			8.99063i			0.321710i
$782$		0			0
$783$			6.26801i			0.224001i
$784$		0			0
$785$	3.65626			0.130498
$786$		0			0
$787$	−13.3671			−0.476487			−0.238243	−	0.971205i	$-0.576572\pi$
$787$	−13.3671			−0.476487			−0.238243	+	0.971205i	$0.576572\pi$
$788$		0			0
$789$			31.3534i			1.11621i
$790$		0			0
$791$			41.1808i			1.46422i
$792$		0			0
$793$		−	57.8865i		−	2.05561i
$794$		0			0
$795$		−	12.2599i		−	0.434814i
$796$		0			0
$797$			31.7592i			1.12497i	0.826808	+	0.562485i	$0.190154\pi$
$797$			31.7592i			1.12497i	−0.826808	+	0.562485i	$0.809846\pi$
$798$		0			0
$799$	64.4701			2.28079
$800$		0			0
$801$		−	2.77495i		−	0.0980481i
$802$		0			0
$803$	4.92079			0.173651
$804$		0			0
$805$	10.8635	+	2.02930i	0.382890	+	0.0715233i
$806$		0			0
$807$			16.8328i			0.592544i
$808$		0			0
$809$	−2.22070			−0.0780757			−0.0390379	−	0.999238i	$-0.512429\pi$
$809$	−2.22070			−0.0780757			−0.0390379	+	0.999238i	$0.512429\pi$
$810$		0			0
$811$			14.9896i			0.526355i	0.964747	+	0.263178i	$0.0847705\pi$
$811$			14.9896i			0.526355i	−0.964747	+	0.263178i	$0.915229\pi$
$812$		0			0
$813$	−21.3986			−0.750482
$814$		0			0
$815$	17.6263			0.617422
$816$		0			0
$817$	−23.9995			−0.839636
$818$		0			0
$819$	12.4088			0.433598
$820$		0			0
$821$	13.2687			0.463080			0.231540	−	0.972825i	$-0.425624\pi$
$821$	13.2687			0.463080			0.231540	+	0.972825i	$0.425624\pi$
$822$		0			0
$823$		−	27.2694i		−	0.950550i	−0.879837	−	0.475275i	$-0.842348\pi$
$823$		−	27.2694i		−	0.950550i	0.879837	−	0.475275i	$-0.157652\pi$
$824$		0			0
$825$		−	0.945051i		−	0.0329025i
$826$		0			0
$827$	54.5307			1.89622			0.948109	−	0.317946i	$-0.102993\pi$
$827$	54.5307			1.89622			0.948109	+	0.317946i	$0.102993\pi$
$828$		0			0
$829$	15.5233			0.539146			0.269573	−	0.962980i	$-0.413117\pi$
$829$	15.5233			0.539146			0.269573	+	0.962980i	$0.413117\pi$
$830$		0			0
$831$			28.4043i			0.985333i
$832$		0			0
$833$			11.0968i			0.384480i
$834$		0			0
$835$	7.18702			0.248717
$836$		0			0
$837$	5.84526			0.202042
$838$		0			0
$839$	34.9144			1.20538			0.602689	−	0.797976i	$-0.294096\pi$
$839$	34.9144			1.20538			0.602689	+	0.797976i	$0.294096\pi$
$840$		0			0
$841$	10.2880			0.354759
$842$		0			0
$843$	−16.8544			−0.580495
$844$		0			0
$845$			15.9966i			0.550299i
$846$		0			0
$847$	−23.2902			−0.800260
$848$		0			0
$849$		−	0.578653i		−	0.0198593i
$850$		0			0
$851$	29.3731	+	5.48686i	1.00690	+	0.188087i
$852$		0			0
$853$	−5.20393			−0.178179			−0.0890896	−	0.996024i	$-0.528396\pi$
$853$	−5.20393			−0.178179			−0.0890896	+	0.996024i	$0.528396\pi$
$854$		0			0
$855$		−	2.72915i		−	0.0933349i
$856$		0			0
$857$	2.91288			0.0995022			0.0497511	−	0.998762i	$-0.484157\pi$
$857$	2.91288			0.0995022			0.0497511	+	0.998762i	$0.484157\pi$
$858$		0			0
$859$			4.00255i			0.136565i	0.997666	+	0.0682826i	$0.0217520\pi$
$859$			4.00255i			0.136565i	−0.997666	+	0.0682826i	$0.978248\pi$
$860$		0			0
$861$			27.1077i			0.923829i
$862$		0			0
$863$		−	34.8184i		−	1.18523i	−0.805485	−	0.592616i	$-0.798095\pi$
$863$		−	34.8184i		−	1.18523i	0.805485	−	0.592616i	$-0.201905\pi$
$864$		0			0
$865$		−	3.18259i		−	0.108211i
$866$		0			0
$867$		−	26.1246i		−	0.887238i
$868$		0			0
$869$	−1.93579			−0.0656670
$870$		0			0
$871$	32.5943			1.10442
$872$		0			0
$873$		−	9.24518i		−	0.312902i
$874$		0			0
$875$		−	2.30439i		−	0.0779026i
$876$		0			0
$877$	−30.3803			−1.02587			−0.512935	−	0.858428i	$-0.671442\pi$
$877$	−30.3803			−1.02587			−0.512935	+	0.858428i	$0.671442\pi$
$878$		0			0
$879$	−4.43678			−0.149649
$880$		0			0
$881$			19.3163i			0.650783i	0.945579	+	0.325392i	$0.105496\pi$
$881$			19.3163i			0.650783i	−0.945579	+	0.325392i	$0.894504\pi$
$882$		0			0
$883$		−	42.0213i		−	1.41413i	−0.707149	−	0.707065i	$-0.750019\pi$
$883$		−	42.0213i		−	1.41413i	0.707149	−	0.707065i	$-0.249981\pi$
$884$		0			0
$885$		−	5.84785i		−	0.196573i
$886$		0			0
$887$		−	18.8127i		−	0.631670i	−0.948814	−	0.315835i	$-0.897715\pi$
$887$		−	18.8127i		−	0.631670i	0.948814	−	0.315835i	$-0.102285\pi$
$888$		0			0
$889$		−	17.5695i		−	0.589263i
$890$		0			0
$891$	0.945051			0.0316604
$892$		0			0
$893$			26.7931i			0.896597i
$894$		0			0
$895$	−1.33852			−0.0447418
$896$		0			0
$897$	−25.3857	−	4.74202i	−0.847604	−	0.158331i
$898$		0			0
$899$		−	36.6381i		−	1.22195i
$900$		0			0
$901$	80.5100			2.68218
$902$		0			0
$903$		−	20.2642i		−	0.674352i
$904$		0			0
$905$	−5.39261			−0.179256
$906$		0			0
$907$	−44.5605			−1.47961			−0.739804	−	0.672822i	$-0.765082\pi$
$907$	−44.5605			−1.47961			−0.739804	+	0.672822i	$0.765082\pi$
$908$		0			0
$909$	−5.52273			−0.183177
$910$		0			0
$911$	13.4637			0.446072			0.223036	−	0.974810i	$-0.428403\pi$
$911$	13.4637			0.446072			0.223036	+	0.974810i	$0.428403\pi$
$912$		0			0
$913$	−3.73025			−0.123453
$914$		0			0
$915$		−	10.7499i		−	0.355380i
$916$		0			0
$917$			22.8726i			0.755321i
$918$		0			0
$919$	−29.8335			−0.984116			−0.492058	−	0.870562i	$-0.663755\pi$
$919$	−29.8335			−0.984116			−0.492058	+	0.870562i	$0.663755\pi$
$920$		0			0
$921$	−13.3426			−0.439654
$922$		0			0
$923$		−	51.2281i		−	1.68619i
$924$		0			0
$925$		−	6.23066i		−	0.204863i
$926$		0			0
$927$	−4.36891			−0.143494
$928$		0			0
$929$	−43.1622			−1.41611			−0.708053	−	0.706159i	$-0.750426\pi$
$929$	−43.1622			−1.41611			−0.708053	+	0.706159i	$0.750426\pi$
$930$		0			0
$931$	−4.61170			−0.151142
$932$		0			0
$933$	11.4921			0.376236
$934$		0			0
$935$	6.20609			0.202961
$936$		0			0
$937$			5.42509i			0.177230i	0.996066	+	0.0886149i	$0.0282441\pi$
$937$			5.42509i			0.177230i	−0.996066	+	0.0886149i	$0.971756\pi$
$938$		0			0
$939$	15.8304			0.516605
$940$		0			0
$941$		−	9.03955i		−	0.294681i	−0.989086	−	0.147340i	$-0.952929\pi$
$941$		−	9.03955i		−	0.294681i	0.989086	−	0.147340i	$-0.0470713\pi$
$942$		0			0
$943$	10.3592	−	55.4566i	0.337342	−	1.80592i
$944$		0			0
$945$	2.30439			0.0749618
$946$		0			0
$947$			5.84958i			0.190086i	0.995473	+	0.0950429i	$0.0302988\pi$
$947$			5.84958i			0.190086i	−0.995473	+	0.0950429i	$0.969701\pi$
$948$		0			0
$949$	−28.0384			−0.910164
$950$		0			0
$951$			1.17611i			0.0381379i
$952$		0			0
$953$			35.2447i			1.14169i	0.821059	+	0.570844i	$0.193384\pi$
$953$			35.2447i			1.14169i	−0.821059	+	0.570844i	$0.806616\pi$
$954$		0			0
$955$			1.10757i			0.0358401i
$956$		0			0
$957$		−	5.92359i		−	0.191483i
$958$		0			0
$959$			16.4295i			0.530536i
$960$		0			0
$961$	−3.16701			−0.102162
$962$		0			0
$963$	12.8331			0.413540
$964$		0			0
$965$			23.3586i			0.751939i
$966$		0			0
$967$		−	49.5802i		−	1.59439i	−0.603720	−	0.797197i	$-0.706315\pi$
$967$		−	49.5802i		−	1.59439i	0.603720	−	0.797197i	$-0.293685\pi$
$968$		0			0
$969$	17.9221			0.575742
$970$		0			0
$971$	12.3427			0.396097			0.198049	−	0.980192i	$-0.436540\pi$
$971$	12.3427			0.396097			0.198049	+	0.980192i	$0.436540\pi$
$972$		0			0
$973$		−	9.29011i		−	0.297827i
$974$		0			0
$975$			5.38485i			0.172453i
$976$		0			0
$977$		−	21.5829i		−	0.690498i	−0.938511	−	0.345249i	$-0.887795\pi$
$977$		−	21.5829i		−	0.690498i	0.938511	−	0.345249i	$-0.112205\pi$
$978$		0			0
$979$			2.62247i			0.0838145i
$980$		0			0
$981$			10.6709i			0.340697i
$982$		0			0
$983$	14.1590			0.451602			0.225801	−	0.974174i	$-0.427500\pi$
$983$	14.1590			0.451602			0.225801	+	0.974174i	$0.427500\pi$
$984$		0			0
$985$		−	4.45281i		−	0.141878i
$986$		0			0
$987$	−22.6231			−0.720100
$988$		0			0
$989$	−7.74398	+	41.4563i	−0.246244	+	1.31823i
$990$		0			0
$991$			46.7408i			1.48477i	0.669974	+	0.742385i	$0.266305\pi$
$991$			46.7408i			1.48477i	−0.669974	+	0.742385i	$0.733695\pi$
$992$		0			0
$993$	−23.1828			−0.735684
$994$		0			0
$995$		−	11.9066i		−	0.377465i
$996$		0			0
$997$	47.1951			1.49468			0.747342	−	0.664439i	$-0.231329\pi$
$997$	47.1951			1.49468			0.747342	+	0.664439i	$0.231329\pi$
$998$		0			0
$999$	6.23066			0.197129

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5520.2.be.c.1471.32	yes	32
4.3	odd	2		5520.2.be.d.1471.31	yes	32
23.22	odd	2		5520.2.be.d.1471.32	yes	32
92.91	even	2	inner	5520.2.be.c.1471.31	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5520.2.be.c.1471.31	✓	32	92.91	even	2	inner
5520.2.be.c.1471.32	yes	32	1.1	even	1	trivial
5520.2.be.d.1471.31	yes	32	4.3	odd	2
5520.2.be.d.1471.32	yes	32	23.22	odd	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:	\( N \)	\(=\)	\( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5520.be (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +1.00000i q^{5} +2.30439 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +1.00000i q^{5} +2.30439 q^{7} -1.00000 q^{9} +0.945051 q^{11} -5.38485 q^{13} -1.00000 q^{15} -6.56693i q^{17} +2.72915 q^{19} +2.30439i q^{21} +(0.880622 - 4.71429i) q^{23} -1.00000 q^{25} -1.00000i q^{27} -6.26801 q^{29} +5.84526i q^{31} +0.945051i q^{33} +2.30439i q^{35} +6.23066i q^{37} -5.38485i q^{39} +11.7635 q^{41} -8.79376 q^{43} -1.00000i q^{45} +9.81739i q^{47} -1.68979 q^{49} +6.56693 q^{51} +12.2599i q^{53} +0.945051i q^{55} +2.72915i q^{57} +5.84785i q^{59} +10.7499i q^{61} -2.30439 q^{63} -5.38485i q^{65} -6.05298 q^{67} +(4.71429 + 0.880622i) q^{69} +9.51338i q^{71} +5.20690 q^{73} -1.00000i q^{75} +2.17776 q^{77} -2.04834 q^{79} +1.00000 q^{81} -3.94714 q^{83} +6.56693 q^{85} -6.26801i q^{87} +2.77495i q^{89} -12.4088 q^{91} -5.84526 q^{93} +2.72915i q^{95} +9.24518i q^{97} -0.945051 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 8 q^{7} - 32 q^{9}+O(q^{10}) \) 32 * q - 8 * q^7 - 32 * q^9	\( 32 q - 8 q^{7} - 32 q^{9} + 8 q^{11} - 8 q^{13} - 32 q^{15} - 32 q^{25} + 4 q^{29} + 20 q^{41} + 52 q^{49} - 4 q^{51} + 8 q^{63} + 32 q^{67} - 40 q^{73} - 24 q^{77} + 32 q^{79} + 32 q^{81} - 4 q^{85} - 48 q^{91} - 8 q^{99}+O(q^{100}) \) 32 * q - 8 * q^7 - 32 * q^9 + 8 * q^11 - 8 * q^13 - 32 * q^15 - 32 * q^25 + 4 * q^29 + 20 * q^41 + 52 * q^49 - 4 * q^51 + 8 * q^63 + 32 * q^67 - 40 * q^73 - 24 * q^77 + 32 * q^79 + 32 * q^81 - 4 * q^85 - 48 * q^91 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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