LMFDB - Embedded newform 1725.2.b.m.1174.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1725,2,Mod(1174,1725)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1725, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1725.1174");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1725 = 3 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1725.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:	$13.7741943487$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 345)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1174.3
Root			$1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1725.1174
Dual form			1725.2.b.m.1174.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.44949i q^{2} -1.00000i q^{3} -4.00000 q^{4} +2.44949 q^{6} -1.00000i q^{7} -4.89898i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+2.44949i q^{2} -1.00000i q^{3} -4.00000 q^{4} +2.44949 q^{6} -1.00000i q^{7} -4.89898i q^{8} -1.00000 q^{9} -2.44949 q^{11} +4.00000i q^{12} +0.449490i q^{13} +2.44949 q^{14} +4.00000 q^{16} -0.550510i q^{17} -2.44949i q^{18} +0.449490 q^{19} -1.00000 q^{21} -6.00000i q^{22} +1.00000i q^{23} -4.89898 q^{24} -1.10102 q^{26} +1.00000i q^{27} +4.00000i q^{28} +4.34847 q^{29} +9.89898 q^{31} +2.44949i q^{33} +1.34847 q^{34} +4.00000 q^{36} -5.89898i q^{37} +1.10102i q^{38} +0.449490 q^{39} +0.550510 q^{41} -2.44949i q^{42} -2.00000i q^{43} +9.79796 q^{44} -2.44949 q^{46} +3.55051i q^{47} -4.00000i q^{48} +6.00000 q^{49} -0.550510 q^{51} -1.79796i q^{52} -5.44949i q^{53} -2.44949 q^{54} -4.89898 q^{56} -0.449490i q^{57} +10.6515i q^{58} +4.34847 q^{59} +15.3485 q^{61} +24.2474i q^{62} +1.00000i q^{63} +8.00000 q^{64} -6.00000 q^{66} -7.00000i q^{67} +2.20204i q^{68} +1.00000 q^{69} +10.3485 q^{71} +4.89898i q^{72} -9.34847i q^{73} +14.4495 q^{74} -1.79796 q^{76} +2.44949i q^{77} +1.10102i q^{78} +4.00000 q^{79} +1.00000 q^{81} +1.34847i q^{82} +9.24745i q^{83} +4.00000 q^{84} +4.89898 q^{86} -4.34847i q^{87} +12.0000i q^{88} +7.10102 q^{89} +0.449490 q^{91} -4.00000i q^{92} -9.89898i q^{93} -8.69694 q^{94} +12.8990i q^{97} +14.6969i q^{98} +2.44949 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 16 q^{4} - 4 q^{9}+O(q^{10}) $ 4 * q - 16 * q^4 - 4 * q^9	$ 4 q - 16 q^{4} - 4 q^{9} + 16 q^{16} - 8 q^{19} - 4 q^{21} - 24 q^{26} - 12 q^{29} + 20 q^{31} - 24 q^{34} + 16 q^{36} - 8 q^{39} + 12 q^{41} + 24 q^{49} - 12 q^{51} - 12 q^{59} + 32 q^{61} + 32 q^{64} - 24 q^{66} + 4 q^{69} + 12 q^{71} + 48 q^{74} + 32 q^{76} + 16 q^{79} + 4 q^{81} + 16 q^{84} + 48 q^{89} - 8 q^{91} + 24 q^{94}+O(q^{100}) $ 4 * q - 16 * q^4 - 4 * q^9 + 16 * q^16 - 8 * q^19 - 4 * q^21 - 24 * q^26 - 12 * q^29 + 20 * q^31 - 24 * q^34 + 16 * q^36 - 8 * q^39 + 12 * q^41 + 24 * q^49 - 12 * q^51 - 12 * q^59 + 32 * q^61 + 32 * q^64 - 24 * q^66 + 4 * q^69 + 12 * q^71 + 48 * q^74 + 32 * q^76 + 16 * q^79 + 4 * q^81 + 16 * q^84 + 48 * q^89 - 8 * q^91 + 24 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1725\mathbb{Z}\right)^\times$.

$n$	$277$	$1151$	$1201$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	2.44949i	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$2$	2.44949i	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	−4.00000	−2.00000
$5$	0	0
$5$	0	0
$6$	2.44949	1.00000
$7$	− 1.00000i	− 0.377964i	−0.981981	−	0.188982i	$-0.939481\pi$
$7$	− 1.00000i	− 0.377964i	0.981981	−	0.188982i	$-0.0605189\pi$
$8$	− 4.89898i	− 1.73205i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−2.44949	−0.738549	−0.369274	−	0.929320i	$-0.620394\pi$
$11$	−2.44949	−0.738549	−0.369274	+	0.929320i	$0.620394\pi$
$12$	4.00000i	1.15470i
$13$	0.449490i	0.124666i	0.998055	+	0.0623330i	$0.0198541\pi$
$13$	0.449490i	0.124666i	−0.998055	+	0.0623330i	$0.980146\pi$
$14$	2.44949	0.654654
$15$	0	0
$16$	4.00000	1.00000
$17$	− 0.550510i	− 0.133518i	−0.997769	−	0.0667592i	$-0.978734\pi$
$17$	− 0.550510i	− 0.133518i	0.997769	−	0.0667592i	$-0.0212659\pi$
$18$	− 2.44949i	− 0.577350i
$19$	0.449490	0.103120	0.0515600	−	0.998670i	$-0.483581\pi$
$19$	0.449490	0.103120	0.0515600	+	0.998670i	$0.483581\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	− 6.00000i	− 1.27920i
$23$	1.00000i	0.208514i
$23$	1.00000i	0.208514i
$24$	−4.89898	−1.00000
$25$	0	0
$26$	−1.10102	−0.215928
$27$	1.00000i	0.192450i
$28$	4.00000i	0.755929i
$29$	4.34847	0.807490	0.403745	−	0.914871i	$-0.367708\pi$
$29$	4.34847	0.807490	0.403745	+	0.914871i	$0.367708\pi$
$30$	0	0
$31$	9.89898	1.77791	0.888955	−	0.457995i	$-0.151432\pi$
$31$	9.89898	1.77791	0.888955	+	0.457995i	$0.151432\pi$
$32$	0	0
$33$	2.44949i	0.426401i
$34$	1.34847	0.231261
$35$	0	0
$36$	4.00000	0.666667
$37$	− 5.89898i	− 0.969786i	−0.874573	−	0.484893i	$-0.838858\pi$
$37$	− 5.89898i	− 0.969786i	0.874573	−	0.484893i	$-0.161142\pi$
$38$	1.10102i	0.178609i
$39$	0.449490	0.0719760
$40$	0	0
$41$	0.550510	0.0859753	0.0429876	−	0.999076i	$-0.486312\pi$
$41$	0.550510	0.0859753	0.0429876	+	0.999076i	$0.486312\pi$
$42$	− 2.44949i	− 0.377964i
$43$	− 2.00000i	− 0.304997i	−0.988304	−	0.152499i	$-0.951268\pi$
$43$	− 2.00000i	− 0.304997i	0.988304	−	0.152499i	$-0.0487319\pi$
$44$	9.79796	1.47710
$45$	0	0
$46$	−2.44949	−0.361158
$47$	3.55051i	0.517895i	0.965891	+	0.258948i	$0.0833757\pi$
$47$	3.55051i	0.517895i	−0.965891	+	0.258948i	$0.916624\pi$
$48$	− 4.00000i	− 0.577350i
$49$	6.00000	0.857143
$50$	0	0
$51$	−0.550510	−0.0770869
$52$	− 1.79796i	− 0.249332i
$53$	− 5.44949i	− 0.748545i	−0.927319	−	0.374272i	$-0.877892\pi$
$53$	− 5.44949i	− 0.748545i	0.927319	−	0.374272i	$-0.122108\pi$
$54$	−2.44949	−0.333333
$55$	0	0
$56$	−4.89898	−0.654654
$57$	− 0.449490i	− 0.0595364i
$58$	10.6515i	1.39861i
$59$	4.34847	0.566122	0.283061	−	0.959102i	$-0.408650\pi$
$59$	4.34847	0.566122	0.283061	+	0.959102i	$0.408650\pi$
$60$	0	0
$61$	15.3485	1.96517	0.982585	−	0.185813i	$-0.0594920\pi$
$61$	15.3485	1.96517	0.982585	+	0.185813i	$0.0594920\pi$
$62$	24.2474i	3.07943i
$63$	1.00000i	0.125988i
$64$	8.00000	1.00000
$65$	0	0
$66$	−6.00000	−0.738549
$67$	− 7.00000i	− 0.855186i	−0.903971	−	0.427593i	$-0.859362\pi$
$67$	− 7.00000i	− 0.855186i	0.903971	−	0.427593i	$-0.140638\pi$
$68$	2.20204i	0.267037i
$69$	1.00000	0.120386
$70$	0	0
$71$	10.3485	1.22814	0.614069	−	0.789253i	$-0.289532\pi$
$71$	10.3485	1.22814	0.614069	+	0.789253i	$0.289532\pi$
$72$	4.89898i	0.577350i
$73$	− 9.34847i	− 1.09416i	−0.837082	−	0.547078i	$-0.815740\pi$
$73$	− 9.34847i	− 1.09416i	0.837082	−	0.547078i	$-0.184260\pi$
$74$	14.4495	1.67972
$75$	0	0
$76$	−1.79796	−0.206240
$77$	2.44949i	0.279145i
$78$	1.10102i	0.124666i
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	1.34847i	0.148914i
$83$	9.24745i	1.01504i	0.861640	+	0.507520i	$0.169438\pi$
$83$	9.24745i	1.01504i	−0.861640	+	0.507520i	$0.830562\pi$
$84$	4.00000	0.436436
$85$	0	0
$86$	4.89898	0.528271
$87$	− 4.34847i	− 0.466205i
$88$	12.0000i	1.27920i
$89$	7.10102	0.752707	0.376353	−	0.926476i	$-0.377178\pi$
$89$	7.10102	0.752707	0.376353	+	0.926476i	$0.377178\pi$
$90$	0	0
$91$	0.449490	0.0471193
$92$	− 4.00000i	− 0.417029i
$93$	− 9.89898i	− 1.02648i
$94$	−8.69694	−0.897021
$95$	0	0
$96$	0	0
$97$	12.8990i	1.30969i	0.755762	+	0.654846i	$0.227267\pi$
$97$	12.8990i	1.30969i	−0.755762	+	0.654846i	$0.772733\pi$
$98$	14.6969i	1.48461i
$99$	2.44949	0.246183
$100$	0	0
$101$	−17.4495	−1.73629	−0.868145	−	0.496311i	$-0.834687\pi$
$101$	−17.4495	−1.73629	−0.868145	+	0.496311i	$0.834687\pi$
$102$	− 1.34847i	− 0.133518i
$103$	− 16.6969i	− 1.64520i	−0.568622	−	0.822599i	$-0.692523\pi$
$103$	− 16.6969i	− 1.64520i	0.568622	−	0.822599i	$-0.307477\pi$
$104$	2.20204	0.215928
$105$	0	0
$106$	13.3485	1.29652
$107$	− 0.550510i	− 0.0532198i	−0.999646	−	0.0266099i	$-0.991529\pi$
$107$	− 0.550510i	− 0.0532198i	0.999646	−	0.0266099i	$-0.00847120\pi$
$108$	− 4.00000i	− 0.384900i
$109$	−16.4495	−1.57558	−0.787788	−	0.615947i	$-0.788774\pi$
$109$	−16.4495	−1.57558	−0.787788	+	0.615947i	$0.788774\pi$
$110$	0	0
$111$	−5.89898	−0.559906
$112$	− 4.00000i	− 0.377964i
$113$	16.3485i	1.53793i	0.639288	+	0.768967i	$0.279229\pi$
$113$	16.3485i	1.53793i	−0.639288	+	0.768967i	$0.720771\pi$
$114$	1.10102	0.103120
$115$	0	0
$116$	−17.3939	−1.61498
$117$	− 0.449490i	− 0.0415553i
$118$	10.6515i	0.980553i
$119$	−0.550510	−0.0504652
$120$	0	0
$121$	−5.00000	−0.454545
$122$	37.5959i	3.40377i
$123$	− 0.550510i	− 0.0496378i
$124$	−39.5959	−3.55582
$125$	0	0
$126$	−2.44949	−0.218218
$127$	− 6.44949i	− 0.572300i	−0.958185	−	0.286150i	$-0.907624\pi$
$127$	− 6.44949i	− 0.572300i	0.958185	−	0.286150i	$-0.0923755\pi$
$128$	19.5959i	1.73205i
$129$	−2.00000	−0.176090
$130$	0	0
$131$	19.5959	1.71210	0.856052	−	0.516890i	$-0.172910\pi$
$131$	19.5959	1.71210	0.856052	+	0.516890i	$0.172910\pi$
$132$	− 9.79796i	− 0.852803i
$133$	− 0.449490i	− 0.0389757i
$134$	17.1464	1.48123
$135$	0	0
$136$	−2.69694	−0.231261
$137$	− 7.10102i	− 0.606681i	−0.952882	−	0.303341i	$-0.901898\pi$
$137$	− 7.10102i	− 0.606681i	0.952882	−	0.303341i	$-0.0981020\pi$
$138$	2.44949i	0.208514i
$139$	−14.7980	−1.25515	−0.627573	−	0.778558i	$-0.715952\pi$
$139$	−14.7980	−1.25515	−0.627573	+	0.778558i	$0.715952\pi$
$140$	0	0
$141$	3.55051	0.299007
$142$	25.3485i	2.12720i
$143$	− 1.10102i	− 0.0920720i
$144$	−4.00000	−0.333333
$145$	0	0
$146$	22.8990	1.89513
$147$	− 6.00000i	− 0.494872i
$148$	23.5959i	1.93957i
$149$	−13.3485	−1.09355	−0.546775	−	0.837280i	$-0.684145\pi$
$149$	−13.3485	−1.09355	−0.546775	+	0.837280i	$0.684145\pi$
$150$	0	0
$151$	14.0000	1.13930	0.569652	−	0.821886i	$-0.307078\pi$
$151$	14.0000	1.13930	0.569652	+	0.821886i	$0.307078\pi$
$152$	− 2.20204i	− 0.178609i
$153$	0.550510i	0.0445061i
$154$	−6.00000	−0.483494
$155$	0	0
$156$	−1.79796	−0.143952
$157$	− 20.5959i	− 1.64373i	−0.569680	−	0.821867i	$-0.692933\pi$
$157$	− 20.5959i	− 1.64373i	0.569680	−	0.821867i	$-0.307067\pi$
$158$	9.79796i	0.779484i
$159$	−5.44949	−0.432173
$160$	0	0
$161$	1.00000	0.0788110
$162$	2.44949i	0.192450i
$163$	10.0000i	0.783260i	0.920123	+	0.391630i	$0.128089\pi$
$163$	10.0000i	0.783260i	−0.920123	+	0.391630i	$0.871911\pi$
$164$	−2.20204	−0.171951
$165$	0	0
$166$	−22.6515	−1.75810
$167$	− 1.34847i	− 0.104348i	−0.998638	−	0.0521738i	$-0.983385\pi$
$167$	− 1.34847i	− 0.104348i	0.998638	−	0.0521738i	$-0.0166150\pi$
$168$	4.89898i	0.377964i
$169$	12.7980	0.984458
$170$	0	0
$171$	−0.449490	−0.0343733
$172$	8.00000i	0.609994i
$173$	19.5959i	1.48985i	0.667148	+	0.744925i	$0.267515\pi$
$173$	19.5959i	1.48985i	−0.667148	+	0.744925i	$0.732485\pi$
$174$	10.6515	0.807490
$175$	0	0
$176$	−9.79796	−0.738549
$177$	− 4.34847i	− 0.326851i
$178$	17.3939i	1.30373i
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	0	0
$181$	0.898979	0.0668206	0.0334103	−	0.999442i	$-0.489363\pi$
$181$	0.898979	0.0668206	0.0334103	+	0.999442i	$0.489363\pi$
$182$	1.10102i	0.0816131i
$183$	− 15.3485i	− 1.13459i
$184$	4.89898	0.361158
$185$	0	0
$186$	24.2474	1.77791
$187$	1.34847i	0.0986098i
$188$	− 14.2020i	− 1.03579i
$189$	1.00000	0.0727393
$190$	0	0
$191$	6.24745	0.452050	0.226025	−	0.974122i	$-0.427427\pi$
$191$	6.24745	0.452050	0.226025	+	0.974122i	$0.427427\pi$
$192$	− 8.00000i	− 0.577350i
$193$	− 22.6969i	− 1.63376i	−0.576807	−	0.816881i	$-0.695701\pi$
$193$	− 22.6969i	− 1.63376i	0.576807	−	0.816881i	$-0.304299\pi$
$194$	−31.5959	−2.26845
$195$	0	0
$196$	−24.0000	−1.71429
$197$	− 13.1010i	− 0.933409i	−0.884413	−	0.466705i	$-0.845441\pi$
$197$	− 13.1010i	− 0.933409i	0.884413	−	0.466705i	$-0.154559\pi$
$198$	6.00000i	0.426401i
$199$	−6.89898	−0.489056	−0.244528	−	0.969642i	$-0.578633\pi$
$199$	−6.89898	−0.489056	−0.244528	+	0.969642i	$0.578633\pi$
$200$	0	0
$201$	−7.00000	−0.493742
$202$	− 42.7423i	− 3.00734i
$203$	− 4.34847i	− 0.305203i
$204$	2.20204	0.154174
$205$	0	0
$206$	40.8990	2.84957
$207$	− 1.00000i	− 0.0695048i
$208$	1.79796i	0.124666i
$209$	−1.10102	−0.0761592
$210$	0	0
$211$	11.0000	0.757271	0.378636	−	0.925546i	$-0.376393\pi$
$211$	11.0000	0.757271	0.378636	+	0.925546i	$0.376393\pi$
$212$	21.7980i	1.49709i
$213$	− 10.3485i	− 0.709065i
$214$	1.34847	0.0921795
$215$	0	0
$216$	4.89898	0.333333
$217$	− 9.89898i	− 0.671987i
$218$	− 40.2929i	− 2.72898i
$219$	−9.34847	−0.631711
$220$	0	0
$221$	0.247449	0.0166452
$222$	− 14.4495i	− 0.969786i
$223$	17.5959i	1.17831i	0.808020	+	0.589155i	$0.200539\pi$
$223$	17.5959i	1.17831i	−0.808020	+	0.589155i	$0.799461\pi$
$224$	0	0
$225$	0	0
$226$	−40.0454	−2.66378
$227$	2.20204i	0.146155i	0.997326	+	0.0730773i	$0.0232820\pi$
$227$	2.20204i	0.146155i	−0.997326	+	0.0730773i	$0.976718\pi$
$228$	1.79796i	0.119073i
$229$	−20.4949	−1.35434	−0.677170	−	0.735826i	$-0.736794\pi$
$229$	−20.4949	−1.35434	−0.677170	+	0.735826i	$0.736794\pi$
$230$	0	0
$231$	2.44949	0.161165
$232$	− 21.3031i	− 1.39861i
$233$	− 2.20204i	− 0.144261i	−0.997395	−	0.0721303i	$-0.977020\pi$
$233$	− 2.20204i	− 0.144261i	0.997395	−	0.0721303i	$-0.0229797\pi$
$234$	1.10102	0.0719760
$235$	0	0
$236$	−17.3939	−1.13224
$237$	− 4.00000i	− 0.259828i
$238$	− 1.34847i	− 0.0874083i
$239$	−11.4495	−0.740606	−0.370303	−	0.928911i	$-0.620746\pi$
$239$	−11.4495	−0.740606	−0.370303	+	0.928911i	$0.620746\pi$
$240$	0	0
$241$	−14.6515	−0.943788	−0.471894	−	0.881655i	$-0.656430\pi$
$241$	−14.6515	−0.943788	−0.471894	+	0.881655i	$0.656430\pi$
$242$	− 12.2474i	− 0.787296i
$243$	− 1.00000i	− 0.0641500i
$244$	−61.3939	−3.93034
$245$	0	0
$246$	1.34847	0.0859753
$247$	0.202041i	0.0128556i
$248$	− 48.4949i	− 3.07943i
$249$	9.24745	0.586033
$250$	0	0
$251$	18.0000	1.13615	0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000	1.13615	0.568075	+	0.822977i	$0.307688\pi$
$252$	− 4.00000i	− 0.251976i
$253$	− 2.44949i	− 0.153998i
$254$	15.7980	0.991252
$255$	0	0
$256$	−32.0000	−2.00000
$257$	− 10.6515i	− 0.664424i	−0.943205	−	0.332212i	$-0.892205\pi$
$257$	− 10.6515i	− 0.664424i	0.943205	−	0.332212i	$-0.107795\pi$
$258$	− 4.89898i	− 0.304997i
$259$	−5.89898	−0.366545
$260$	0	0
$261$	−4.34847	−0.269163
$262$	48.0000i	2.96545i
$263$	− 2.75255i	− 0.169730i	−0.996392	−	0.0848648i	$-0.972954\pi$
$263$	− 2.75255i	− 0.169730i	0.996392	−	0.0848648i	$-0.0270458\pi$
$264$	12.0000	0.738549
$265$	0	0
$266$	1.10102	0.0675079
$267$	− 7.10102i	− 0.434575i
$268$	28.0000i	1.71037i
$269$	0.550510	0.0335652	0.0167826	−	0.999859i	$-0.494658\pi$
$269$	0.550510	0.0335652	0.0167826	+	0.999859i	$0.494658\pi$
$270$	0	0
$271$	2.30306	0.139901	0.0699505	−	0.997550i	$-0.477716\pi$
$271$	2.30306	0.139901	0.0699505	+	0.997550i	$0.477716\pi$
$272$	− 2.20204i	− 0.133518i
$273$	− 0.449490i	− 0.0272044i
$274$	17.3939	1.05080
$275$	0	0
$276$	−4.00000	−0.240772
$277$	− 10.0000i	− 0.600842i	−0.953807	−	0.300421i	$-0.902873\pi$
$277$	− 10.0000i	− 0.600842i	0.953807	−	0.300421i	$-0.0971271\pi$
$278$	− 36.2474i	− 2.17398i
$279$	−9.89898	−0.592636
$280$	0	0
$281$	−4.65153	−0.277487	−0.138744	−	0.990328i	$-0.544306\pi$
$281$	−4.65153	−0.277487	−0.138744	+	0.990328i	$0.544306\pi$
$282$	8.69694i	0.517895i
$283$	11.8990i	0.707321i	0.935374	+	0.353660i	$0.115063\pi$
$283$	11.8990i	0.707321i	−0.935374	+	0.353660i	$0.884937\pi$
$284$	−41.3939	−2.45627
$285$	0	0
$286$	2.69694	0.159473
$287$	− 0.550510i	− 0.0324956i
$288$	0	0
$289$	16.6969	0.982173
$290$	0	0
$291$	12.8990	0.756152
$292$	37.3939i	2.18831i
$293$	− 3.24745i	− 0.189718i	−0.995491	−	0.0948590i	$-0.969760\pi$
$293$	− 3.24745i	− 0.189718i	0.995491	−	0.0948590i	$-0.0302400\pi$
$294$	14.6969	0.857143
$295$	0	0
$296$	−28.8990	−1.67972
$297$	− 2.44949i	− 0.142134i
$298$	− 32.6969i	− 1.89408i
$299$	−0.449490	−0.0259947
$300$	0	0
$301$	−2.00000	−0.115278
$302$	34.2929i	1.97333i
$303$	17.4495i	1.00245i
$304$	1.79796	0.103120
$305$	0	0
$306$	−1.34847	−0.0770869
$307$	− 17.3485i	− 0.990129i	−0.868856	−	0.495065i	$-0.835144\pi$
$307$	− 17.3485i	− 0.990129i	0.868856	−	0.495065i	$-0.164856\pi$
$308$	− 9.79796i	− 0.558291i
$309$	−16.6969	−0.949856
$310$	0	0
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	− 2.20204i	− 0.124666i
$313$	9.69694i	0.548103i	0.961715	+	0.274052i	$0.0883639\pi$
$313$	9.69694i	0.548103i	−0.961715	+	0.274052i	$0.911636\pi$
$314$	50.4495	2.84703
$315$	0	0
$316$	−16.0000	−0.900070
$317$	− 18.2474i	− 1.02488i	−0.858723	−	0.512439i	$-0.828742\pi$
$317$	− 18.2474i	− 1.02488i	0.858723	−	0.512439i	$-0.171258\pi$
$318$	− 13.3485i	− 0.748545i
$319$	−10.6515	−0.596371
$320$	0	0
$321$	−0.550510	−0.0307265
$322$	2.44949i	0.136505i
$323$	− 0.247449i	− 0.0137684i
$324$	−4.00000	−0.222222
$325$	0	0
$326$	−24.4949	−1.35665
$327$	16.4495i	0.909659i
$328$	− 2.69694i	− 0.148914i
$329$	3.55051	0.195746
$330$	0	0
$331$	−14.5959	−0.802264	−0.401132	−	0.916020i	$-0.631383\pi$
$331$	−14.5959	−0.802264	−0.401132	+	0.916020i	$0.631383\pi$
$332$	− 36.9898i	− 2.03008i
$333$	5.89898i	0.323262i
$334$	3.30306	0.180735
$335$	0	0
$336$	−4.00000	−0.218218
$337$	− 0.202041i	− 0.0110059i	−0.999985	−	0.00550294i	$-0.998248\pi$
$337$	− 0.202041i	− 0.0110059i	0.999985	−	0.00550294i	$-0.00175165\pi$
$338$	31.3485i	1.70513i
$339$	16.3485	0.887927
$340$	0	0
$341$	−24.2474	−1.31307
$342$	− 1.10102i	− 0.0595364i
$343$	− 13.0000i	− 0.701934i
$344$	−9.79796	−0.528271
$345$	0	0
$346$	−48.0000	−2.58050
$347$	− 19.1010i	− 1.02540i	−0.858569	−	0.512698i	$-0.828646\pi$
$347$	− 19.1010i	− 1.02540i	0.858569	−	0.512698i	$-0.171354\pi$
$348$	17.3939i	0.932410i
$349$	25.4949	1.36471	0.682355	−	0.731021i	$-0.260956\pi$
$349$	25.4949	1.36471	0.682355	+	0.731021i	$0.260956\pi$
$350$	0	0
$351$	−0.449490	−0.0239920
$352$	0	0
$353$	20.4495i	1.08842i	0.838950	+	0.544208i	$0.183170\pi$
$353$	20.4495i	1.08842i	−0.838950	+	0.544208i	$0.816830\pi$
$354$	10.6515	0.566122
$355$	0	0
$356$	−28.4041	−1.50541
$357$	0.550510i	0.0291361i
$358$	14.6969i	0.776757i
$359$	8.44949	0.445947	0.222974	−	0.974825i	$-0.428424\pi$
$359$	8.44949	0.445947	0.222974	+	0.974825i	$0.428424\pi$
$360$	0	0
$361$	−18.7980	−0.989366
$362$	2.20204i	0.115737i
$363$	5.00000i	0.262432i
$364$	−1.79796	−0.0942387
$365$	0	0
$366$	37.5959	1.96517
$367$	− 15.6969i	− 0.819374i	−0.912226	−	0.409687i	$-0.865638\pi$
$367$	− 15.6969i	− 0.819374i	0.912226	−	0.409687i	$-0.134362\pi$
$368$	4.00000i	0.208514i
$369$	−0.550510	−0.0286584
$370$	0	0
$371$	−5.44949	−0.282923
$372$	39.5959i	2.05295i
$373$	11.5959i	0.600414i	0.953874	+	0.300207i	$0.0970557\pi$
$373$	11.5959i	0.600414i	−0.953874	+	0.300207i	$0.902944\pi$
$374$	−3.30306	−0.170797
$375$	0	0
$376$	17.3939	0.897021
$377$	1.95459i	0.100667i
$378$	2.44949i	0.125988i
$379$	33.3939	1.71533	0.857664	−	0.514210i	$-0.171915\pi$
$379$	33.3939	1.71533	0.857664	+	0.514210i	$0.171915\pi$
$380$	0	0
$381$	−6.44949	−0.330417
$382$	15.3031i	0.782973i
$383$	− 16.3485i	− 0.835368i	−0.908592	−	0.417684i	$-0.862842\pi$
$383$	− 16.3485i	− 0.835368i	0.908592	−	0.417684i	$-0.137158\pi$
$384$	19.5959	1.00000
$385$	0	0
$386$	55.5959	2.82976
$387$	2.00000i	0.101666i
$388$	− 51.5959i	− 2.61939i
$389$	32.6969	1.65780	0.828900	−	0.559396i	$-0.188967\pi$
$389$	32.6969	1.65780	0.828900	+	0.559396i	$0.188967\pi$
$390$	0	0
$391$	0.550510	0.0278405
$392$	− 29.3939i	− 1.48461i
$393$	− 19.5959i	− 0.988483i
$394$	32.0908	1.61671
$395$	0	0
$396$	−9.79796	−0.492366
$397$	5.79796i	0.290991i	0.989359	+	0.145496i	$0.0464777\pi$
$397$	5.79796i	0.290991i	−0.989359	+	0.145496i	$0.953522\pi$
$398$	− 16.8990i	− 0.847069i
$399$	−0.449490	−0.0225026
$400$	0	0
$401$	−22.8990	−1.14352	−0.571760	−	0.820421i	$-0.693739\pi$
$401$	−22.8990	−1.14352	−0.571760	+	0.820421i	$0.693739\pi$
$402$	− 17.1464i	− 0.855186i
$403$	4.44949i	0.221645i
$404$	69.7980	3.47258
$405$	0	0
$406$	10.6515	0.528627
$407$	14.4495i	0.716235i
$408$	2.69694i	0.133518i
$409$	−12.1010	−0.598357	−0.299178	−	0.954197i	$-0.596713\pi$
$409$	−12.1010	−0.598357	−0.299178	+	0.954197i	$0.596713\pi$
$410$	0	0
$411$	−7.10102	−0.350268
$412$	66.7878i	3.29040i
$413$	− 4.34847i	− 0.213974i
$414$	2.44949	0.120386
$415$	0	0
$416$	0	0
$417$	14.7980i	0.724659i
$418$	− 2.69694i	− 0.131912i
$419$	−38.4495	−1.87838	−0.939190	−	0.343397i	$-0.888422\pi$
$419$	−38.4495	−1.87838	−0.939190	+	0.343397i	$0.888422\pi$
$420$	0	0
$421$	2.24745	0.109534	0.0547670	−	0.998499i	$-0.482558\pi$
$421$	2.24745	0.109534	0.0547670	+	0.998499i	$0.482558\pi$
$422$	26.9444i	1.31163i
$423$	− 3.55051i	− 0.172632i
$424$	−26.6969	−1.29652
$425$	0	0
$426$	25.3485	1.22814
$427$	− 15.3485i	− 0.742764i
$428$	2.20204i	0.106440i
$429$	−1.10102	−0.0531578
$430$	0	0
$431$	24.4949	1.17988	0.589939	−	0.807448i	$-0.299152\pi$
$431$	24.4949	1.17988	0.589939	+	0.807448i	$0.299152\pi$
$432$	4.00000i	0.192450i
$433$	16.7980i	0.807258i	0.914923	+	0.403629i	$0.132251\pi$
$433$	16.7980i	0.807258i	−0.914923	+	0.403629i	$0.867749\pi$
$434$	24.2474	1.16391
$435$	0	0
$436$	65.7980	3.15115
$437$	0.449490i	0.0215020i
$438$	− 22.8990i	− 1.09416i
$439$	7.30306	0.348556	0.174278	−	0.984696i	$-0.444241\pi$
$439$	7.30306	0.348556	0.174278	+	0.984696i	$0.444241\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	0.606123i	0.0288303i
$443$	15.5505i	0.738827i	0.929265	+	0.369414i	$0.120441\pi$
$443$	15.5505i	0.738827i	−0.929265	+	0.369414i	$0.879559\pi$
$444$	23.5959	1.11981
$445$	0	0
$446$	−43.1010	−2.04089
$447$	13.3485i	0.631361i
$448$	− 8.00000i	− 0.377964i
$449$	−3.24745	−0.153257	−0.0766283	−	0.997060i	$-0.524415\pi$
$449$	−3.24745	−0.153257	−0.0766283	+	0.997060i	$0.524415\pi$
$450$	0	0
$451$	−1.34847	−0.0634969
$452$	− 65.3939i	− 3.07587i
$453$	− 14.0000i	− 0.657777i
$454$	−5.39388	−0.253147
$455$	0	0
$456$	−2.20204	−0.103120
$457$	9.89898i	0.463055i	0.972828	+	0.231527i	$0.0743723\pi$
$457$	9.89898i	0.463055i	−0.972828	+	0.231527i	$0.925628\pi$
$458$	− 50.2020i	− 2.34579i
$459$	0.550510	0.0256956
$460$	0	0
$461$	30.4949	1.42029	0.710144	−	0.704056i	$-0.248630\pi$
$461$	30.4949	1.42029	0.710144	+	0.704056i	$0.248630\pi$
$462$	6.00000i	0.279145i
$463$	− 28.9444i	− 1.34516i	−0.740025	−	0.672580i	$-0.765186\pi$
$463$	− 28.9444i	− 1.34516i	0.740025	−	0.672580i	$-0.234814\pi$
$464$	17.3939	0.807490
$465$	0	0
$466$	5.39388	0.249867
$467$	28.8434i	1.33471i	0.744739	+	0.667356i	$0.232574\pi$
$467$	28.8434i	1.33471i	−0.744739	+	0.667356i	$0.767426\pi$
$468$	1.79796i	0.0831107i
$469$	−7.00000	−0.323230
$470$	0	0
$471$	−20.5959	−0.949010
$472$	− 21.3031i	− 0.980553i
$473$	4.89898i	0.225255i
$474$	9.79796	0.450035
$475$	0	0
$476$	2.20204	0.100930
$477$	5.44949i	0.249515i
$478$	− 28.0454i	− 1.28277i
$479$	20.9444	0.956973	0.478487	−	0.878095i	$-0.341185\pi$
$479$	20.9444	0.956973	0.478487	+	0.878095i	$0.341185\pi$
$480$	0	0
$481$	2.65153	0.120899
$482$	− 35.8888i	− 1.63469i
$483$	− 1.00000i	− 0.0455016i
$484$	20.0000	0.909091
$485$	0	0
$486$	2.44949	0.111111
$487$	34.9444i	1.58348i	0.610857	+	0.791741i	$0.290825\pi$
$487$	34.9444i	1.58348i	−0.610857	+	0.791741i	$0.709175\pi$
$488$	− 75.1918i	− 3.40377i
$489$	10.0000	0.452216
$490$	0	0
$491$	17.4495	0.787484	0.393742	−	0.919221i	$-0.371180\pi$
$491$	17.4495	0.787484	0.393742	+	0.919221i	$0.371180\pi$
$492$	2.20204i	0.0992757i
$493$	− 2.39388i	− 0.107815i
$494$	−0.494897	−0.0222665
$495$	0	0
$496$	39.5959	1.77791
$497$	− 10.3485i	− 0.464192i
$498$	22.6515i	1.01504i
$499$	1.00000	0.0447661	0.0223831	−	0.999749i	$-0.492875\pi$
$499$	1.00000	0.0447661	0.0223831	+	0.999749i	$0.492875\pi$
$500$	0	0
$501$	−1.34847	−0.0602452
$502$	44.0908i	1.96787i
$503$	37.0454i	1.65177i	0.563836	+	0.825887i	$0.309325\pi$
$503$	37.0454i	1.65177i	−0.563836	+	0.825887i	$0.690675\pi$
$504$	4.89898	0.218218
$505$	0	0
$506$	6.00000	0.266733
$507$	− 12.7980i	− 0.568377i
$508$	25.7980i	1.14460i
$509$	−19.1010	−0.846638	−0.423319	−	0.905981i	$-0.639135\pi$
$509$	−19.1010	−0.846638	−0.423319	+	0.905981i	$0.639135\pi$
$510$	0	0
$511$	−9.34847	−0.413552
$512$	− 39.1918i	− 1.73205i
$513$	0.449490i	0.0198455i
$514$	26.0908	1.15082
$515$	0	0
$516$	8.00000	0.352180
$517$	− 8.69694i	− 0.382491i
$518$	− 14.4495i	− 0.634874i
$519$	19.5959	0.860165
$520$	0	0
$521$	19.8434	0.869354	0.434677	−	0.900587i	$-0.356863\pi$
$521$	19.8434	0.869354	0.434677	+	0.900587i	$0.356863\pi$
$522$	− 10.6515i	− 0.466205i
$523$	− 25.3939i	− 1.11040i	−0.831718	−	0.555198i	$-0.812642\pi$
$523$	− 25.3939i	− 1.11040i	0.831718	−	0.555198i	$-0.187358\pi$
$524$	−78.3837	−3.42421
$525$	0	0
$526$	6.74235	0.293980
$527$	− 5.44949i	− 0.237384i
$528$	9.79796i	0.426401i
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	−4.34847	−0.188707
$532$	1.79796i	0.0779514i
$533$	0.247449i	0.0107182i
$534$	17.3939	0.752707
$535$	0	0
$536$	−34.2929	−1.48123
$537$	− 6.00000i	− 0.258919i
$538$	1.34847i	0.0581366i
$539$	−14.6969	−0.633042
$540$	0	0
$541$	3.10102	0.133323	0.0666616	−	0.997776i	$-0.478765\pi$
$541$	3.10102	0.133323	0.0666616	+	0.997776i	$0.478765\pi$
$542$	5.64133i	0.242316i
$543$	− 0.898979i	− 0.0385789i
$544$	0	0
$545$	0	0
$546$	1.10102	0.0471193
$547$	− 28.0000i	− 1.19719i	−0.801050	−	0.598597i	$-0.795725\pi$
$547$	− 28.0000i	− 1.19719i	0.801050	−	0.598597i	$-0.204275\pi$
$548$	28.4041i	1.21336i
$549$	−15.3485	−0.655057
$550$	0	0
$551$	1.95459	0.0832684
$552$	− 4.89898i	− 0.208514i
$553$	− 4.00000i	− 0.170097i
$554$	24.4949	1.04069
$555$	0	0
$556$	59.1918	2.51029
$557$	25.0454i	1.06121i	0.847620	+	0.530604i	$0.178035\pi$
$557$	25.0454i	1.06121i	−0.847620	+	0.530604i	$0.821965\pi$
$558$	− 24.2474i	− 1.02648i
$559$	0.898979	0.0380228
$560$	0	0
$561$	1.34847	0.0569324
$562$	− 11.3939i	− 0.480622i
$563$	− 0.550510i	− 0.0232012i	−0.999933	−	0.0116006i	$-0.996307\pi$
$563$	− 0.550510i	− 0.0232012i	0.999933	−	0.0116006i	$-0.00369268\pi$
$564$	−14.2020	−0.598014
$565$	0	0
$566$	−29.1464	−1.22512
$567$	− 1.00000i	− 0.0419961i
$568$	− 50.6969i	− 2.12720i
$569$	−40.2929	−1.68916	−0.844582	−	0.535426i	$-0.820151\pi$
$569$	−40.2929	−1.68916	−0.844582	+	0.535426i	$0.820151\pi$
$570$	0	0
$571$	−21.1464	−0.884950	−0.442475	−	0.896781i	$-0.645900\pi$
$571$	−21.1464	−0.884950	−0.442475	+	0.896781i	$0.645900\pi$
$572$	4.40408i	0.184144i
$573$	− 6.24745i	− 0.260991i
$574$	1.34847	0.0562840
$575$	0	0
$576$	−8.00000	−0.333333
$577$	− 28.0000i	− 1.16566i	−0.812596	−	0.582828i	$-0.801946\pi$
$577$	− 28.0000i	− 1.16566i	0.812596	−	0.582828i	$-0.198054\pi$
$578$	40.8990i	1.70117i
$579$	−22.6969	−0.943253
$580$	0	0
$581$	9.24745	0.383649
$582$	31.5959i	1.30969i
$583$	13.3485i	0.552837i
$584$	−45.7980	−1.89513
$585$	0	0
$586$	7.95459	0.328601
$587$	− 6.00000i	− 0.247647i	−0.992304	−	0.123823i	$-0.960484\pi$
$587$	− 6.00000i	− 0.247647i	0.992304	−	0.123823i	$-0.0395156\pi$
$588$	24.0000i	0.989743i
$589$	4.44949	0.183338
$590$	0	0
$591$	−13.1010	−0.538904
$592$	− 23.5959i	− 0.969786i
$593$	− 32.4495i	− 1.33254i	−0.745710	−	0.666270i	$-0.767890\pi$
$593$	− 32.4495i	− 1.33254i	0.745710	−	0.666270i	$-0.232110\pi$
$594$	6.00000	0.246183
$595$	0	0
$596$	53.3939	2.18710
$597$	6.89898i	0.282356i
$598$	− 1.10102i	− 0.0450241i
$599$	−26.6969	−1.09081	−0.545404	−	0.838174i	$-0.683624\pi$
$599$	−26.6969	−1.09081	−0.545404	+	0.838174i	$0.683624\pi$
$600$	0	0
$601$	5.00000	0.203954	0.101977	−	0.994787i	$-0.467483\pi$
$601$	5.00000	0.203954	0.101977	+	0.994787i	$0.467483\pi$
$602$	− 4.89898i	− 0.199667i
$603$	7.00000i	0.285062i
$604$	−56.0000	−2.27861
$605$	0	0
$606$	−42.7423	−1.73629
$607$	8.24745i	0.334754i	0.985893	+	0.167377i	$0.0535296\pi$
$607$	8.24745i	0.334754i	−0.985893	+	0.167377i	$0.946470\pi$
$608$	0	0
$609$	−4.34847	−0.176209
$610$	0	0
$611$	−1.59592	−0.0645639
$612$	− 2.20204i	− 0.0890122i
$613$	− 14.0000i	− 0.565455i	−0.959200	−	0.282727i	$-0.908761\pi$
$613$	− 14.0000i	− 0.565455i	0.959200	−	0.282727i	$-0.0912392\pi$
$614$	42.4949	1.71495
$615$	0	0
$616$	12.0000	0.483494
$617$	17.4495i	0.702490i	0.936284	+	0.351245i	$0.114242\pi$
$617$	17.4495i	0.702490i	−0.936284	+	0.351245i	$0.885758\pi$
$618$	− 40.8990i	− 1.64520i
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	29.3939i	1.17859i
$623$	− 7.10102i	− 0.284496i
$624$	1.79796	0.0719760
$625$	0	0
$626$	−23.7526	−0.949343
$627$	1.10102i	0.0439705i
$628$	82.3837i	3.28747i
$629$	−3.24745	−0.129484
$630$	0	0
$631$	34.4495	1.37141	0.685706	−	0.727878i	$-0.259493\pi$
$631$	34.4495	1.37141	0.685706	+	0.727878i	$0.259493\pi$
$632$	− 19.5959i	− 0.779484i
$633$	− 11.0000i	− 0.437211i
$634$	44.6969	1.77514
$635$	0	0
$636$	21.7980	0.864345
$637$	2.69694i	0.106857i
$638$	− 26.0908i	− 1.03295i
$639$	−10.3485	−0.409379
$640$	0	0
$641$	−26.9444	−1.06424	−0.532120	−	0.846669i	$-0.678604\pi$
$641$	−26.9444	−1.06424	−0.532120	+	0.846669i	$0.678604\pi$
$642$	− 1.34847i	− 0.0532198i
$643$	− 19.6969i	− 0.776771i	−0.921497	−	0.388386i	$-0.873033\pi$
$643$	− 19.6969i	− 0.776771i	0.921497	−	0.388386i	$-0.126967\pi$
$644$	−4.00000	−0.157622
$645$	0	0
$646$	0.606123	0.0238476
$647$	46.0454i	1.81023i	0.425165	+	0.905116i	$0.360216\pi$
$647$	46.0454i	1.81023i	−0.425165	+	0.905116i	$0.639784\pi$
$648$	− 4.89898i	− 0.192450i
$649$	−10.6515	−0.418109
$650$	0	0
$651$	−9.89898	−0.387972
$652$	− 40.0000i	− 1.56652i
$653$	19.8434i	0.776531i	0.921548	+	0.388265i	$0.126926\pi$
$653$	19.8434i	0.776531i	−0.921548	+	0.388265i	$0.873074\pi$
$654$	−40.2929	−1.57558
$655$	0	0
$656$	2.20204	0.0859753
$657$	9.34847i	0.364719i
$658$	8.69694i	0.339042i
$659$	−29.1464	−1.13538	−0.567692	−	0.823241i	$-0.692163\pi$
$659$	−29.1464	−1.13538	−0.567692	+	0.823241i	$0.692163\pi$
$660$	0	0
$661$	−42.0908	−1.63714	−0.818571	−	0.574405i	$-0.805234\pi$
$661$	−42.0908	−1.63714	−0.818571	+	0.574405i	$0.805234\pi$
$662$	− 35.7526i	− 1.38956i
$663$	− 0.247449i	− 0.00961011i
$664$	45.3031	1.75810
$665$	0	0
$666$	−14.4495	−0.559906
$667$	4.34847i	0.168373i
$668$	5.39388i	0.208695i
$669$	17.5959	0.680297
$670$	0	0
$671$	−37.5959	−1.45137
$672$	0	0
$673$	− 17.5505i	− 0.676522i	−0.941052	−	0.338261i	$-0.890161\pi$
$673$	− 17.5505i	− 0.676522i	0.941052	−	0.338261i	$-0.109839\pi$
$674$	0.494897	0.0190627
$675$	0	0
$676$	−51.1918	−1.96892
$677$	− 17.4495i	− 0.670638i	−0.942105	−	0.335319i	$-0.891156\pi$
$677$	− 17.4495i	− 0.670638i	0.942105	−	0.335319i	$-0.108844\pi$
$678$	40.0454i	1.53793i
$679$	12.8990	0.495017
$680$	0	0
$681$	2.20204	0.0843824
$682$	− 59.3939i	− 2.27431i
$683$	6.24745i	0.239052i	0.992831	+	0.119526i	$0.0381375\pi$
$683$	6.24745i	0.239052i	−0.992831	+	0.119526i	$0.961863\pi$
$684$	1.79796	0.0687467
$685$	0	0
$686$	31.8434	1.21579
$687$	20.4949i	0.781929i
$688$	− 8.00000i	− 0.304997i
$689$	2.44949	0.0933181
$690$	0	0
$691$	35.7980	1.36182	0.680909	−	0.732368i	$-0.261585\pi$
$691$	35.7980	1.36182	0.680909	+	0.732368i	$0.261585\pi$
$692$	− 78.3837i	− 2.97970i
$693$	− 2.44949i	− 0.0930484i
$694$	46.7878	1.77604
$695$	0	0
$696$	−21.3031	−0.807490
$697$	− 0.303062i	− 0.0114793i
$698$	62.4495i	2.36375i
$699$	−2.20204	−0.0832888
$700$	0	0
$701$	−7.95459	−0.300441	−0.150220	−	0.988653i	$-0.547998\pi$
$701$	−7.95459	−0.300441	−0.150220	+	0.988653i	$0.547998\pi$
$702$	− 1.10102i	− 0.0415553i
$703$	− 2.65153i	− 0.100004i
$704$	−19.5959	−0.738549
$705$	0	0
$706$	−50.0908	−1.88519
$707$	17.4495i	0.656256i
$708$	17.3939i	0.653702i
$709$	−44.7423	−1.68033	−0.840167	−	0.542328i	$-0.817543\pi$
$709$	−44.7423	−1.68033	−0.840167	+	0.542328i	$0.817543\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	− 34.7878i	− 1.30373i
$713$	9.89898i	0.370720i
$714$	−1.34847	−0.0504652
$715$	0	0
$716$	−24.0000	−0.896922
$717$	11.4495i	0.427589i
$718$	20.6969i	0.772403i
$719$	−41.9444	−1.56426	−0.782131	−	0.623114i	$-0.785867\pi$
$719$	−41.9444	−1.56426	−0.782131	+	0.623114i	$0.785867\pi$
$720$	0	0
$721$	−16.6969	−0.621826
$722$	− 46.0454i	− 1.71363i
$723$	14.6515i	0.544896i
$724$	−3.59592	−0.133641
$725$	0	0
$726$	−12.2474	−0.454545
$727$	− 45.6969i	− 1.69481i	−0.530951	−	0.847403i	$-0.678165\pi$
$727$	− 45.6969i	− 1.69481i	0.530951	−	0.847403i	$-0.321835\pi$
$728$	− 2.20204i	− 0.0816131i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−1.10102	−0.0407227
$732$	61.3939i	2.26918i
$733$	− 30.5959i	− 1.13009i	−0.825061	−	0.565043i	$-0.808860\pi$
$733$	− 30.5959i	− 1.13009i	0.825061	−	0.565043i	$-0.191140\pi$
$734$	38.4495	1.41920
$735$	0	0
$736$	0	0
$737$	17.1464i	0.631597i
$738$	− 1.34847i	− 0.0496378i
$739$	28.7980	1.05935	0.529675	−	0.848201i	$-0.322314\pi$
$739$	28.7980	1.05935	0.529675	+	0.848201i	$0.322314\pi$
$740$	0	0
$741$	0.202041	0.00742216
$742$	− 13.3485i	− 0.490038i
$743$	2.20204i	0.0807851i	0.999184	+	0.0403925i	$0.0128608\pi$
$743$	2.20204i	0.0807851i	−0.999184	+	0.0403925i	$0.987139\pi$
$744$	−48.4949	−1.77791
$745$	0	0
$746$	−28.4041	−1.03995
$747$	− 9.24745i	− 0.338346i
$748$	− 5.39388i	− 0.197220i
$749$	−0.550510	−0.0201152
$750$	0	0
$751$	45.3485	1.65479	0.827395	−	0.561621i	$-0.189822\pi$
$751$	45.3485	1.65479	0.827395	+	0.561621i	$0.189822\pi$
$752$	14.2020i	0.517895i
$753$	− 18.0000i	− 0.655956i
$754$	−4.78775	−0.174360
$755$	0	0
$756$	−4.00000	−0.145479
$757$	25.6969i	0.933971i	0.884265	+	0.466986i	$0.154660\pi$
$757$	25.6969i	0.933971i	−0.884265	+	0.466986i	$0.845340\pi$
$758$	81.7980i	2.97104i
$759$	−2.44949	−0.0889108
$760$	0	0
$761$	−20.1464	−0.730307	−0.365154	−	0.930947i	$-0.618984\pi$
$761$	−20.1464	−0.730307	−0.365154	+	0.930947i	$0.618984\pi$
$762$	− 15.7980i	− 0.572300i
$763$	16.4495i	0.595512i
$764$	−24.9898	−0.904099
$765$	0	0
$766$	40.0454	1.44690
$767$	1.95459i	0.0705762i
$768$	32.0000i	1.15470i
$769$	−5.55051	−0.200157	−0.100078	−	0.994980i	$-0.531909\pi$
$769$	−5.55051	−0.200157	−0.100078	+	0.994980i	$0.531909\pi$
$770$	0	0
$771$	−10.6515	−0.383606
$772$	90.7878i	3.26752i
$773$	− 52.2929i	− 1.88084i	−0.340010	−	0.940422i	$-0.610431\pi$
$773$	− 52.2929i	− 1.88084i	0.340010	−	0.940422i	$-0.389569\pi$
$774$	−4.89898	−0.176090
$775$	0	0
$776$	63.1918	2.26845
$777$	5.89898i	0.211625i
$778$	80.0908i	2.87139i
$779$	0.247449	0.00886577
$780$	0	0
$781$	−25.3485	−0.907040
$782$	1.34847i	0.0482212i
$783$	4.34847i	0.155402i
$784$	24.0000	0.857143
$785$	0	0
$786$	48.0000	1.71210
$787$	7.69694i	0.274366i	0.990546	+	0.137183i	$0.0438049\pi$
$787$	7.69694i	0.274366i	−0.990546	+	0.137183i	$0.956195\pi$
$788$	52.4041i	1.86682i
$789$	−2.75255	−0.0979934
$790$	0	0
$791$	16.3485	0.581285
$792$	− 12.0000i	− 0.426401i
$793$	6.89898i	0.244990i
$794$	−14.2020	−0.504012
$795$	0	0
$796$	27.5959	0.978111
$797$	− 15.2474i	− 0.540092i	−0.962847	−	0.270046i	$-0.912961\pi$
$797$	− 15.2474i	− 0.540092i	0.962847	−	0.270046i	$-0.0870390\pi$
$798$	− 1.10102i	− 0.0389757i
$799$	1.95459	0.0691485
$800$	0	0
$801$	−7.10102	−0.250902
$802$	− 56.0908i	− 1.98064i
$803$	22.8990i	0.808087i
$804$	28.0000	0.987484
$805$	0	0
$806$	−10.8990	−0.383900
$807$	− 0.550510i	− 0.0193789i
$808$	85.4847i	3.00734i
$809$	−11.4495	−0.402543	−0.201271	−	0.979536i	$-0.564507\pi$
$809$	−11.4495	−0.402543	−0.201271	+	0.979536i	$0.564507\pi$
$810$	0	0
$811$	16.3939	0.575667	0.287833	−	0.957680i	$-0.407065\pi$
$811$	16.3939	0.575667	0.287833	+	0.957680i	$0.407065\pi$
$812$	17.3939i	0.610405i
$813$	− 2.30306i	− 0.0807719i
$814$	−35.3939	−1.24055
$815$	0	0
$816$	−2.20204	−0.0770869
$817$	− 0.898979i	− 0.0314513i
$818$	− 29.6413i	− 1.03638i
$819$	−0.449490	−0.0157064
$820$	0	0
$821$	50.6969	1.76934	0.884668	−	0.466222i	$-0.154385\pi$
$821$	50.6969	1.76934	0.884668	+	0.466222i	$0.154385\pi$
$822$	− 17.3939i	− 0.606681i
$823$	35.5959i	1.24080i	0.784287	+	0.620398i	$0.213029\pi$
$823$	35.5959i	1.24080i	−0.784287	+	0.620398i	$0.786971\pi$
$824$	−81.7980	−2.84957
$825$	0	0
$826$	10.6515	0.370614
$827$	− 2.14643i	− 0.0746386i	−0.999303	−	0.0373193i	$-0.988118\pi$
$827$	− 2.14643i	− 0.0746386i	0.999303	−	0.0373193i	$-0.0118819\pi$
$828$	4.00000i	0.139010i
$829$	−1.69694	−0.0589371	−0.0294686	−	0.999566i	$-0.509381\pi$
$829$	−1.69694	−0.0589371	−0.0294686	+	0.999566i	$0.509381\pi$
$830$	0	0
$831$	−10.0000	−0.346896
$832$	3.59592i	0.124666i
$833$	− 3.30306i	− 0.114444i
$834$	−36.2474	−1.25515
$835$	0	0
$836$	4.40408	0.152318
$837$	9.89898i	0.342159i
$838$	− 94.1816i	− 3.25345i
$839$	−33.1918	−1.14591	−0.572955	−	0.819587i	$-0.694203\pi$
$839$	−33.1918	−1.14591	−0.572955	+	0.819587i	$0.694203\pi$
$840$	0	0
$841$	−10.0908	−0.347959
$842$	5.50510i	0.189718i
$843$	4.65153i	0.160207i
$844$	−44.0000	−1.51454
$845$	0	0
$846$	8.69694	0.299007
$847$	5.00000i	0.171802i
$848$	− 21.7980i	− 0.748545i
$849$	11.8990	0.408372
$850$	0	0
$851$	5.89898	0.202214
$852$	41.3939i	1.41813i
$853$	42.0908i	1.44116i	0.693371	+	0.720581i	$0.256125\pi$
$853$	42.0908i	1.44116i	−0.693371	+	0.720581i	$0.743875\pi$
$854$	37.5959	1.28651
$855$	0	0
$856$	−2.69694	−0.0921795
$857$	25.5959i	0.874340i	0.899379	+	0.437170i	$0.144019\pi$
$857$	25.5959i	0.874340i	−0.899379	+	0.437170i	$0.855981\pi$
$858$	− 2.69694i	− 0.0920720i
$859$	40.1918	1.37133	0.685664	−	0.727918i	$-0.259512\pi$
$859$	40.1918	1.37133	0.685664	+	0.727918i	$0.259512\pi$
$860$	0	0
$861$	−0.550510	−0.0187613
$862$	60.0000i	2.04361i
$863$	42.4949i	1.44654i	0.690564	+	0.723272i	$0.257363\pi$
$863$	42.4949i	1.44654i	−0.690564	+	0.723272i	$0.742637\pi$
$864$	0	0
$865$	0	0
$866$	−41.1464	−1.39821
$867$	− 16.6969i	− 0.567058i
$868$	39.5959i	1.34397i
$869$	−9.79796	−0.332373
$870$	0	0
$871$	3.14643	0.106613
$872$	80.5857i	2.72898i
$873$	− 12.8990i	− 0.436564i
$874$	−1.10102	−0.0372426
$875$	0	0
$876$	37.3939	1.26342
$877$	38.4949i	1.29988i	0.759985	+	0.649940i	$0.225206\pi$
$877$	38.4949i	1.29988i	−0.759985	+	0.649940i	$0.774794\pi$
$878$	17.8888i	0.603717i
$879$	−3.24745	−0.109534
$880$	0	0
$881$	3.55051	0.119620	0.0598099	−	0.998210i	$-0.480951\pi$
$881$	3.55051	0.119620	0.0598099	+	0.998210i	$0.480951\pi$
$882$	− 14.6969i	− 0.494872i
$883$	42.4495i	1.42854i	0.699871	+	0.714270i	$0.253241\pi$
$883$	42.4495i	1.42854i	−0.699871	+	0.714270i	$0.746759\pi$
$884$	−0.989795	−0.0332904
$885$	0	0
$886$	−38.0908	−1.27969
$887$	42.4949i	1.42684i	0.700737	+	0.713420i	$0.252855\pi$
$887$	42.4949i	1.42684i	−0.700737	+	0.713420i	$0.747145\pi$
$888$	28.8990i	0.969786i
$889$	−6.44949	−0.216309
$890$	0	0
$891$	−2.44949	−0.0820610
$892$	− 70.3837i	− 2.35662i
$893$	1.59592i	0.0534054i
$894$	−32.6969	−1.09355
$895$	0	0
$896$	19.5959	0.654654
$897$	0.449490i	0.0150080i
$898$	− 7.95459i	− 0.265448i
$899$	43.0454	1.43564
$900$	0	0
$901$	−3.00000	−0.0999445
$902$	− 3.30306i	− 0.109980i
$903$	2.00000i	0.0665558i
$904$	80.0908	2.66378
$905$	0	0
$906$	34.2929	1.13930
$907$	19.2020i	0.637593i	0.947823	+	0.318797i	$0.103279\pi$
$907$	19.2020i	0.637593i	−0.947823	+	0.318797i	$0.896721\pi$
$908$	− 8.80816i	− 0.292309i
$909$	17.4495	0.578763
$910$	0	0
$911$	−33.7980	−1.11978	−0.559888	−	0.828568i	$-0.689156\pi$
$911$	−33.7980	−1.11978	−0.559888	+	0.828568i	$0.689156\pi$
$912$	− 1.79796i	− 0.0595364i
$913$	− 22.6515i	− 0.749656i
$914$	−24.2474	−0.802034
$915$	0	0
$916$	81.9796	2.70868
$917$	− 19.5959i	− 0.647114i
$918$	1.34847i	0.0445061i
$919$	−13.3939	−0.441823	−0.220912	−	0.975294i	$-0.570903\pi$
$919$	−13.3939	−0.441823	−0.220912	+	0.975294i	$0.570903\pi$
$920$	0	0
$921$	−17.3485	−0.571651
$922$	74.6969i	2.46001i
$923$	4.65153i	0.153107i
$924$	−9.79796	−0.322329
$925$	0	0
$926$	70.8990	2.32989
$927$	16.6969i	0.548399i
$928$	0	0
$929$	40.8434	1.34003	0.670014	−	0.742349i	$-0.266288\pi$
$929$	40.8434	1.34003	0.670014	+	0.742349i	$0.266288\pi$
$930$	0	0
$931$	2.69694	0.0883886
$932$	8.80816i	0.288521i
$933$	− 12.0000i	− 0.392862i
$934$	−70.6515	−2.31179
$935$	0	0
$936$	−2.20204	−0.0719760
$937$	0.898979i	0.0293684i	0.999892	+	0.0146842i	$0.00467429\pi$
$937$	0.898979i	0.0293684i	−0.999892	+	0.0146842i	$0.995326\pi$
$938$	− 17.1464i	− 0.559851i
$939$	9.69694	0.316448
$940$	0	0
$941$	−59.1464	−1.92812	−0.964059	−	0.265687i	$-0.914401\pi$
$941$	−59.1464	−1.92812	−0.964059	+	0.265687i	$0.914401\pi$
$942$	− 50.4495i	− 1.64373i
$943$	0.550510i	0.0179271i
$944$	17.3939	0.566122
$945$	0	0
$946$	−12.0000	−0.390154
$947$	48.4949i	1.57587i	0.615757	+	0.787936i	$0.288850\pi$
$947$	48.4949i	1.57587i	−0.615757	+	0.787936i	$0.711150\pi$
$948$	16.0000i	0.519656i
$949$	4.20204	0.136404
$950$	0	0
$951$	−18.2474	−0.591714
$952$	2.69694i	0.0874083i
$953$	− 49.5959i	− 1.60657i	−0.595595	−	0.803285i	$-0.703084\pi$
$953$	− 49.5959i	− 1.60657i	0.595595	−	0.803285i	$-0.296916\pi$
$954$	−13.3485	−0.432173
$955$	0	0
$956$	45.7980	1.48121
$957$	10.6515i	0.344315i
$958$	51.3031i	1.65753i
$959$	−7.10102	−0.229304
$960$	0	0
$961$	66.9898	2.16096
$962$	6.49490i	0.209404i
$963$	0.550510i	0.0177399i
$964$	58.6061	1.88758
$965$	0	0
$966$	2.44949	0.0788110
$967$	9.95459i	0.320118i	0.987107	+	0.160059i	$0.0511685\pi$
$967$	9.95459i	0.320118i	−0.987107	+	0.160059i	$0.948832\pi$
$968$	24.4949i	0.787296i
$969$	−0.247449	−0.00794920
$970$	0	0
$971$	−46.2929	−1.48561	−0.742804	−	0.669509i	$-0.766505\pi$
$971$	−46.2929	−1.48561	−0.742804	+	0.669509i	$0.766505\pi$
$972$	4.00000i	0.128300i
$973$	14.7980i	0.474401i
$974$	−85.5959	−2.74267
$975$	0	0
$976$	61.3939	1.96517
$977$	34.3485i	1.09890i	0.835525	+	0.549452i	$0.185164\pi$
$977$	34.3485i	1.09890i	−0.835525	+	0.549452i	$0.814836\pi$
$978$	24.4949i	0.783260i
$979$	−17.3939	−0.555911
$980$	0	0
$981$	16.4495	0.525192
$982$	42.7423i	1.36396i
$983$	− 39.7423i	− 1.26758i	−0.773504	−	0.633792i	$-0.781498\pi$
$983$	− 39.7423i	− 1.26758i	0.773504	−	0.633792i	$-0.218502\pi$
$984$	−2.69694	−0.0859753
$985$	0	0
$986$	5.86378	0.186741
$987$	− 3.55051i	− 0.113014i
$988$	− 0.808164i	− 0.0257111i
$989$	2.00000	0.0635963
$990$	0	0
$991$	−59.7878	−1.89922	−0.949610	−	0.313433i	$-0.898521\pi$
$991$	−59.7878	−1.89922	−0.949610	+	0.313433i	$0.898521\pi$
$992$	0	0
$993$	14.5959i	0.463187i
$994$	25.3485	0.804005
$995$	0	0
$996$	−36.9898	−1.17207
$997$	− 48.0908i	− 1.52305i	−0.648135	−	0.761526i	$-0.724451\pi$
$997$	− 48.0908i	− 1.52305i	0.648135	−	0.761526i	$-0.275549\pi$
$998$	2.44949i	0.0775372i
$999$	5.89898	0.186635

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1725.2.b.m.1174.3		4
5.2	odd	4		1725.2.a.y.1.1		2
5.3	odd	4		345.2.a.i.1.2	✓	2
5.4	even	2	inner	1725.2.b.m.1174.2		4
15.2	even	4		5175.2.a.bl.1.2		2
15.8	even	4		1035.2.a.k.1.1		2
20.3	even	4		5520.2.a.bi.1.2		2
115.68	even	4		7935.2.a.t.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
345.2.a.i.1.2	✓	2	5.3	odd	4
1035.2.a.k.1.1		2	15.8	even	4
1725.2.a.y.1.1		2	5.2	odd	4
1725.2.b.m.1174.2		4	5.4	even	2	inner
1725.2.b.m.1174.3		4	1.1	even	1	trivial
5175.2.a.bl.1.2		2	15.2	even	4
5520.2.a.bi.1.2		2	20.3	even	4
7935.2.a.t.1.2		2	115.68	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:	\( N \)	\(=\)	\( 1725 = 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1725.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.44949i q^{2} -1.00000i q^{3} -4.00000 q^{4} +2.44949 q^{6} -1.00000i q^{7} -4.89898i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+2.44949i q^{2} -1.00000i q^{3} -4.00000 q^{4} +2.44949 q^{6} -1.00000i q^{7} -4.89898i q^{8} -1.00000 q^{9} -2.44949 q^{11} +4.00000i q^{12} +0.449490i q^{13} +2.44949 q^{14} +4.00000 q^{16} -0.550510i q^{17} -2.44949i q^{18} +0.449490 q^{19} -1.00000 q^{21} -6.00000i q^{22} +1.00000i q^{23} -4.89898 q^{24} -1.10102 q^{26} +1.00000i q^{27} +4.00000i q^{28} +4.34847 q^{29} +9.89898 q^{31} +2.44949i q^{33} +1.34847 q^{34} +4.00000 q^{36} -5.89898i q^{37} +1.10102i q^{38} +0.449490 q^{39} +0.550510 q^{41} -2.44949i q^{42} -2.00000i q^{43} +9.79796 q^{44} -2.44949 q^{46} +3.55051i q^{47} -4.00000i q^{48} +6.00000 q^{49} -0.550510 q^{51} -1.79796i q^{52} -5.44949i q^{53} -2.44949 q^{54} -4.89898 q^{56} -0.449490i q^{57} +10.6515i q^{58} +4.34847 q^{59} +15.3485 q^{61} +24.2474i q^{62} +1.00000i q^{63} +8.00000 q^{64} -6.00000 q^{66} -7.00000i q^{67} +2.20204i q^{68} +1.00000 q^{69} +10.3485 q^{71} +4.89898i q^{72} -9.34847i q^{73} +14.4495 q^{74} -1.79796 q^{76} +2.44949i q^{77} +1.10102i q^{78} +4.00000 q^{79} +1.00000 q^{81} +1.34847i q^{82} +9.24745i q^{83} +4.00000 q^{84} +4.89898 q^{86} -4.34847i q^{87} +12.0000i q^{88} +7.10102 q^{89} +0.449490 q^{91} -4.00000i q^{92} -9.89898i q^{93} -8.69694 q^{94} +12.8990i q^{97} +14.6969i q^{98} +2.44949 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 16 q^{4} - 4 q^{9}+O(q^{10}) \) 4 * q - 16 * q^4 - 4 * q^9	\( 4 q - 16 q^{4} - 4 q^{9} + 16 q^{16} - 8 q^{19} - 4 q^{21} - 24 q^{26} - 12 q^{29} + 20 q^{31} - 24 q^{34} + 16 q^{36} - 8 q^{39} + 12 q^{41} + 24 q^{49} - 12 q^{51} - 12 q^{59} + 32 q^{61} + 32 q^{64} - 24 q^{66} + 4 q^{69} + 12 q^{71} + 48 q^{74} + 32 q^{76} + 16 q^{79} + 4 q^{81} + 16 q^{84} + 48 q^{89} - 8 q^{91} + 24 q^{94}+O(q^{100}) \) 4 * q - 16 * q^4 - 4 * q^9 + 16 * q^16 - 8 * q^19 - 4 * q^21 - 24 * q^26 - 12 * q^29 + 20 * q^31 - 24 * q^34 + 16 * q^36 - 8 * q^39 + 12 * q^41 + 24 * q^49 - 12 * q^51 - 12 * q^59 + 32 * q^61 + 32 * q^64 - 24 * q^66 + 4 * q^69 + 12 * q^71 + 48 * q^74 + 32 * q^76 + 16 * q^79 + 4 * q^81 + 16 * q^84 + 48 * q^89 - 8 * q^91 + 24 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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