LMFDB - Embedded newform 1075.2.b.f.474.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1075,2,Mod(474,1075)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1075, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

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

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

chi := DirichletCharacter("1075.474");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1075 = 5^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1075.b (of order $2$, degree $1$, not 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:	$8.58391821729$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 43)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			474.1
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1075.474
Dual form			1075.2.b.f.474.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} -1.41421i q^{3} -2.00000 q^{6} -3.41421i q^{7} -2.82843i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.41421i q^{2} -1.41421i q^{3} -2.00000 q^{6} -3.41421i q^{7} -2.82843i q^{8} +1.00000 q^{9} -3.82843 q^{11} +1.82843i q^{13} -4.82843 q^{14} -4.00000 q^{16} +2.17157i q^{17} -1.41421i q^{18} -0.828427 q^{19} -4.82843 q^{21} +5.41421i q^{22} -6.65685i q^{23} -4.00000 q^{24} +2.58579 q^{26} -5.65685i q^{27} +4.24264 q^{29} -3.00000 q^{31} +5.41421i q^{33} +3.07107 q^{34} +8.48528i q^{37} +1.17157i q^{38} +2.58579 q^{39} +1.82843 q^{41} +6.82843i q^{42} -1.00000i q^{43} -9.41421 q^{46} +6.00000i q^{47} +5.65685i q^{48} -4.65685 q^{49} +3.07107 q^{51} -13.8284i q^{53} -8.00000 q^{54} -9.65685 q^{56} +1.17157i q^{57} -6.00000i q^{58} +4.82843 q^{59} -0.242641 q^{61} +4.24264i q^{62} -3.41421i q^{63} -8.00000 q^{64} +7.65685 q^{66} -7.48528i q^{67} -9.41421 q^{69} -3.17157 q^{71} -2.82843i q^{72} +16.2426i q^{73} +12.0000 q^{74} +13.0711i q^{77} -3.65685i q^{78} -4.82843 q^{79} -5.00000 q^{81} -2.58579i q^{82} -3.34315i q^{83} -1.41421 q^{86} -6.00000i q^{87} +10.8284i q^{88} +1.75736 q^{89} +6.24264 q^{91} +4.24264i q^{93} +8.48528 q^{94} +1.82843i q^{97} +6.58579i q^{98} -3.82843 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{6} + 4 q^{9}+O(q^{10}) $ 4 * q - 8 * q^6 + 4 * q^9	$ 4 q - 8 q^{6} + 4 q^{9} - 4 q^{11} - 8 q^{14} - 16 q^{16} + 8 q^{19} - 8 q^{21} - 16 q^{24} + 16 q^{26} - 12 q^{31} - 16 q^{34} + 16 q^{39} - 4 q^{41} - 32 q^{46} + 4 q^{49} - 16 q^{51} - 32 q^{54} - 16 q^{56} + 8 q^{59} + 16 q^{61} - 32 q^{64} + 8 q^{66} - 32 q^{69} - 24 q^{71} + 48 q^{74} - 8 q^{79} - 20 q^{81} + 24 q^{89} + 8 q^{91} - 4 q^{99}+O(q^{100}) $ 4 * q - 8 * q^6 + 4 * q^9 - 4 * q^11 - 8 * q^14 - 16 * q^16 + 8 * q^19 - 8 * q^21 - 16 * q^24 + 16 * q^26 - 12 * q^31 - 16 * q^34 + 16 * q^39 - 4 * q^41 - 32 * q^46 + 4 * q^49 - 16 * q^51 - 32 * q^54 - 16 * q^56 + 8 * q^59 + 16 * q^61 - 32 * q^64 + 8 * q^66 - 32 * q^69 - 24 * q^71 + 48 * q^74 - 8 * q^79 - 20 * q^81 + 24 * q^89 + 8 * q^91 - 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1075\mathbb{Z}\right)^\times$.

$n$	$302$	$476$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 1.41421i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$2$	− 1.41421i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$3$	− 1.41421i	− 0.816497i	−0.912871	−	0.408248i	$-0.866140\pi$
$3$	− 1.41421i	− 0.816497i	0.912871	−	0.408248i	$-0.133860\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	−2.00000	−0.816497
$7$	− 3.41421i	− 1.29045i	−0.763992	−	0.645226i	$-0.776763\pi$
$7$	− 3.41421i	− 1.29045i	0.763992	−	0.645226i	$-0.223237\pi$
$8$	− 2.82843i	− 1.00000i
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.82843	−1.15431	−0.577157	−	0.816633i	$-0.695838\pi$
$11$	−3.82843	−1.15431	−0.577157	+	0.816633i	$0.695838\pi$
$12$	0	0
$13$	1.82843i	0.507114i	0.967320	+	0.253557i	$0.0816006\pi$
$13$	1.82843i	0.507114i	−0.967320	+	0.253557i	$0.918399\pi$
$14$	−4.82843	−1.29045
$15$	0	0
$16$	−4.00000	−1.00000
$17$	2.17157i	0.526684i	0.964703	+	0.263342i	$0.0848247\pi$
$17$	2.17157i	0.526684i	−0.964703	+	0.263342i	$0.915175\pi$
$18$	− 1.41421i	− 0.333333i
$19$	−0.828427	−0.190054	−0.0950271	−	0.995475i	$-0.530294\pi$
$19$	−0.828427	−0.190054	−0.0950271	+	0.995475i	$0.530294\pi$
$20$	0	0
$21$	−4.82843	−1.05365
$22$	5.41421i	1.15431i
$23$	− 6.65685i	− 1.38805i	−0.719951	−	0.694025i	$-0.755836\pi$
$23$	− 6.65685i	− 1.38805i	0.719951	−	0.694025i	$-0.244164\pi$
$24$	−4.00000	−0.816497
$25$	0	0
$26$	2.58579	0.507114
$27$	− 5.65685i	− 1.08866i
$28$	0	0
$29$	4.24264	0.787839	0.393919	−	0.919145i	$-0.371119\pi$
$29$	4.24264	0.787839	0.393919	+	0.919145i	$0.371119\pi$
$30$	0	0
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	0	0
$33$	5.41421i	0.942494i
$34$	3.07107	0.526684
$35$	0	0
$36$	0	0
$37$	8.48528i	1.39497i	0.716599	+	0.697486i	$0.245698\pi$
$37$	8.48528i	1.39497i	−0.716599	+	0.697486i	$0.754302\pi$
$38$	1.17157i	0.190054i
$39$	2.58579	0.414057
$40$	0	0
$41$	1.82843	0.285552	0.142776	−	0.989755i	$-0.454397\pi$
$41$	1.82843	0.285552	0.142776	+	0.989755i	$0.454397\pi$
$42$	6.82843i	1.05365i
$43$	− 1.00000i	− 0.152499i
$43$	− 1.00000i	− 0.152499i
$44$	0	0
$45$	0	0
$46$	−9.41421	−1.38805
$47$	6.00000i	0.875190i	0.899172	+	0.437595i	$0.144170\pi$
$47$	6.00000i	0.875190i	−0.899172	+	0.437595i	$0.855830\pi$
$48$	5.65685i	0.816497i
$49$	−4.65685	−0.665265
$50$	0	0
$51$	3.07107	0.430036
$52$	0	0
$53$	− 13.8284i	− 1.89948i	−0.313039	−	0.949740i	$-0.601347\pi$
$53$	− 13.8284i	− 1.89948i	0.313039	−	0.949740i	$-0.398653\pi$
$54$	−8.00000	−1.08866
$55$	0	0
$56$	−9.65685	−1.29045
$57$	1.17157i	0.155179i
$58$	− 6.00000i	− 0.787839i
$59$	4.82843	0.628608	0.314304	−	0.949322i	$-0.398229\pi$
$59$	4.82843	0.628608	0.314304	+	0.949322i	$0.398229\pi$
$60$	0	0
$61$	−0.242641	−0.0310670	−0.0155335	−	0.999879i	$-0.504945\pi$
$61$	−0.242641	−0.0310670	−0.0155335	+	0.999879i	$0.504945\pi$
$62$	4.24264i	0.538816i
$63$	− 3.41421i	− 0.430150i
$64$	−8.00000	−1.00000
$65$	0	0
$66$	7.65685	0.942494
$67$	− 7.48528i	− 0.914473i	−0.889345	−	0.457236i	$-0.848839\pi$
$67$	− 7.48528i	− 0.914473i	0.889345	−	0.457236i	$-0.151161\pi$
$68$	0	0
$69$	−9.41421	−1.13334
$70$	0	0
$71$	−3.17157	−0.376396	−0.188198	−	0.982131i	$-0.560265\pi$
$71$	−3.17157	−0.376396	−0.188198	+	0.982131i	$0.560265\pi$
$72$	− 2.82843i	− 0.333333i
$73$	16.2426i	1.90106i	0.310637	+	0.950529i	$0.399458\pi$
$73$	16.2426i	1.90106i	−0.310637	+	0.950529i	$0.600542\pi$
$74$	12.0000	1.39497
$75$	0	0
$76$	0	0
$77$	13.0711i	1.48959i
$78$	− 3.65685i	− 0.414057i
$79$	−4.82843	−0.543240	−0.271620	−	0.962405i	$-0.587559\pi$
$79$	−4.82843	−0.543240	−0.271620	+	0.962405i	$0.587559\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	− 2.58579i	− 0.285552i
$83$	− 3.34315i	− 0.366958i	−0.983024	−	0.183479i	$-0.941264\pi$
$83$	− 3.34315i	− 0.366958i	0.983024	−	0.183479i	$-0.0587359\pi$
$84$	0	0
$85$	0	0
$86$	−1.41421	−0.152499
$87$	− 6.00000i	− 0.643268i
$88$	10.8284i	1.15431i
$89$	1.75736	0.186280	0.0931399	−	0.995653i	$-0.470310\pi$
$89$	1.75736	0.186280	0.0931399	+	0.995653i	$0.470310\pi$
$90$	0	0
$91$	6.24264	0.654407
$92$	0	0
$93$	4.24264i	0.439941i
$94$	8.48528	0.875190
$95$	0	0
$96$	0	0
$97$	1.82843i	0.185649i	0.995683	+	0.0928243i	$0.0295895\pi$
$97$	1.82843i	0.185649i	−0.995683	+	0.0928243i	$0.970411\pi$
$98$	6.58579i	0.665265i
$99$	−3.82843	−0.384771
$100$	0	0
$101$	5.82843	0.579950	0.289975	−	0.957034i	$-0.406353\pi$
$101$	5.82843	0.579950	0.289975	+	0.957034i	$0.406353\pi$
$102$	− 4.34315i	− 0.430036i
$103$	− 0.514719i	− 0.0507167i	−0.999678	−	0.0253584i	$-0.991927\pi$
$103$	− 0.514719i	− 0.0507167i	0.999678	−	0.0253584i	$-0.00807268\pi$
$104$	5.17157	0.507114
$105$	0	0
$106$	−19.5563	−1.89948
$107$	− 0.343146i	− 0.0331732i	−0.999862	−	0.0165866i	$-0.994720\pi$
$107$	− 0.343146i	− 0.0331732i	0.999862	−	0.0165866i	$-0.00527991\pi$
$108$	0	0
$109$	19.9706	1.91283	0.956416	−	0.292006i	$-0.0943227\pi$
$109$	19.9706	1.91283	0.956416	+	0.292006i	$0.0943227\pi$
$110$	0	0
$111$	12.0000	1.13899
$112$	13.6569i	1.29045i
$113$	6.82843i	0.642364i	0.947017	+	0.321182i	$0.104080\pi$
$113$	6.82843i	0.642364i	−0.947017	+	0.321182i	$0.895920\pi$
$114$	1.65685	0.155179
$115$	0	0
$116$	0	0
$117$	1.82843i	0.169038i
$118$	− 6.82843i	− 0.628608i
$119$	7.41421	0.679660
$120$	0	0
$121$	3.65685	0.332441
$122$	0.343146i	0.0310670i
$123$	− 2.58579i	− 0.233153i
$124$	0	0
$125$	0	0
$126$	−4.82843	−0.430150
$127$	3.82843i	0.339718i	0.985468	+	0.169859i	$0.0543312\pi$
$127$	3.82843i	0.339718i	−0.985468	+	0.169859i	$0.945669\pi$
$128$	11.3137i	1.00000i
$129$	−1.41421	−0.124515
$130$	0	0
$131$	9.65685	0.843723	0.421862	−	0.906660i	$-0.361377\pi$
$131$	9.65685	0.843723	0.421862	+	0.906660i	$0.361377\pi$
$132$	0	0
$133$	2.82843i	0.245256i
$134$	−10.5858	−0.914473
$135$	0	0
$136$	6.14214	0.526684
$137$	− 14.4853i	− 1.23756i	−0.785564	−	0.618781i	$-0.787627\pi$
$137$	− 14.4853i	− 1.23756i	0.785564	−	0.618781i	$-0.212373\pi$
$138$	13.3137i	1.13334i
$139$	−5.48528	−0.465255	−0.232628	−	0.972566i	$-0.574732\pi$
$139$	−5.48528	−0.465255	−0.232628	+	0.972566i	$0.574732\pi$
$140$	0	0
$141$	8.48528	0.714590
$142$	4.48528i	0.376396i
$143$	− 7.00000i	− 0.585369i
$144$	−4.00000	−0.333333
$145$	0	0
$146$	22.9706	1.90106
$147$	6.58579i	0.543187i
$148$	0	0
$149$	−8.48528	−0.695141	−0.347571	−	0.937654i	$-0.612993\pi$
$149$	−8.48528	−0.695141	−0.347571	+	0.937654i	$0.612993\pi$
$150$	0	0
$151$	18.2426	1.48457	0.742283	−	0.670087i	$-0.233743\pi$
$151$	18.2426	1.48457	0.742283	+	0.670087i	$0.233743\pi$
$152$	2.34315i	0.190054i
$153$	2.17157i	0.175561i
$154$	18.4853	1.48959
$155$	0	0
$156$	0	0
$157$	− 10.0000i	− 0.798087i	−0.916932	−	0.399043i	$-0.869342\pi$
$157$	− 10.0000i	− 0.798087i	0.916932	−	0.399043i	$-0.130658\pi$
$158$	6.82843i	0.543240i
$159$	−19.5563	−1.55092
$160$	0	0
$161$	−22.7279	−1.79121
$162$	7.07107i	0.555556i
$163$	11.7574i	0.920907i	0.887684	+	0.460454i	$0.152313\pi$
$163$	11.7574i	0.920907i	−0.887684	+	0.460454i	$0.847687\pi$
$164$	0	0
$165$	0	0
$166$	−4.72792	−0.366958
$167$	− 14.3137i	− 1.10763i	−0.832640	−	0.553814i	$-0.813172\pi$
$167$	− 14.3137i	− 1.10763i	0.832640	−	0.553814i	$-0.186828\pi$
$168$	13.6569i	1.05365i
$169$	9.65685	0.742835
$170$	0	0
$171$	−0.828427	−0.0633514
$172$	0	0
$173$	− 23.6569i	− 1.79860i	−0.437335	−	0.899299i	$-0.644078\pi$
$173$	− 23.6569i	− 1.79860i	0.437335	−	0.899299i	$-0.355922\pi$
$174$	−8.48528	−0.643268
$175$	0	0
$176$	15.3137	1.15431
$177$	− 6.82843i	− 0.513256i
$178$	− 2.48528i	− 0.186280i
$179$	4.58579	0.342758	0.171379	−	0.985205i	$-0.445178\pi$
$179$	4.58579	0.342758	0.171379	+	0.985205i	$0.445178\pi$
$180$	0	0
$181$	7.31371	0.543624	0.271812	−	0.962350i	$-0.412377\pi$
$181$	7.31371	0.543624	0.271812	+	0.962350i	$0.412377\pi$
$182$	− 8.82843i	− 0.654407i
$183$	0.343146i	0.0253661i
$184$	−18.8284	−1.38805
$185$	0	0
$186$	6.00000	0.439941
$187$	− 8.31371i	− 0.607959i
$188$	0	0
$189$	−19.3137	−1.40487
$190$	0	0
$191$	−6.14214	−0.444429	−0.222215	−	0.974998i	$-0.571329\pi$
$191$	−6.14214	−0.444429	−0.222215	+	0.974998i	$0.571329\pi$
$192$	11.3137i	0.816497i
$193$	− 15.9706i	− 1.14959i	−0.818299	−	0.574793i	$-0.805083\pi$
$193$	− 15.9706i	− 1.14959i	0.818299	−	0.574793i	$-0.194917\pi$
$194$	2.58579	0.185649
$195$	0	0
$196$	0	0
$197$	− 14.1421i	− 1.00759i	−0.863825	−	0.503793i	$-0.831938\pi$
$197$	− 14.1421i	− 1.00759i	0.863825	−	0.503793i	$-0.168062\pi$
$198$	5.41421i	0.384771i
$199$	−7.65685	−0.542780	−0.271390	−	0.962469i	$-0.587483\pi$
$199$	−7.65685	−0.542780	−0.271390	+	0.962469i	$0.587483\pi$
$200$	0	0
$201$	−10.5858	−0.746664
$202$	− 8.24264i	− 0.579950i
$203$	− 14.4853i	− 1.01667i
$204$	0	0
$205$	0	0
$206$	−0.727922	−0.0507167
$207$	− 6.65685i	− 0.462683i
$208$	− 7.31371i	− 0.507114i
$209$	3.17157	0.219382
$210$	0	0
$211$	−0.142136	−0.00978502	−0.00489251	−	0.999988i	$-0.501557\pi$
$211$	−0.142136	−0.00978502	−0.00489251	+	0.999988i	$0.501557\pi$
$212$	0	0
$213$	4.48528i	0.307326i
$214$	−0.485281	−0.0331732
$215$	0	0
$216$	−16.0000	−1.08866
$217$	10.2426i	0.695316i
$218$	− 28.2426i	− 1.91283i
$219$	22.9706	1.55221
$220$	0	0
$221$	−3.97056	−0.267089
$222$	− 16.9706i	− 1.13899i
$223$	− 4.10051i	− 0.274590i	−0.990530	−	0.137295i	$-0.956159\pi$
$223$	− 4.10051i	− 0.274590i	0.990530	−	0.137295i	$-0.0438409\pi$
$224$	0	0
$225$	0	0
$226$	9.65685	0.642364
$227$	− 6.34315i	− 0.421009i	−0.977593	−	0.210505i	$-0.932489\pi$
$227$	− 6.34315i	− 0.421009i	0.977593	−	0.210505i	$-0.0675107\pi$
$228$	0	0
$229$	25.9706	1.71618	0.858092	−	0.513497i	$-0.171650\pi$
$229$	25.9706	1.71618	0.858092	+	0.513497i	$0.171650\pi$
$230$	0	0
$231$	18.4853	1.21624
$232$	− 12.0000i	− 0.787839i
$233$	− 4.82843i	− 0.316321i	−0.987413	−	0.158160i	$-0.949444\pi$
$233$	− 4.82843i	− 0.316321i	0.987413	−	0.158160i	$-0.0505563\pi$
$234$	2.58579	0.169038
$235$	0	0
$236$	0	0
$237$	6.82843i	0.443554i
$238$	− 10.4853i	− 0.679660i
$239$	−14.4853	−0.936975	−0.468487	−	0.883470i	$-0.655201\pi$
$239$	−14.4853	−0.936975	−0.468487	+	0.883470i	$0.655201\pi$
$240$	0	0
$241$	4.00000	0.257663	0.128831	−	0.991667i	$-0.458877\pi$
$241$	4.00000	0.257663	0.128831	+	0.991667i	$0.458877\pi$
$242$	− 5.17157i	− 0.332441i
$243$	− 9.89949i	− 0.635053i
$244$	0	0
$245$	0	0
$246$	−3.65685	−0.233153
$247$	− 1.51472i	− 0.0963792i
$248$	8.48528i	0.538816i
$249$	−4.72792	−0.299620
$250$	0	0
$251$	23.1421	1.46072	0.730359	−	0.683063i	$-0.239353\pi$
$251$	23.1421	1.46072	0.730359	+	0.683063i	$0.239353\pi$
$252$	0	0
$253$	25.4853i	1.60225i
$254$	5.41421	0.339718
$255$	0	0
$256$	0	0
$257$	− 1.75736i	− 0.109621i	−0.998497	−	0.0548105i	$-0.982545\pi$
$257$	− 1.75736i	− 0.109621i	0.998497	−	0.0548105i	$-0.0174555\pi$
$258$	2.00000i	0.124515i
$259$	28.9706	1.80014
$260$	0	0
$261$	4.24264	0.262613
$262$	− 13.6569i	− 0.843723i
$263$	− 0.142136i	− 0.00876446i	−0.999990	−	0.00438223i	$-0.998605\pi$
$263$	− 0.142136i	− 0.00876446i	0.999990	−	0.00438223i	$-0.00139491\pi$
$264$	15.3137	0.942494
$265$	0	0
$266$	4.00000	0.245256
$267$	− 2.48528i	− 0.152097i
$268$	0	0
$269$	3.00000	0.182913	0.0914566	−	0.995809i	$-0.470848\pi$
$269$	3.00000	0.182913	0.0914566	+	0.995809i	$0.470848\pi$
$270$	0	0
$271$	−25.9706	−1.57760	−0.788800	−	0.614650i	$-0.789297\pi$
$271$	−25.9706	−1.57760	−0.788800	+	0.614650i	$0.789297\pi$
$272$	− 8.68629i	− 0.526684i
$273$	− 8.82843i	− 0.534321i
$274$	−20.4853	−1.23756
$275$	0	0
$276$	0	0
$277$	7.89949i	0.474635i	0.971432	+	0.237317i	$0.0762681\pi$
$277$	7.89949i	0.474635i	−0.971432	+	0.237317i	$0.923732\pi$
$278$	7.75736i	0.465255i
$279$	−3.00000	−0.179605
$280$	0	0
$281$	−8.65685	−0.516425	−0.258212	−	0.966088i	$-0.583133\pi$
$281$	−8.65685	−0.516425	−0.258212	+	0.966088i	$0.583133\pi$
$282$	− 12.0000i	− 0.714590i
$283$	− 18.3137i	− 1.08864i	−0.838879	−	0.544318i	$-0.816788\pi$
$283$	− 18.3137i	− 1.08864i	0.838879	−	0.544318i	$-0.183212\pi$
$284$	0	0
$285$	0	0
$286$	−9.89949	−0.585369
$287$	− 6.24264i	− 0.368491i
$288$	0	0
$289$	12.2843	0.722604
$290$	0	0
$291$	2.58579	0.151581
$292$	0	0
$293$	− 10.3431i	− 0.604253i	−0.953268	−	0.302127i	$-0.902303\pi$
$293$	− 10.3431i	− 0.604253i	0.953268	−	0.302127i	$-0.0976965\pi$
$294$	9.31371	0.543187
$295$	0	0
$296$	24.0000	1.39497
$297$	21.6569i	1.25666i
$298$	12.0000i	0.695141i
$299$	12.1716	0.703900
$300$	0	0
$301$	−3.41421	−0.196792
$302$	− 25.7990i	− 1.48457i
$303$	− 8.24264i	− 0.473527i
$304$	3.31371	0.190054
$305$	0	0
$306$	3.07107	0.175561
$307$	26.7990i	1.52950i	0.644328	+	0.764750i	$0.277137\pi$
$307$	26.7990i	1.52950i	−0.644328	+	0.764750i	$0.722863\pi$
$308$	0	0
$309$	−0.727922	−0.0414100
$310$	0	0
$311$	−13.9706	−0.792198	−0.396099	−	0.918208i	$-0.629636\pi$
$311$	−13.9706	−0.792198	−0.396099	+	0.918208i	$0.629636\pi$
$312$	− 7.31371i	− 0.414057i
$313$	25.2132i	1.42513i	0.701604	+	0.712567i	$0.252468\pi$
$313$	25.2132i	1.42513i	−0.701604	+	0.712567i	$0.747532\pi$
$314$	−14.1421	−0.798087
$315$	0	0
$316$	0	0
$317$	25.8284i	1.45067i	0.688397	+	0.725334i	$0.258315\pi$
$317$	25.8284i	1.45067i	−0.688397	+	0.725334i	$0.741685\pi$
$318$	27.6569i	1.55092i
$319$	−16.2426	−0.909413
$320$	0	0
$321$	−0.485281	−0.0270858
$322$	32.1421i	1.79121i
$323$	− 1.79899i	− 0.100098i
$324$	0	0
$325$	0	0
$326$	16.6274	0.920907
$327$	− 28.2426i	− 1.56182i
$328$	− 5.17157i	− 0.285552i
$329$	20.4853	1.12939
$330$	0	0
$331$	21.4142	1.17703	0.588516	−	0.808486i	$-0.299712\pi$
$331$	21.4142	1.17703	0.588516	+	0.808486i	$0.299712\pi$
$332$	0	0
$333$	8.48528i	0.464991i
$334$	−20.2426	−1.10763
$335$	0	0
$336$	19.3137	1.05365
$337$	− 5.00000i	− 0.272367i	−0.990684	−	0.136184i	$-0.956516\pi$
$337$	− 5.00000i	− 0.272367i	0.990684	−	0.136184i	$-0.0434837\pi$
$338$	− 13.6569i	− 0.742835i
$339$	9.65685	0.524488
$340$	0	0
$341$	11.4853	0.621963
$342$	1.17157i	0.0633514i
$343$	− 8.00000i	− 0.431959i
$344$	−2.82843	−0.152499
$345$	0	0
$346$	−33.4558	−1.79860
$347$	− 4.58579i	− 0.246178i	−0.992396	−	0.123089i	$-0.960720\pi$
$347$	− 4.58579i	− 0.246178i	0.992396	−	0.123089i	$-0.0392800\pi$
$348$	0	0
$349$	20.2426	1.08356	0.541782	−	0.840519i	$-0.317750\pi$
$349$	20.2426	1.08356	0.541782	+	0.840519i	$0.317750\pi$
$350$	0	0
$351$	10.3431	0.552076
$352$	0	0
$353$	− 5.82843i	− 0.310216i	−0.987898	−	0.155108i	$-0.950427\pi$
$353$	− 5.82843i	− 0.310216i	0.987898	−	0.155108i	$-0.0495725\pi$
$354$	−9.65685	−0.513256
$355$	0	0
$356$	0	0
$357$	− 10.4853i	− 0.554940i
$358$	− 6.48528i	− 0.342758i
$359$	21.3431	1.12645	0.563224	−	0.826304i	$-0.309561\pi$
$359$	21.3431	1.12645	0.563224	+	0.826304i	$0.309561\pi$
$360$	0	0
$361$	−18.3137	−0.963879
$362$	− 10.3431i	− 0.543624i
$363$	− 5.17157i	− 0.271437i
$364$	0	0
$365$	0	0
$366$	0.485281	0.0253661
$367$	9.79899i	0.511503i	0.966743	+	0.255752i	$0.0823229\pi$
$367$	9.79899i	0.511503i	−0.966743	+	0.255752i	$0.917677\pi$
$368$	26.6274i	1.38805i
$369$	1.82843	0.0951841
$370$	0	0
$371$	−47.2132	−2.45119
$372$	0	0
$373$	− 8.48528i	− 0.439351i	−0.975573	−	0.219676i	$-0.929500\pi$
$373$	− 8.48528i	− 0.439351i	0.975573	−	0.219676i	$-0.0704999\pi$
$374$	−11.7574	−0.607959
$375$	0	0
$376$	16.9706	0.875190
$377$	7.75736i	0.399524i
$378$	27.3137i	1.40487i
$379$	−30.3137	−1.55711	−0.778555	−	0.627576i	$-0.784047\pi$
$379$	−30.3137	−1.55711	−0.778555	+	0.627576i	$0.784047\pi$
$380$	0	0
$381$	5.41421	0.277379
$382$	8.68629i	0.444429i
$383$	20.4853i	1.04675i	0.852103	+	0.523374i	$0.175327\pi$
$383$	20.4853i	1.04675i	−0.852103	+	0.523374i	$0.824673\pi$
$384$	16.0000	0.816497
$385$	0	0
$386$	−22.5858	−1.14959
$387$	− 1.00000i	− 0.0508329i
$388$	0	0
$389$	16.6274	0.843044	0.421522	−	0.906818i	$-0.361496\pi$
$389$	16.6274	0.843044	0.421522	+	0.906818i	$0.361496\pi$
$390$	0	0
$391$	14.4558	0.731063
$392$	13.1716i	0.665265i
$393$	− 13.6569i	− 0.688897i
$394$	−20.0000	−1.00759
$395$	0	0
$396$	0	0
$397$	29.4558i	1.47835i	0.673515	+	0.739173i	$0.264784\pi$
$397$	29.4558i	1.47835i	−0.673515	+	0.739173i	$0.735216\pi$
$398$	10.8284i	0.542780i
$399$	4.00000	0.200250
$400$	0	0
$401$	−12.5147	−0.624955	−0.312478	−	0.949925i	$-0.601159\pi$
$401$	−12.5147	−0.624955	−0.312478	+	0.949925i	$0.601159\pi$
$402$	14.9706i	0.746664i
$403$	− 5.48528i	− 0.273241i
$404$	0	0
$405$	0	0
$406$	−20.4853	−1.01667
$407$	− 32.4853i	− 1.61024i
$408$	− 8.68629i	− 0.430036i
$409$	16.8701	0.834171	0.417085	−	0.908867i	$-0.363052\pi$
$409$	16.8701	0.834171	0.417085	+	0.908867i	$0.363052\pi$
$410$	0	0
$411$	−20.4853	−1.01046
$412$	0	0
$413$	− 16.4853i	− 0.811188i
$414$	−9.41421	−0.462683
$415$	0	0
$416$	0	0
$417$	7.75736i	0.379880i
$418$	− 4.48528i	− 0.219382i
$419$	23.8995	1.16757	0.583783	−	0.811909i	$-0.301572\pi$
$419$	23.8995	1.16757	0.583783	+	0.811909i	$0.301572\pi$
$420$	0	0
$421$	−15.6569	−0.763068	−0.381534	−	0.924355i	$-0.624604\pi$
$421$	−15.6569	−0.763068	−0.381534	+	0.924355i	$0.624604\pi$
$422$	0.201010i	0.00978502i
$423$	6.00000i	0.291730i
$424$	−39.1127	−1.89948
$425$	0	0
$426$	6.34315	0.307326
$427$	0.828427i	0.0400904i
$428$	0	0
$429$	−9.89949	−0.477952
$430$	0	0
$431$	23.2843	1.12156	0.560782	−	0.827964i	$-0.310501\pi$
$431$	23.2843	1.12156	0.560782	+	0.827964i	$0.310501\pi$
$432$	22.6274i	1.08866i
$433$	− 21.7574i	− 1.04559i	−0.852458	−	0.522796i	$-0.824889\pi$
$433$	− 21.7574i	− 1.04559i	0.852458	−	0.522796i	$-0.175111\pi$
$434$	14.4853	0.695316
$435$	0	0
$436$	0	0
$437$	5.51472i	0.263805i
$438$	− 32.4853i	− 1.55221i
$439$	−11.4853	−0.548163	−0.274081	−	0.961707i	$-0.588374\pi$
$439$	−11.4853	−0.548163	−0.274081	+	0.961707i	$0.588374\pi$
$440$	0	0
$441$	−4.65685	−0.221755
$442$	5.61522i	0.267089i
$443$	− 7.85786i	− 0.373338i	−0.982423	−	0.186669i	$-0.940231\pi$
$443$	− 7.85786i	− 0.373338i	0.982423	−	0.186669i	$-0.0597693\pi$
$444$	0	0
$445$	0	0
$446$	−5.79899	−0.274590
$447$	12.0000i	0.567581i
$448$	27.3137i	1.29045i
$449$	−33.2132	−1.56743	−0.783714	−	0.621122i	$-0.786677\pi$
$449$	−33.2132	−1.56743	−0.783714	+	0.621122i	$0.786677\pi$
$450$	0	0
$451$	−7.00000	−0.329617
$452$	0	0
$453$	− 25.7990i	− 1.21214i
$454$	−8.97056	−0.421009
$455$	0	0
$456$	3.31371	0.155179
$457$	0.727922i	0.0340508i	0.999855	+	0.0170254i	$0.00541961\pi$
$457$	0.727922i	0.0340508i	−0.999855	+	0.0170254i	$0.994580\pi$
$458$	− 36.7279i	− 1.71618i
$459$	12.2843	0.573381
$460$	0	0
$461$	42.6274	1.98536	0.992678	−	0.120788i	$-0.0385420\pi$
$461$	42.6274	1.98536	0.992678	+	0.120788i	$0.0385420\pi$
$462$	− 26.1421i	− 1.21624i
$463$	30.7279i	1.42805i	0.700121	+	0.714024i	$0.253129\pi$
$463$	30.7279i	1.42805i	−0.700121	+	0.714024i	$0.746871\pi$
$464$	−16.9706	−0.787839
$465$	0	0
$466$	−6.82843	−0.316321
$467$	23.6569i	1.09471i	0.836901	+	0.547354i	$0.184365\pi$
$467$	23.6569i	1.09471i	−0.836901	+	0.547354i	$0.815635\pi$
$468$	0	0
$469$	−25.5563	−1.18008
$470$	0	0
$471$	−14.1421	−0.651635
$472$	− 13.6569i	− 0.628608i
$473$	3.82843i	0.176031i
$474$	9.65685	0.443554
$475$	0	0
$476$	0	0
$477$	− 13.8284i	− 0.633160i
$478$	20.4853i	0.936975i
$479$	6.65685	0.304159	0.152080	−	0.988368i	$-0.451403\pi$
$479$	6.65685	0.304159	0.152080	+	0.988368i	$0.451403\pi$
$480$	0	0
$481$	−15.5147	−0.707410
$482$	− 5.65685i	− 0.257663i
$483$	32.1421i	1.46252i
$484$	0	0
$485$	0	0
$486$	−14.0000	−0.635053
$487$	22.3431i	1.01246i	0.862397	+	0.506232i	$0.168962\pi$
$487$	22.3431i	1.01246i	−0.862397	+	0.506232i	$0.831038\pi$
$488$	0.686292i	0.0310670i
$489$	16.6274	0.751918
$490$	0	0
$491$	13.4142	0.605375	0.302687	−	0.953090i	$-0.402116\pi$
$491$	13.4142	0.605375	0.302687	+	0.953090i	$0.402116\pi$
$492$	0	0
$493$	9.21320i	0.414942i
$494$	−2.14214	−0.0963792
$495$	0	0
$496$	12.0000	0.538816
$497$	10.8284i	0.485721i
$498$	6.68629i	0.299620i
$499$	25.7574	1.15306	0.576529	−	0.817077i	$-0.304407\pi$
$499$	25.7574	1.15306	0.576529	+	0.817077i	$0.304407\pi$
$500$	0	0
$501$	−20.2426	−0.904374
$502$	− 32.7279i	− 1.46072i
$503$	30.7696i	1.37195i	0.727627	+	0.685973i	$0.240623\pi$
$503$	30.7696i	1.37195i	−0.727627	+	0.685973i	$0.759377\pi$
$504$	−9.65685	−0.430150
$505$	0	0
$506$	36.0416	1.60225
$507$	− 13.6569i	− 0.606522i
$508$	0	0
$509$	−0.514719	−0.0228145	−0.0114073	−	0.999935i	$-0.503631\pi$
$509$	−0.514719	−0.0228145	−0.0114073	+	0.999935i	$0.503631\pi$
$510$	0	0
$511$	55.4558	2.45322
$512$	22.6274i	1.00000i
$513$	4.68629i	0.206905i
$514$	−2.48528	−0.109621
$515$	0	0
$516$	0	0
$517$	− 22.9706i	− 1.01024i
$518$	− 40.9706i	− 1.80014i
$519$	−33.4558	−1.46855
$520$	0	0
$521$	16.9289	0.741670	0.370835	−	0.928699i	$-0.379072\pi$
$521$	16.9289	0.741670	0.370835	+	0.928699i	$0.379072\pi$
$522$	− 6.00000i	− 0.262613i
$523$	39.2132i	1.71467i	0.514756	+	0.857337i	$0.327883\pi$
$523$	39.2132i	1.71467i	−0.514756	+	0.857337i	$0.672117\pi$
$524$	0	0
$525$	0	0
$526$	−0.201010	−0.00876446
$527$	− 6.51472i	− 0.283786i
$528$	− 21.6569i	− 0.942494i
$529$	−21.3137	−0.926683
$530$	0	0
$531$	4.82843	0.209536
$532$	0	0
$533$	3.34315i	0.144808i
$534$	−3.51472	−0.152097
$535$	0	0
$536$	−21.1716	−0.914473
$537$	− 6.48528i	− 0.279861i
$538$	− 4.24264i	− 0.182913i
$539$	17.8284	0.767925
$540$	0	0
$541$	8.79899	0.378298	0.189149	−	0.981948i	$-0.439427\pi$
$541$	8.79899	0.378298	0.189149	+	0.981948i	$0.439427\pi$
$542$	36.7279i	1.57760i
$543$	− 10.3431i	− 0.443867i
$544$	0	0
$545$	0	0
$546$	−12.4853	−0.534321
$547$	9.00000i	0.384812i	0.981315	+	0.192406i	$0.0616291\pi$
$547$	9.00000i	0.384812i	−0.981315	+	0.192406i	$0.938371\pi$
$548$	0	0
$549$	−0.242641	−0.0103557
$550$	0	0
$551$	−3.51472	−0.149732
$552$	26.6274i	1.13334i
$553$	16.4853i	0.701025i
$554$	11.1716	0.474635
$555$	0	0
$556$	0	0
$557$	− 9.00000i	− 0.381342i	−0.981654	−	0.190671i	$-0.938934\pi$
$557$	− 9.00000i	− 0.381342i	0.981654	−	0.190671i	$-0.0610664\pi$
$558$	4.24264i	0.179605i
$559$	1.82843	0.0773342
$560$	0	0
$561$	−11.7574	−0.496396
$562$	12.2426i	0.516425i
$563$	− 39.6274i	− 1.67010i	−0.550177	−	0.835048i	$-0.685440\pi$
$563$	− 39.6274i	− 1.67010i	0.550177	−	0.835048i	$-0.314560\pi$
$564$	0	0
$565$	0	0
$566$	−25.8995	−1.08864
$567$	17.0711i	0.716917i
$568$	8.97056i	0.376396i
$569$	−13.3431	−0.559374	−0.279687	−	0.960091i	$-0.590231\pi$
$569$	−13.3431	−0.559374	−0.279687	+	0.960091i	$0.590231\pi$
$570$	0	0
$571$	−3.07107	−0.128520	−0.0642601	−	0.997933i	$-0.520469\pi$
$571$	−3.07107	−0.128520	−0.0642601	+	0.997933i	$0.520469\pi$
$572$	0	0
$573$	8.68629i	0.362875i
$574$	−8.82843	−0.368491
$575$	0	0
$576$	−8.00000	−0.333333
$577$	33.0711i	1.37677i	0.725347	+	0.688383i	$0.241679\pi$
$577$	33.0711i	1.37677i	−0.725347	+	0.688383i	$0.758321\pi$
$578$	− 17.3726i	− 0.722604i
$579$	−22.5858	−0.938633
$580$	0	0
$581$	−11.4142	−0.473541
$582$	− 3.65685i	− 0.151581i
$583$	52.9411i	2.19260i
$584$	45.9411	1.90106
$585$	0	0
$586$	−14.6274	−0.604253
$587$	33.7990i	1.39503i	0.716568	+	0.697517i	$0.245712\pi$
$587$	33.7990i	1.39503i	−0.716568	+	0.697517i	$0.754288\pi$
$588$	0	0
$589$	2.48528	0.102404
$590$	0	0
$591$	−20.0000	−0.822690
$592$	− 33.9411i	− 1.39497i
$593$	9.07107i	0.372504i	0.982502	+	0.186252i	$0.0596341\pi$
$593$	9.07107i	0.372504i	−0.982502	+	0.186252i	$0.940366\pi$
$594$	30.6274	1.25666
$595$	0	0
$596$	0	0
$597$	10.8284i	0.443178i
$598$	− 17.2132i	− 0.703900i
$599$	−36.6569	−1.49776	−0.748879	−	0.662706i	$-0.769408\pi$
$599$	−36.6569	−1.49776	−0.748879	+	0.662706i	$0.769408\pi$
$600$	0	0
$601$	−24.9706	−1.01857	−0.509285	−	0.860598i	$-0.670090\pi$
$601$	−24.9706	−1.01857	−0.509285	+	0.860598i	$0.670090\pi$
$602$	4.82843i	0.196792i
$603$	− 7.48528i	− 0.304824i
$604$	0	0
$605$	0	0
$606$	−11.6569	−0.473527
$607$	− 32.9706i	− 1.33823i	−0.743157	−	0.669117i	$-0.766673\pi$
$607$	− 32.9706i	− 1.33823i	0.743157	−	0.669117i	$-0.233327\pi$
$608$	0	0
$609$	−20.4853	−0.830105
$610$	0	0
$611$	−10.9706	−0.443821
$612$	0	0
$613$	− 6.82843i	− 0.275798i	−0.990446	−	0.137899i	$-0.955965\pi$
$613$	− 6.82843i	− 0.275798i	0.990446	−	0.137899i	$-0.0440349\pi$
$614$	37.8995	1.52950
$615$	0	0
$616$	36.9706	1.48959
$617$	19.9706i	0.803985i	0.915643	+	0.401992i	$0.131682\pi$
$617$	19.9706i	0.803985i	−0.915643	+	0.401992i	$0.868318\pi$
$618$	1.02944i	0.0414100i
$619$	−28.9706	−1.16443	−0.582213	−	0.813037i	$-0.697813\pi$
$619$	−28.9706	−1.16443	−0.582213	+	0.813037i	$0.697813\pi$
$620$	0	0
$621$	−37.6569	−1.51112
$622$	19.7574i	0.792198i
$623$	− 6.00000i	− 0.240385i
$624$	−10.3431	−0.414057
$625$	0	0
$626$	35.6569	1.42513
$627$	− 4.48528i	− 0.179125i
$628$	0	0
$629$	−18.4264	−0.734709
$630$	0	0
$631$	46.7696	1.86187	0.930933	−	0.365189i	$-0.118996\pi$
$631$	46.7696	1.86187	0.930933	+	0.365189i	$0.118996\pi$
$632$	13.6569i	0.543240i
$633$	0.201010i	0.00798944i
$634$	36.5269	1.45067
$635$	0	0
$636$	0	0
$637$	− 8.51472i	− 0.337365i
$638$	22.9706i	0.909413i
$639$	−3.17157	−0.125465
$640$	0	0
$641$	−29.5563	−1.16741	−0.583703	−	0.811967i	$-0.698397\pi$
$641$	−29.5563	−1.16741	−0.583703	+	0.811967i	$0.698397\pi$
$642$	0.686292i	0.0270858i
$643$	26.4853i	1.04448i	0.852799	+	0.522239i	$0.174903\pi$
$643$	26.4853i	1.04448i	−0.852799	+	0.522239i	$0.825097\pi$
$644$	0	0
$645$	0	0
$646$	−2.54416	−0.100098
$647$	− 1.17157i	− 0.0460593i	−0.999735	−	0.0230296i	$-0.992669\pi$
$647$	− 1.17157i	− 0.0460593i	0.999735	−	0.0230296i	$-0.00733121\pi$
$648$	14.1421i	0.555556i
$649$	−18.4853	−0.725611
$650$	0	0
$651$	14.4853	0.567723
$652$	0	0
$653$	− 15.1716i	− 0.593710i	−0.954923	−	0.296855i	$-0.904062\pi$
$653$	− 15.1716i	− 0.593710i	0.954923	−	0.296855i	$-0.0959377\pi$
$654$	−39.9411	−1.56182
$655$	0	0
$656$	−7.31371	−0.285552
$657$	16.2426i	0.633686i
$658$	− 28.9706i	− 1.12939i
$659$	−16.3137	−0.635492	−0.317746	−	0.948176i	$-0.602926\pi$
$659$	−16.3137	−0.635492	−0.317746	+	0.948176i	$0.602926\pi$
$660$	0	0
$661$	−17.0000	−0.661223	−0.330612	−	0.943767i	$-0.607255\pi$
$661$	−17.0000	−0.661223	−0.330612	+	0.943767i	$0.607255\pi$
$662$	− 30.2843i	− 1.17703i
$663$	5.61522i	0.218077i
$664$	−9.45584	−0.366958
$665$	0	0
$666$	12.0000	0.464991
$667$	− 28.2426i	− 1.09356i
$668$	0	0
$669$	−5.79899	−0.224202
$670$	0	0
$671$	0.928932	0.0358610
$672$	0	0
$673$	16.7279i	0.644814i	0.946601	+	0.322407i	$0.104492\pi$
$673$	16.7279i	0.644814i	−0.946601	+	0.322407i	$0.895508\pi$
$674$	−7.07107	−0.272367
$675$	0	0
$676$	0	0
$677$	11.1716i	0.429358i	0.976685	+	0.214679i	$0.0688706\pi$
$677$	11.1716i	0.429358i	−0.976685	+	0.214679i	$0.931129\pi$
$678$	− 13.6569i	− 0.524488i
$679$	6.24264	0.239571
$680$	0	0
$681$	−8.97056	−0.343753
$682$	− 16.2426i	− 0.621963i
$683$	46.4558i	1.77758i	0.458311	+	0.888792i	$0.348454\pi$
$683$	46.4558i	1.77758i	−0.458311	+	0.888792i	$0.651546\pi$
$684$	0	0
$685$	0	0
$686$	−11.3137	−0.431959
$687$	− 36.7279i	− 1.40126i
$688$	4.00000i	0.152499i
$689$	25.2843	0.963254
$690$	0	0
$691$	−26.7279	−1.01678	−0.508389	−	0.861128i	$-0.669759\pi$
$691$	−26.7279	−1.01678	−0.508389	+	0.861128i	$0.669759\pi$
$692$	0	0
$693$	13.0711i	0.496529i
$694$	−6.48528	−0.246178
$695$	0	0
$696$	−16.9706	−0.643268
$697$	3.97056i	0.150396i
$698$	− 28.6274i	− 1.08356i
$699$	−6.82843	−0.258275
$700$	0	0
$701$	−29.6569	−1.12012	−0.560062	−	0.828451i	$-0.689223\pi$
$701$	−29.6569	−1.12012	−0.560062	+	0.828451i	$0.689223\pi$
$702$	− 14.6274i	− 0.552076i
$703$	− 7.02944i	− 0.265120i
$704$	30.6274	1.15431
$705$	0	0
$706$	−8.24264	−0.310216
$707$	− 19.8995i	− 0.748398i
$708$	0	0
$709$	38.1127	1.43135	0.715676	−	0.698432i	$-0.246119\pi$
$709$	38.1127	1.43135	0.715676	+	0.698432i	$0.246119\pi$
$710$	0	0
$711$	−4.82843	−0.181080
$712$	− 4.97056i	− 0.186280i
$713$	19.9706i	0.747903i
$714$	−14.8284	−0.554940
$715$	0	0
$716$	0	0
$717$	20.4853i	0.765037i
$718$	− 30.1838i	− 1.12645i
$719$	22.6274	0.843860	0.421930	−	0.906628i	$-0.361353\pi$
$719$	22.6274	0.843860	0.421930	+	0.906628i	$0.361353\pi$
$720$	0	0
$721$	−1.75736	−0.0654475
$722$	25.8995i	0.963879i
$723$	− 5.65685i	− 0.210381i
$724$	0	0
$725$	0	0
$726$	−7.31371	−0.271437
$727$	− 24.9706i	− 0.926107i	−0.886330	−	0.463053i	$-0.846754\pi$
$727$	− 24.9706i	− 0.926107i	0.886330	−	0.463053i	$-0.153246\pi$
$728$	− 17.6569i	− 0.654407i
$729$	−29.0000	−1.07407
$730$	0	0
$731$	2.17157	0.0803185
$732$	0	0
$733$	− 16.9706i	− 0.626822i	−0.949618	−	0.313411i	$-0.898528\pi$
$733$	− 16.9706i	− 0.626822i	0.949618	−	0.313411i	$-0.101472\pi$
$734$	13.8579	0.511503
$735$	0	0
$736$	0	0
$737$	28.6569i	1.05559i
$738$	− 2.58579i	− 0.0951841i
$739$	39.4558	1.45141	0.725703	−	0.688008i	$-0.241515\pi$
$739$	39.4558	1.45141	0.725703	+	0.688008i	$0.241515\pi$
$740$	0	0
$741$	−2.14214	−0.0786933
$742$	66.7696i	2.45119i
$743$	− 15.1127i	− 0.554431i	−0.960808	−	0.277216i	$-0.910588\pi$
$743$	− 15.1127i	− 0.554431i	0.960808	−	0.277216i	$-0.0894116\pi$
$744$	12.0000	0.439941
$745$	0	0
$746$	−12.0000	−0.439351
$747$	− 3.34315i	− 0.122319i
$748$	0	0
$749$	−1.17157	−0.0428083
$750$	0	0
$751$	−11.7574	−0.429032	−0.214516	−	0.976720i	$-0.568817\pi$
$751$	−11.7574	−0.429032	−0.214516	+	0.976720i	$0.568817\pi$
$752$	− 24.0000i	− 0.875190i
$753$	− 32.7279i	− 1.19267i
$754$	10.9706	0.399524
$755$	0	0
$756$	0	0
$757$	3.51472i	0.127745i	0.997958	+	0.0638723i	$0.0203450\pi$
$757$	3.51472i	0.127745i	−0.997958	+	0.0638723i	$0.979655\pi$
$758$	42.8701i	1.55711i
$759$	36.0416	1.30823
$760$	0	0
$761$	19.1127	0.692835	0.346417	−	0.938080i	$-0.387398\pi$
$761$	19.1127	0.692835	0.346417	+	0.938080i	$0.387398\pi$
$762$	− 7.65685i	− 0.277379i
$763$	− 68.1838i	− 2.46842i
$764$	0	0
$765$	0	0
$766$	28.9706	1.04675
$767$	8.82843i	0.318776i
$768$	0	0
$769$	−44.7696	−1.61443	−0.807216	−	0.590257i	$-0.799027\pi$
$769$	−44.7696	−1.61443	−0.807216	+	0.590257i	$0.799027\pi$
$770$	0	0
$771$	−2.48528	−0.0895052
$772$	0	0
$773$	− 8.10051i	− 0.291355i	−0.989332	−	0.145677i	$-0.953464\pi$
$773$	− 8.10051i	− 0.291355i	0.989332	−	0.145677i	$-0.0465362\pi$
$774$	−1.41421	−0.0508329
$775$	0	0
$776$	5.17157	0.185649
$777$	− 40.9706i	− 1.46981i
$778$	− 23.5147i	− 0.843044i
$779$	−1.51472	−0.0542704
$780$	0	0
$781$	12.1421	0.434480
$782$	− 20.4437i	− 0.731063i
$783$	− 24.0000i	− 0.857690i
$784$	18.6274	0.665265
$785$	0	0
$786$	−19.3137	−0.688897
$787$	9.79899i	0.349296i	0.984631	+	0.174648i	$0.0558788\pi$
$787$	9.79899i	0.349296i	−0.984631	+	0.174648i	$0.944121\pi$
$788$	0	0
$789$	−0.201010	−0.00715615
$790$	0	0
$791$	23.3137	0.828940
$792$	10.8284i	0.384771i
$793$	− 0.443651i	− 0.0157545i
$794$	41.6569	1.47835
$795$	0	0
$796$	0	0
$797$	35.3137i	1.25088i	0.780274	+	0.625438i	$0.215080\pi$
$797$	35.3137i	1.25088i	−0.780274	+	0.625438i	$0.784920\pi$
$798$	− 5.65685i	− 0.200250i
$799$	−13.0294	−0.460948
$800$	0	0
$801$	1.75736	0.0620932
$802$	17.6985i	0.624955i
$803$	− 62.1838i	− 2.19442i
$804$	0	0
$805$	0	0
$806$	−7.75736	−0.273241
$807$	− 4.24264i	− 0.149348i
$808$	− 16.4853i	− 0.579950i
$809$	5.65685	0.198884	0.0994422	−	0.995043i	$-0.468294\pi$
$809$	5.65685	0.198884	0.0994422	+	0.995043i	$0.468294\pi$
$810$	0	0
$811$	23.2721	0.817193	0.408597	−	0.912715i	$-0.366018\pi$
$811$	23.2721	0.817193	0.408597	+	0.912715i	$0.366018\pi$
$812$	0	0
$813$	36.7279i	1.28810i
$814$	−45.9411	−1.61024
$815$	0	0
$816$	−12.2843	−0.430036
$817$	0.828427i	0.0289830i
$818$	− 23.8579i	− 0.834171i
$819$	6.24264	0.218136
$820$	0	0
$821$	−52.1127	−1.81875	−0.909373	−	0.415982i	$-0.863438\pi$
$821$	−52.1127	−1.81875	−0.909373	+	0.415982i	$0.863438\pi$
$822$	28.9706i	1.01046i
$823$	54.6569i	1.90522i	0.304197	+	0.952609i	$0.401612\pi$
$823$	54.6569i	1.90522i	−0.304197	+	0.952609i	$0.598388\pi$
$824$	−1.45584	−0.0507167
$825$	0	0
$826$	−23.3137	−0.811188
$827$	− 1.65685i	− 0.0576145i	−0.999585	−	0.0288072i	$-0.990829\pi$
$827$	− 1.65685i	− 0.0576145i	0.999585	−	0.0288072i	$-0.00917090\pi$
$828$	0	0
$829$	−23.7990	−0.826573	−0.413287	−	0.910601i	$-0.635619\pi$
$829$	−23.7990	−0.826573	−0.413287	+	0.910601i	$0.635619\pi$
$830$	0	0
$831$	11.1716	0.387538
$832$	− 14.6274i	− 0.507114i
$833$	− 10.1127i	− 0.350384i
$834$	10.9706	0.379880
$835$	0	0
$836$	0	0
$837$	16.9706i	0.586588i
$838$	− 33.7990i	− 1.16757i
$839$	48.8701	1.68718	0.843591	−	0.536986i	$-0.180437\pi$
$839$	48.8701	1.68718	0.843591	+	0.536986i	$0.180437\pi$
$840$	0	0
$841$	−11.0000	−0.379310
$842$	22.1421i	0.763068i
$843$	12.2426i	0.421659i
$844$	0	0
$845$	0	0
$846$	8.48528	0.291730
$847$	− 12.4853i	− 0.428999i
$848$	55.3137i	1.89948i
$849$	−25.8995	−0.888868
$850$	0	0
$851$	56.4853	1.93629
$852$	0	0
$853$	− 46.5980i	− 1.59548i	−0.602999	−	0.797742i	$-0.706028\pi$
$853$	− 46.5980i	− 1.59548i	0.602999	−	0.797742i	$-0.293972\pi$
$854$	1.17157	0.0400904
$855$	0	0
$856$	−0.970563	−0.0331732
$857$	7.65685i	0.261553i	0.991412	+	0.130777i	$0.0417471\pi$
$857$	7.65685i	0.261553i	−0.991412	+	0.130777i	$0.958253\pi$
$858$	14.0000i	0.477952i
$859$	−16.9706	−0.579028	−0.289514	−	0.957174i	$-0.593494\pi$
$859$	−16.9706	−0.579028	−0.289514	+	0.957174i	$0.593494\pi$
$860$	0	0
$861$	−8.82843	−0.300872
$862$	− 32.9289i	− 1.12156i
$863$	47.2548i	1.60857i	0.594242	+	0.804287i	$0.297452\pi$
$863$	47.2548i	1.60857i	−0.594242	+	0.804287i	$0.702548\pi$
$864$	0	0
$865$	0	0
$866$	−30.7696	−1.04559
$867$	− 17.3726i	− 0.590004i
$868$	0	0
$869$	18.4853	0.627070
$870$	0	0
$871$	13.6863	0.463742
$872$	− 56.4853i	− 1.91283i
$873$	1.82843i	0.0618829i
$874$	7.79899	0.263805
$875$	0	0
$876$	0	0
$877$	36.7990i	1.24261i	0.783567	+	0.621307i	$0.213398\pi$
$877$	36.7990i	1.24261i	−0.783567	+	0.621307i	$0.786602\pi$
$878$	16.2426i	0.548163i
$879$	−14.6274	−0.493371
$880$	0	0
$881$	36.2548	1.22146	0.610728	−	0.791840i	$-0.290877\pi$
$881$	36.2548	1.22146	0.610728	+	0.791840i	$0.290877\pi$
$882$	6.58579i	0.221755i
$883$	8.02944i	0.270212i	0.990831	+	0.135106i	$0.0431375\pi$
$883$	8.02944i	0.270212i	−0.990831	+	0.135106i	$0.956862\pi$
$884$	0	0
$885$	0	0
$886$	−11.1127	−0.373338
$887$	58.9706i	1.98004i	0.140935	+	0.990019i	$0.454989\pi$
$887$	58.9706i	1.98004i	−0.140935	+	0.990019i	$0.545011\pi$
$888$	− 33.9411i	− 1.13899i
$889$	13.0711	0.438390
$890$	0	0
$891$	19.1421	0.641286
$892$	0	0
$893$	− 4.97056i	− 0.166334i
$894$	16.9706	0.567581
$895$	0	0
$896$	38.6274	1.29045
$897$	− 17.2132i	− 0.574732i
$898$	46.9706i	1.56743i
$899$	−12.7279	−0.424500
$900$	0	0
$901$	30.0294	1.00043
$902$	9.89949i	0.329617i
$903$	4.82843i	0.160680i
$904$	19.3137	0.642364
$905$	0	0
$906$	−36.4853	−1.21214
$907$	− 19.9706i	− 0.663112i	−0.943436	−	0.331556i	$-0.892426\pi$
$907$	− 19.9706i	− 0.663112i	0.943436	−	0.331556i	$-0.107574\pi$
$908$	0	0
$909$	5.82843	0.193317
$910$	0	0
$911$	−4.24264	−0.140565	−0.0702825	−	0.997527i	$-0.522390\pi$
$911$	−4.24264	−0.140565	−0.0702825	+	0.997527i	$0.522390\pi$
$912$	− 4.68629i	− 0.155179i
$913$	12.7990i	0.423585i
$914$	1.02944	0.0340508
$915$	0	0
$916$	0	0
$917$	− 32.9706i	− 1.08878i
$918$	− 17.3726i	− 0.573381i
$919$	12.5147	0.412822	0.206411	−	0.978465i	$-0.433822\pi$
$919$	12.5147	0.412822	0.206411	+	0.978465i	$0.433822\pi$
$920$	0	0
$921$	37.8995	1.24883
$922$	− 60.2843i	− 1.98536i
$923$	− 5.79899i	− 0.190876i
$924$	0	0
$925$	0	0
$926$	43.4558	1.42805
$927$	− 0.514719i	− 0.0169056i
$928$	0	0
$929$	−39.1716	−1.28518	−0.642589	−	0.766211i	$-0.722140\pi$
$929$	−39.1716	−1.28518	−0.642589	+	0.766211i	$0.722140\pi$
$930$	0	0
$931$	3.85786	0.126436
$932$	0	0
$933$	19.7574i	0.646827i
$934$	33.4558	1.09471
$935$	0	0
$936$	5.17157	0.169038
$937$	− 51.0122i	− 1.66650i	−0.552900	−	0.833248i	$-0.686479\pi$
$937$	− 51.0122i	− 1.66650i	0.552900	−	0.833248i	$-0.313521\pi$
$938$	36.1421i	1.18008i
$939$	35.6569	1.16362
$940$	0	0
$941$	−31.6274	−1.03102	−0.515512	−	0.856882i	$-0.672398\pi$
$941$	−31.6274	−1.03102	−0.515512	+	0.856882i	$0.672398\pi$
$942$	20.0000i	0.651635i
$943$	− 12.1716i	− 0.396361i
$944$	−19.3137	−0.628608
$945$	0	0
$946$	5.41421	0.176031
$947$	− 5.82843i	− 0.189398i	−0.995506	−	0.0946992i	$-0.969811\pi$
$947$	− 5.82843i	− 0.189398i	0.995506	−	0.0946992i	$-0.0301889\pi$
$948$	0	0
$949$	−29.6985	−0.964054
$950$	0	0
$951$	36.5269	1.18447
$952$	− 20.9706i	− 0.679660i
$953$	24.0416i	0.778785i	0.921072	+	0.389392i	$0.127315\pi$
$953$	24.0416i	0.778785i	−0.921072	+	0.389392i	$0.872685\pi$
$954$	−19.5563	−0.633160
$955$	0	0
$956$	0	0
$957$	22.9706i	0.742533i
$958$	− 9.41421i	− 0.304159i
$959$	−49.4558	−1.59701
$960$	0	0
$961$	−22.0000	−0.709677
$962$	21.9411i	0.707410i
$963$	− 0.343146i	− 0.0110577i
$964$	0	0
$965$	0	0
$966$	45.4558	1.46252
$967$	− 40.9411i	− 1.31658i	−0.752765	−	0.658289i	$-0.771281\pi$
$967$	− 40.9411i	− 1.31658i	0.752765	−	0.658289i	$-0.228719\pi$
$968$	− 10.3431i	− 0.332441i
$969$	−2.54416	−0.0817301
$970$	0	0
$971$	−3.14214	−0.100836	−0.0504180	−	0.998728i	$-0.516055\pi$
$971$	−3.14214	−0.100836	−0.0504180	+	0.998728i	$0.516055\pi$
$972$	0	0
$973$	18.7279i	0.600390i
$974$	31.5980	1.01246
$975$	0	0
$976$	0.970563	0.0310670
$977$	− 23.3137i	− 0.745872i	−0.927857	−	0.372936i	$-0.878351\pi$
$977$	− 23.3137i	− 0.745872i	0.927857	−	0.372936i	$-0.121649\pi$
$978$	− 23.5147i	− 0.751918i
$979$	−6.72792	−0.215025
$980$	0	0
$981$	19.9706	0.637611
$982$	− 18.9706i	− 0.605375i
$983$	62.5269i	1.99430i	0.0754527	+	0.997149i	$0.475960\pi$
$983$	62.5269i	1.99430i	−0.0754527	+	0.997149i	$0.524040\pi$
$984$	−7.31371	−0.233153
$985$	0	0
$986$	13.0294	0.414942
$987$	− 28.9706i	− 0.922143i
$988$	0	0
$989$	−6.65685	−0.211676
$990$	0	0
$991$	11.5563	0.367100	0.183550	−	0.983010i	$-0.441241\pi$
$991$	11.5563	0.367100	0.183550	+	0.983010i	$0.441241\pi$
$992$	0	0
$993$	− 30.2843i	− 0.961042i
$994$	15.3137	0.485721
$995$	0	0
$996$	0	0
$997$	2.92893i	0.0927602i	0.998924	+	0.0463801i	$0.0147685\pi$
$997$	2.92893i	0.0927602i	−0.998924	+	0.0463801i	$0.985231\pi$
$998$	− 36.4264i	− 1.15306i
$999$	48.0000	1.51865

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1075.2.b.f.474.1		4
5.2	odd	4		1075.2.a.i.1.2		2
5.3	odd	4		43.2.a.b.1.1	✓	2
5.4	even	2	inner	1075.2.b.f.474.4		4
15.2	even	4		9675.2.a.bf.1.1		2
15.8	even	4		387.2.a.h.1.2		2
20.3	even	4		688.2.a.f.1.1		2
35.13	even	4		2107.2.a.b.1.1		2
40.3	even	4		2752.2.a.m.1.2		2
40.13	odd	4		2752.2.a.l.1.1		2
55.43	even	4		5203.2.a.f.1.2		2
60.23	odd	4		6192.2.a.bd.1.1		2
65.38	odd	4		7267.2.a.b.1.2		2
215.128	even	4		1849.2.a.f.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.2.a.b.1.1	✓	2	5.3	odd	4
387.2.a.h.1.2		2	15.8	even	4
688.2.a.f.1.1		2	20.3	even	4
1075.2.a.i.1.2		2	5.2	odd	4
1075.2.b.f.474.1		4	1.1	even	1	trivial
1075.2.b.f.474.4		4	5.4	even	2	inner
1849.2.a.f.1.2		2	215.128	even	4
2107.2.a.b.1.1		2	35.13	even	4
2752.2.a.l.1.1		2	40.13	odd	4
2752.2.a.m.1.2		2	40.3	even	4
5203.2.a.f.1.2		2	55.43	even	4
6192.2.a.bd.1.1		2	60.23	odd	4
7267.2.a.b.1.2		2	65.38	odd	4
9675.2.a.bf.1.1		2	15.2	even	4

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:	\( N \)	\(=\)	\( 1075 = 5^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1075.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -1.41421i q^{3} -2.00000 q^{6} -3.41421i q^{7} -2.82843i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.41421i q^{2} -1.41421i q^{3} -2.00000 q^{6} -3.41421i q^{7} -2.82843i q^{8} +1.00000 q^{9} -3.82843 q^{11} +1.82843i q^{13} -4.82843 q^{14} -4.00000 q^{16} +2.17157i q^{17} -1.41421i q^{18} -0.828427 q^{19} -4.82843 q^{21} +5.41421i q^{22} -6.65685i q^{23} -4.00000 q^{24} +2.58579 q^{26} -5.65685i q^{27} +4.24264 q^{29} -3.00000 q^{31} +5.41421i q^{33} +3.07107 q^{34} +8.48528i q^{37} +1.17157i q^{38} +2.58579 q^{39} +1.82843 q^{41} +6.82843i q^{42} -1.00000i q^{43} -9.41421 q^{46} +6.00000i q^{47} +5.65685i q^{48} -4.65685 q^{49} +3.07107 q^{51} -13.8284i q^{53} -8.00000 q^{54} -9.65685 q^{56} +1.17157i q^{57} -6.00000i q^{58} +4.82843 q^{59} -0.242641 q^{61} +4.24264i q^{62} -3.41421i q^{63} -8.00000 q^{64} +7.65685 q^{66} -7.48528i q^{67} -9.41421 q^{69} -3.17157 q^{71} -2.82843i q^{72} +16.2426i q^{73} +12.0000 q^{74} +13.0711i q^{77} -3.65685i q^{78} -4.82843 q^{79} -5.00000 q^{81} -2.58579i q^{82} -3.34315i q^{83} -1.41421 q^{86} -6.00000i q^{87} +10.8284i q^{88} +1.75736 q^{89} +6.24264 q^{91} +4.24264i q^{93} +8.48528 q^{94} +1.82843i q^{97} +6.58579i q^{98} -3.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{6} + 4 q^{9}+O(q^{10}) \) 4 * q - 8 * q^6 + 4 * q^9	\( 4 q - 8 q^{6} + 4 q^{9} - 4 q^{11} - 8 q^{14} - 16 q^{16} + 8 q^{19} - 8 q^{21} - 16 q^{24} + 16 q^{26} - 12 q^{31} - 16 q^{34} + 16 q^{39} - 4 q^{41} - 32 q^{46} + 4 q^{49} - 16 q^{51} - 32 q^{54} - 16 q^{56} + 8 q^{59} + 16 q^{61} - 32 q^{64} + 8 q^{66} - 32 q^{69} - 24 q^{71} + 48 q^{74} - 8 q^{79} - 20 q^{81} + 24 q^{89} + 8 q^{91} - 4 q^{99}+O(q^{100}) \) 4 * q - 8 * q^6 + 4 * q^9 - 4 * q^11 - 8 * q^14 - 16 * q^16 + 8 * q^19 - 8 * q^21 - 16 * q^24 + 16 * q^26 - 12 * q^31 - 16 * q^34 + 16 * q^39 - 4 * q^41 - 32 * q^46 + 4 * q^49 - 16 * q^51 - 32 * q^54 - 16 * q^56 + 8 * q^59 + 16 * q^61 - 32 * q^64 + 8 * q^66 - 32 * q^69 - 24 * q^71 + 48 * q^74 - 8 * q^79 - 20 * q^81 + 24 * q^89 + 8 * q^91 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more