LMFDB - Embedded newform 1725.2.b.j.1174.1

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:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 345)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1174.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1725.1174
Dual form			1725.2.b.j.1174.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +2.00000 q^{4} -3.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +2.00000 q^{4} -3.00000i q^{7} -1.00000 q^{9} -4.00000 q^{11} -2.00000i q^{12} +4.00000 q^{16} -3.00000i q^{17} +8.00000 q^{19} -3.00000 q^{21} -1.00000i q^{23} +1.00000i q^{27} -6.00000i q^{28} -9.00000 q^{29} -5.00000 q^{31} +4.00000i q^{33} -2.00000 q^{36} -9.00000i q^{37} +7.00000 q^{41} -4.00000i q^{43} -8.00000 q^{44} -2.00000i q^{47} -4.00000i q^{48} -2.00000 q^{49} -3.00000 q^{51} -13.0000i q^{53} -8.00000i q^{57} +3.00000 q^{59} -14.0000 q^{61} +3.00000i q^{63} +8.00000 q^{64} +13.0000i q^{67} -6.00000i q^{68} -1.00000 q^{69} -13.0000 q^{71} +4.00000i q^{73} +16.0000 q^{76} +12.0000i q^{77} +1.00000 q^{81} +1.00000i q^{83} -6.00000 q^{84} +9.00000i q^{87} +8.00000 q^{89} -2.00000i q^{92} +5.00000i q^{93} +10.0000i q^{97} +4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{4} - 2 q^{9}+O(q^{10}) $ 2 * q + 4 * q^4 - 2 * q^9	$ 2 q + 4 q^{4} - 2 q^{9} - 8 q^{11} + 8 q^{16} + 16 q^{19} - 6 q^{21} - 18 q^{29} - 10 q^{31} - 4 q^{36} + 14 q^{41} - 16 q^{44} - 4 q^{49} - 6 q^{51} + 6 q^{59} - 28 q^{61} + 16 q^{64} - 2 q^{69} - 26 q^{71} + 32 q^{76} + 2 q^{81} - 12 q^{84} + 16 q^{89} + 8 q^{99}+O(q^{100}) $ 2 * q + 4 * q^4 - 2 * q^9 - 8 * q^11 + 8 * q^16 + 16 * q^19 - 6 * q^21 - 18 * q^29 - 10 * q^31 - 4 * q^36 + 14 * q^41 - 16 * q^44 - 4 * q^49 - 6 * q^51 + 6 * q^59 - 28 * q^61 + 16 * q^64 - 2 * q^69 - 26 * q^71 + 32 * q^76 + 2 * q^81 - 12 * q^84 + 16 * q^89 + 8 * q^99

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 $

$2$	0	0	1.00000			$0$
$2$	0	0	−1.00000			$\pi$
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	2.00000	1.00000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 3.00000i	− 1.13389i	−0.823754	−	0.566947i	$-0.808125\pi$
$7$	− 3.00000i	− 1.13389i	0.823754	−	0.566947i	$-0.191875\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	− 2.00000i	− 0.577350i
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	1.00000
$17$	− 3.00000i	− 0.727607i	−0.931476	−	0.363803i	$-0.881478\pi$
$17$	− 3.00000i	− 0.727607i	0.931476	−	0.363803i	$-0.118522\pi$
$18$	0	0
$19$	8.00000	1.83533	0.917663	−	0.397360i	$-0.130073\pi$
$19$	8.00000	1.83533	0.917663	+	0.397360i	$0.130073\pi$
$20$	0	0
$21$	−3.00000	−0.654654
$22$	0	0
$23$	− 1.00000i	− 0.208514i
$23$	− 1.00000i	− 0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000i	0.192450i
$28$	− 6.00000i	− 1.13389i
$29$	−9.00000	−1.67126	−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000	−1.67126	−0.835629	+	0.549294i	$0.814897\pi$
$30$	0	0
$31$	−5.00000	−0.898027	−0.449013	−	0.893525i	$-0.648224\pi$
$31$	−5.00000	−0.898027	−0.449013	+	0.893525i	$0.648224\pi$
$32$	0	0
$33$	4.00000i	0.696311i
$34$	0	0
$35$	0	0
$36$	−2.00000	−0.333333
$37$	− 9.00000i	− 1.47959i	−0.672832	−	0.739795i	$-0.734922\pi$
$37$	− 9.00000i	− 1.47959i	0.672832	−	0.739795i	$-0.265078\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.00000	1.09322	0.546608	−	0.837389i	$-0.315919\pi$
$41$	7.00000	1.09322	0.546608	+	0.837389i	$0.315919\pi$
$42$	0	0
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	−8.00000	−1.20605
$45$	0	0
$46$	0	0
$47$	− 2.00000i	− 0.291730i	−0.989305	−	0.145865i	$-0.953403\pi$
$47$	− 2.00000i	− 0.291730i	0.989305	−	0.145865i	$-0.0465965\pi$
$48$	− 4.00000i	− 0.577350i
$49$	−2.00000	−0.285714
$50$	0	0
$51$	−3.00000	−0.420084
$52$	0	0
$53$	− 13.0000i	− 1.78569i	−0.450367	−	0.892844i	$-0.648707\pi$
$53$	− 13.0000i	− 1.78569i	0.450367	−	0.892844i	$-0.351293\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 8.00000i	− 1.05963i
$58$	0	0
$59$	3.00000	0.390567	0.195283	−	0.980747i	$-0.437437\pi$
$59$	3.00000	0.390567	0.195283	+	0.980747i	$0.437437\pi$
$60$	0	0
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	0	0
$63$	3.00000i	0.377964i
$64$	8.00000	1.00000
$65$	0	0
$66$	0	0
$67$	13.0000i	1.58820i	0.607785	+	0.794101i	$0.292058\pi$
$67$	13.0000i	1.58820i	−0.607785	+	0.794101i	$0.707942\pi$
$68$	− 6.00000i	− 0.727607i
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−13.0000	−1.54282	−0.771408	−	0.636341i	$-0.780447\pi$
$71$	−13.0000	−1.54282	−0.771408	+	0.636341i	$0.780447\pi$
$72$	0	0
$73$	4.00000i	0.468165i	0.972217	+	0.234082i	$0.0752085\pi$
$73$	4.00000i	0.468165i	−0.972217	+	0.234082i	$0.924791\pi$
$74$	0	0
$75$	0	0
$76$	16.0000	1.83533
$77$	12.0000i	1.36753i
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.00000i	0.109764i	0.998493	+	0.0548821i	$0.0174783\pi$
$83$	1.00000i	0.109764i	−0.998493	+	0.0548821i	$0.982522\pi$
$84$	−6.00000	−0.654654
$85$	0	0
$86$	0	0
$87$	9.00000i	0.964901i
$88$	0	0
$89$	8.00000	0.847998	0.423999	−	0.905663i	$-0.360626\pi$
$89$	8.00000	0.847998	0.423999	+	0.905663i	$0.360626\pi$
$90$	0	0
$91$	0	0
$92$	− 2.00000i	− 0.208514i
$93$	5.00000i	0.518476i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	10.0000i	1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$	10.0000i	1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$	0	0
$99$	4.00000	0.402015
$100$	0	0
$101$	9.00000	0.895533	0.447767	−	0.894150i	$-0.352219\pi$
$101$	9.00000	0.895533	0.447767	+	0.894150i	$0.352219\pi$
$102$	0	0
$103$	− 8.00000i	− 0.788263i	−0.919054	−	0.394132i	$-0.871045\pi$
$103$	− 8.00000i	− 0.788263i	0.919054	−	0.394132i	$-0.128955\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	11.0000i	1.06341i	0.846930	+	0.531705i	$0.178449\pi$
$107$	11.0000i	1.06341i	−0.846930	+	0.531705i	$0.821551\pi$
$108$	2.00000i	0.192450i
$109$	16.0000	1.53252	0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000	1.53252	0.766261	+	0.642529i	$0.222115\pi$
$110$	0	0
$111$	−9.00000	−0.854242
$112$	− 12.0000i	− 1.13389i
$113$	− 1.00000i	− 0.0940721i	−0.998893	−	0.0470360i	$-0.985022\pi$
$113$	− 1.00000i	− 0.0940721i	0.998893	−	0.0470360i	$-0.0149776\pi$
$114$	0	0
$115$	0	0
$116$	−18.0000	−1.67126
$117$	0	0
$118$	0	0
$119$	−9.00000	−0.825029
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	− 7.00000i	− 0.631169i
$124$	−10.0000	−0.898027
$125$	0	0
$126$	0	0
$127$	− 18.0000i	− 1.59724i	−0.601834	−	0.798621i	$-0.705563\pi$
$127$	− 18.0000i	− 1.59724i	0.601834	−	0.798621i	$-0.294437\pi$
$128$	0	0
$129$	−4.00000	−0.352180
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	8.00000i	0.696311i
$133$	− 24.0000i	− 2.08106i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.00000i	0.170872i	0.996344	+	0.0854358i	$0.0272282\pi$
$137$	2.00000i	0.170872i	−0.996344	+	0.0854358i	$0.972772\pi$
$138$	0	0
$139$	3.00000	0.254457	0.127228	−	0.991873i	$-0.459392\pi$
$139$	3.00000	0.254457	0.127228	+	0.991873i	$0.459392\pi$
$140$	0	0
$141$	−2.00000	−0.168430
$142$	0	0
$143$	0	0
$144$	−4.00000	−0.333333
$145$	0	0
$146$	0	0
$147$	2.00000i	0.164957i
$148$	− 18.0000i	− 1.47959i
$149$	14.0000	1.14692	0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000	1.14692	0.573462	+	0.819232i	$0.305600\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	0	0
$153$	3.00000i	0.242536i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 3.00000i	− 0.239426i	−0.992809	−	0.119713i	$-0.961803\pi$
$157$	− 3.00000i	− 0.239426i	0.992809	−	0.119713i	$-0.0381975\pi$
$158$	0	0
$159$	−13.0000	−1.03097
$160$	0	0
$161$	−3.00000	−0.236433
$162$	0	0
$163$	2.00000i	0.156652i	0.996928	+	0.0783260i	$0.0249575\pi$
$163$	2.00000i	0.156652i	−0.996928	+	0.0783260i	$0.975042\pi$
$164$	14.0000	1.09322
$165$	0	0
$166$	0	0
$167$	− 22.0000i	− 1.70241i	−0.524832	−	0.851206i	$-0.675872\pi$
$167$	− 22.0000i	− 1.70241i	0.524832	−	0.851206i	$-0.324128\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	0	0
$171$	−8.00000	−0.611775
$172$	− 8.00000i	− 0.609994i
$173$	24.0000i	1.82469i	0.409426	+	0.912343i	$0.365729\pi$
$173$	24.0000i	1.82469i	−0.409426	+	0.912343i	$0.634271\pi$
$174$	0	0
$175$	0	0
$176$	−16.0000	−1.20605
$177$	− 3.00000i	− 0.225494i
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	14.0000i	1.03491i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	12.0000i	0.877527i
$188$	− 4.00000i	− 0.291730i
$189$	3.00000	0.218218
$190$	0	0
$191$	−2.00000	−0.144715	−0.0723575	−	0.997379i	$-0.523052\pi$
$191$	−2.00000	−0.144715	−0.0723575	+	0.997379i	$0.523052\pi$
$192$	− 8.00000i	− 0.577350i
$193$	− 4.00000i	− 0.287926i	−0.989583	−	0.143963i	$-0.954015\pi$
$193$	− 4.00000i	− 0.287926i	0.989583	−	0.143963i	$-0.0459847\pi$
$194$	0	0
$195$	0	0
$196$	−4.00000	−0.285714
$197$	2.00000i	0.142494i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$	2.00000i	0.142494i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$	0	0
$199$	18.0000	1.27599	0.637993	−	0.770042i	$-0.279765\pi$
$199$	18.0000	1.27599	0.637993	+	0.770042i	$0.279765\pi$
$200$	0	0
$201$	13.0000	0.916949
$202$	0	0
$203$	27.0000i	1.89503i
$204$	−6.00000	−0.420084
$205$	0	0
$206$	0	0
$207$	1.00000i	0.0695048i
$208$	0	0
$209$	−32.0000	−2.21349
$210$	0	0
$211$	7.00000	0.481900	0.240950	−	0.970538i	$-0.422541\pi$
$211$	7.00000	0.481900	0.240950	+	0.970538i	$0.422541\pi$
$212$	− 26.0000i	− 1.78569i
$213$	13.0000i	0.890745i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	15.0000i	1.01827i
$218$	0	0
$219$	4.00000	0.270295
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 10.0000i	− 0.669650i	−0.942280	−	0.334825i	$-0.891323\pi$
$223$	− 10.0000i	− 0.669650i	0.942280	−	0.334825i	$-0.108677\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 4.00000i	− 0.265489i	−0.991150	−	0.132745i	$-0.957621\pi$
$227$	− 4.00000i	− 0.265489i	0.991150	−	0.132745i	$-0.0423790\pi$
$228$	− 16.0000i	− 1.05963i
$229$	−4.00000	−0.264327	−0.132164	−	0.991228i	$-0.542192\pi$
$229$	−4.00000	−0.264327	−0.132164	+	0.991228i	$0.542192\pi$
$230$	0	0
$231$	12.0000	0.789542
$232$	0	0
$233$	8.00000i	0.524097i	0.965055	+	0.262049i	$0.0843981\pi$
$233$	8.00000i	0.524097i	−0.965055	+	0.262049i	$0.915602\pi$
$234$	0	0
$235$	0	0
$236$	6.00000	0.390567
$237$	0	0
$238$	0	0
$239$	−13.0000	−0.840900	−0.420450	−	0.907316i	$-0.638128\pi$
$239$	−13.0000	−0.840900	−0.420450	+	0.907316i	$0.638128\pi$
$240$	0	0
$241$	20.0000	1.28831	0.644157	−	0.764894i	$-0.277208\pi$
$241$	20.0000	1.28831	0.644157	+	0.764894i	$0.277208\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	−28.0000	−1.79252
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	1.00000	0.0633724
$250$	0	0
$251$	18.0000	1.13615	0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000	1.13615	0.568075	+	0.822977i	$0.307688\pi$
$252$	6.00000i	0.377964i
$253$	4.00000i	0.251478i
$254$	0	0
$255$	0	0
$256$	16.0000	1.00000
$257$	18.0000i	1.12281i	0.827541	+	0.561405i	$0.189739\pi$
$257$	18.0000i	1.12281i	−0.827541	+	0.561405i	$0.810261\pi$
$258$	0	0
$259$	−27.0000	−1.67770
$260$	0	0
$261$	9.00000	0.557086
$262$	0	0
$263$	23.0000i	1.41824i	0.705087	+	0.709120i	$0.250908\pi$
$263$	23.0000i	1.41824i	−0.705087	+	0.709120i	$0.749092\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 8.00000i	− 0.489592i
$268$	26.0000i	1.58820i
$269$	17.0000	1.03651	0.518254	−	0.855227i	$-0.326582\pi$
$269$	17.0000	1.03651	0.518254	+	0.855227i	$0.326582\pi$
$270$	0	0
$271$	−23.0000	−1.39715	−0.698575	−	0.715537i	$-0.746182\pi$
$271$	−23.0000	−1.39715	−0.698575	+	0.715537i	$0.746182\pi$
$272$	− 12.0000i	− 0.727607i
$273$	0	0
$274$	0	0
$275$	0	0
$276$	−2.00000	−0.120386
$277$	18.0000i	1.08152i	0.841178	+	0.540758i	$0.181862\pi$
$277$	18.0000i	1.08152i	−0.841178	+	0.540758i	$0.818138\pi$
$278$	0	0
$279$	5.00000	0.299342
$280$	0	0
$281$	24.0000	1.43172	0.715860	−	0.698244i	$-0.246035\pi$
$281$	24.0000	1.43172	0.715860	+	0.698244i	$0.246035\pi$
$282$	0	0
$283$	5.00000i	0.297219i	0.988896	+	0.148610i	$0.0474798\pi$
$283$	5.00000i	0.297219i	−0.988896	+	0.148610i	$0.952520\pi$
$284$	−26.0000	−1.54282
$285$	0	0
$286$	0	0
$287$	− 21.0000i	− 1.23959i
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	10.0000	0.586210
$292$	8.00000i	0.468165i
$293$	− 19.0000i	− 1.10999i	−0.831853	−	0.554996i	$-0.812720\pi$
$293$	− 19.0000i	− 1.10999i	0.831853	−	0.554996i	$-0.187280\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 4.00000i	− 0.232104i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−12.0000	−0.691669
$302$	0	0
$303$	− 9.00000i	− 0.517036i
$304$	32.0000	1.83533
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	24.0000i	1.36753i
$309$	−8.00000	−0.455104
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	− 25.0000i	− 1.41308i	−0.707671	−	0.706542i	$-0.750254\pi$
$313$	− 25.0000i	− 1.41308i	0.707671	−	0.706542i	$-0.249746\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	30.0000i	1.68497i	0.538721	+	0.842484i	$0.318908\pi$
$317$	30.0000i	1.68497i	−0.538721	+	0.842484i	$0.681092\pi$
$318$	0	0
$319$	36.0000	2.01561
$320$	0	0
$321$	11.0000	0.613960
$322$	0	0
$323$	− 24.0000i	− 1.33540i
$324$	2.00000	0.111111
$325$	0	0
$326$	0	0
$327$	− 16.0000i	− 0.884802i
$328$	0	0
$329$	−6.00000	−0.330791
$330$	0	0
$331$	7.00000	0.384755	0.192377	−	0.981321i	$-0.438380\pi$
$331$	7.00000	0.384755	0.192377	+	0.981321i	$0.438380\pi$
$332$	2.00000i	0.109764i
$333$	9.00000i	0.493197i
$334$	0	0
$335$	0	0
$336$	−12.0000	−0.654654
$337$	10.0000i	0.544735i	0.962193	+	0.272367i	$0.0878066\pi$
$337$	10.0000i	0.544735i	−0.962193	+	0.272367i	$0.912193\pi$
$338$	0	0
$339$	−1.00000	−0.0543125
$340$	0	0
$341$	20.0000	1.08306
$342$	0	0
$343$	− 15.0000i	− 0.809924i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 16.0000i	− 0.858925i	−0.903085	−	0.429463i	$-0.858703\pi$
$347$	− 16.0000i	− 0.858925i	0.903085	−	0.429463i	$-0.141297\pi$
$348$	18.0000i	0.964901i
$349$	1.00000	0.0535288	0.0267644	−	0.999642i	$-0.491480\pi$
$349$	1.00000	0.0535288	0.0267644	+	0.999642i	$0.491480\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	6.00000i	0.319348i	0.987170	+	0.159674i	$0.0510443\pi$
$353$	6.00000i	0.319348i	−0.987170	+	0.159674i	$0.948956\pi$
$354$	0	0
$355$	0	0
$356$	16.0000	0.847998
$357$	9.00000i	0.476331i
$358$	0	0
$359$	2.00000	0.105556	0.0527780	−	0.998606i	$-0.483192\pi$
$359$	2.00000	0.105556	0.0527780	+	0.998606i	$0.483192\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	0	0
$363$	− 5.00000i	− 0.262432i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	31.0000i	1.61819i	0.587680	+	0.809093i	$0.300041\pi$
$367$	31.0000i	1.61819i	−0.587680	+	0.809093i	$0.699959\pi$
$368$	− 4.00000i	− 0.208514i
$369$	−7.00000	−0.364405
$370$	0	0
$371$	−39.0000	−2.02478
$372$	10.0000i	0.518476i
$373$	6.00000i	0.310668i	0.987862	+	0.155334i	$0.0496454\pi$
$373$	6.00000i	0.310668i	−0.987862	+	0.155334i	$0.950355\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	8.00000	0.410932	0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000	0.410932	0.205466	+	0.978664i	$0.434129\pi$
$380$	0	0
$381$	−18.0000	−0.922168
$382$	0	0
$383$	23.0000i	1.17525i	0.809135	+	0.587623i	$0.199936\pi$
$383$	23.0000i	1.17525i	−0.809135	+	0.587623i	$0.800064\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.00000i	0.203331i
$388$	20.0000i	1.01535i
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	0	0
$391$	−3.00000	−0.151717
$392$	0	0
$393$	4.00000i	0.201773i
$394$	0	0
$395$	0	0
$396$	8.00000	0.402015
$397$	8.00000i	0.401508i	0.979642	+	0.200754i	$0.0643393\pi$
$397$	8.00000i	0.401508i	−0.979642	+	0.200754i	$0.935661\pi$
$398$	0	0
$399$	−24.0000	−1.20150
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	0	0
$403$	0	0
$404$	18.0000	0.895533
$405$	0	0
$406$	0	0
$407$	36.0000i	1.78445i
$408$	0	0
$409$	−25.0000	−1.23617	−0.618085	−	0.786111i	$-0.712091\pi$
$409$	−25.0000	−1.23617	−0.618085	+	0.786111i	$0.712091\pi$
$410$	0	0
$411$	2.00000	0.0986527
$412$	− 16.0000i	− 0.788263i
$413$	− 9.00000i	− 0.442861i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 3.00000i	− 0.146911i
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	4.00000	0.194948	0.0974740	−	0.995238i	$-0.468924\pi$
$421$	4.00000	0.194948	0.0974740	+	0.995238i	$0.468924\pi$
$422$	0	0
$423$	2.00000i	0.0972433i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	42.0000i	2.03252i
$428$	22.0000i	1.06341i
$429$	0	0
$430$	0	0
$431$	20.0000	0.963366	0.481683	−	0.876346i	$-0.340026\pi$
$431$	20.0000	0.963366	0.481683	+	0.876346i	$0.340026\pi$
$432$	4.00000i	0.192450i
$433$	37.0000i	1.77811i	0.457804	+	0.889053i	$0.348636\pi$
$433$	37.0000i	1.77811i	−0.457804	+	0.889053i	$0.651364\pi$
$434$	0	0
$435$	0	0
$436$	32.0000	1.53252
$437$	− 8.00000i	− 0.382692i
$438$	0	0
$439$	24.0000	1.14546	0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000	1.14546	0.572729	+	0.819745i	$0.305885\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	0	0
$443$	− 6.00000i	− 0.285069i	−0.989790	−	0.142534i	$-0.954475\pi$
$443$	− 6.00000i	− 0.285069i	0.989790	−	0.142534i	$-0.0455251\pi$
$444$	−18.0000	−0.854242
$445$	0	0
$446$	0	0
$447$	− 14.0000i	− 0.662177i
$448$	− 24.0000i	− 1.13389i
$449$	11.0000	0.519122	0.259561	−	0.965727i	$-0.416422\pi$
$449$	11.0000	0.519122	0.259561	+	0.965727i	$0.416422\pi$
$450$	0	0
$451$	−28.0000	−1.31847
$452$	− 2.00000i	− 0.0940721i
$453$	8.00000i	0.375873i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 35.0000i	− 1.63723i	−0.574342	−	0.818615i	$-0.694742\pi$
$457$	− 35.0000i	− 1.63723i	0.574342	−	0.818615i	$-0.305258\pi$
$458$	0	0
$459$	3.00000	0.140028
$460$	0	0
$461$	26.0000	1.21094	0.605470	−	0.795868i	$-0.292985\pi$
$461$	26.0000	1.21094	0.605470	+	0.795868i	$0.292985\pi$
$462$	0	0
$463$	20.0000i	0.929479i	0.885448	+	0.464739i	$0.153852\pi$
$463$	20.0000i	0.929479i	−0.885448	+	0.464739i	$0.846148\pi$
$464$	−36.0000	−1.67126
$465$	0	0
$466$	0	0
$467$	− 13.0000i	− 0.601568i	−0.953692	−	0.300784i	$-0.902752\pi$
$467$	− 13.0000i	− 0.601568i	0.953692	−	0.300784i	$-0.0972484\pi$
$468$	0	0
$469$	39.0000	1.80085
$470$	0	0
$471$	−3.00000	−0.138233
$472$	0	0
$473$	16.0000i	0.735681i
$474$	0	0
$475$	0	0
$476$	−18.0000	−0.825029
$477$	13.0000i	0.595229i
$478$	0	0
$479$	−6.00000	−0.274147	−0.137073	−	0.990561i	$-0.543770\pi$
$479$	−6.00000	−0.274147	−0.137073	+	0.990561i	$0.543770\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	3.00000i	0.136505i
$484$	10.0000	0.454545
$485$	0	0
$486$	0	0
$487$	2.00000i	0.0906287i	0.998973	+	0.0453143i	$0.0144289\pi$
$487$	2.00000i	0.0906287i	−0.998973	+	0.0453143i	$0.985571\pi$
$488$	0	0
$489$	2.00000	0.0904431
$490$	0	0
$491$	3.00000	0.135388	0.0676941	−	0.997706i	$-0.478436\pi$
$491$	3.00000	0.135388	0.0676941	+	0.997706i	$0.478436\pi$
$492$	− 14.0000i	− 0.631169i
$493$	27.0000i	1.21602i
$494$	0	0
$495$	0	0
$496$	−20.0000	−0.898027
$497$	39.0000i	1.74939i
$498$	0	0
$499$	−21.0000	−0.940089	−0.470045	−	0.882643i	$-0.655762\pi$
$499$	−21.0000	−0.940089	−0.470045	+	0.882643i	$0.655762\pi$
$500$	0	0
$501$	−22.0000	−0.982888
$502$	0	0
$503$	− 21.0000i	− 0.936344i	−0.883637	−	0.468172i	$-0.844913\pi$
$503$	− 21.0000i	− 0.936344i	0.883637	−	0.468172i	$-0.155087\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 13.0000i	− 0.577350i
$508$	− 36.0000i	− 1.59724i
$509$	−30.0000	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$510$	0	0
$511$	12.0000	0.530849
$512$	0	0
$513$	8.00000i	0.353209i
$514$	0	0
$515$	0	0
$516$	−8.00000	−0.352180
$517$	8.00000i	0.351840i
$518$	0	0
$519$	24.0000	1.05348
$520$	0	0
$521$	−12.0000	−0.525730	−0.262865	−	0.964833i	$-0.584667\pi$
$521$	−12.0000	−0.525730	−0.262865	+	0.964833i	$0.584667\pi$
$522$	0	0
$523$	20.0000i	0.874539i	0.899331	+	0.437269i	$0.144054\pi$
$523$	20.0000i	0.874539i	−0.899331	+	0.437269i	$0.855946\pi$
$524$	−8.00000	−0.349482
$525$	0	0
$526$	0	0
$527$	15.0000i	0.653410i
$528$	16.0000i	0.696311i
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	−3.00000	−0.130189
$532$	− 48.0000i	− 2.08106i
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	8.00000	0.344584
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	20.0000i	0.855138i	0.903983	+	0.427569i	$0.140630\pi$
$547$	20.0000i	0.855138i	−0.903983	+	0.427569i	$0.859370\pi$
$548$	4.00000i	0.170872i
$549$	14.0000	0.597505
$550$	0	0
$551$	−72.0000	−3.06730
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	6.00000	0.254457
$557$	21.0000i	0.889799i	0.895581	+	0.444899i	$0.146761\pi$
$557$	21.0000i	0.889799i	−0.895581	+	0.444899i	$0.853239\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	12.0000	0.506640
$562$	0	0
$563$	11.0000i	0.463595i	0.972764	+	0.231797i	$0.0744606\pi$
$563$	11.0000i	0.463595i	−0.972764	+	0.231797i	$0.925539\pi$
$564$	−4.00000	−0.168430
$565$	0	0
$566$	0	0
$567$	− 3.00000i	− 0.125988i
$568$	0	0
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	−22.0000	−0.920671	−0.460336	−	0.887745i	$-0.652271\pi$
$571$	−22.0000	−0.920671	−0.460336	+	0.887745i	$0.652271\pi$
$572$	0	0
$573$	2.00000i	0.0835512i
$574$	0	0
$575$	0	0
$576$	−8.00000	−0.333333
$577$	− 20.0000i	− 0.832611i	−0.909225	−	0.416305i	$-0.863325\pi$
$577$	− 20.0000i	− 0.832611i	0.909225	−	0.416305i	$-0.136675\pi$
$578$	0	0
$579$	−4.00000	−0.166234
$580$	0	0
$581$	3.00000	0.124461
$582$	0	0
$583$	52.0000i	2.15362i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	14.0000i	0.577842i	0.957353	+	0.288921i	$0.0932965\pi$
$587$	14.0000i	0.577842i	−0.957353	+	0.288921i	$0.906704\pi$
$588$	4.00000i	0.164957i
$589$	−40.0000	−1.64817
$590$	0	0
$591$	2.00000	0.0822690
$592$	− 36.0000i	− 1.47959i
$593$	− 18.0000i	− 0.739171i	−0.929197	−	0.369586i	$-0.879500\pi$
$593$	− 18.0000i	− 0.739171i	0.929197	−	0.369586i	$-0.120500\pi$
$594$	0	0
$595$	0	0
$596$	28.0000	1.14692
$597$	− 18.0000i	− 0.736691i
$598$	0	0
$599$	4.00000	0.163436	0.0817178	−	0.996656i	$-0.473959\pi$
$599$	4.00000	0.163436	0.0817178	+	0.996656i	$0.473959\pi$
$600$	0	0
$601$	−41.0000	−1.67242	−0.836212	−	0.548406i	$-0.815235\pi$
$601$	−41.0000	−1.67242	−0.836212	+	0.548406i	$0.815235\pi$
$602$	0	0
$603$	− 13.0000i	− 0.529401i
$604$	−16.0000	−0.651031
$605$	0	0
$606$	0	0
$607$	− 10.0000i	− 0.405887i	−0.979190	−	0.202944i	$-0.934949\pi$
$607$	− 10.0000i	− 0.405887i	0.979190	−	0.202944i	$-0.0650509\pi$
$608$	0	0
$609$	27.0000	1.09410
$610$	0	0
$611$	0	0
$612$	6.00000i	0.242536i
$613$	− 14.0000i	− 0.565455i	−0.959200	−	0.282727i	$-0.908761\pi$
$613$	− 14.0000i	− 0.565455i	0.959200	−	0.282727i	$-0.0912392\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 33.0000i	− 1.32853i	−0.747497	−	0.664265i	$-0.768745\pi$
$617$	− 33.0000i	− 1.32853i	0.747497	−	0.664265i	$-0.231255\pi$
$618$	0	0
$619$	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
$622$	0	0
$623$	− 24.0000i	− 0.961540i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	32.0000i	1.27796i
$628$	− 6.00000i	− 0.239426i
$629$	−27.0000	−1.07656
$630$	0	0
$631$	38.0000	1.51276	0.756378	−	0.654135i	$-0.226967\pi$
$631$	38.0000	1.51276	0.756378	+	0.654135i	$0.226967\pi$
$632$	0	0
$633$	− 7.00000i	− 0.278225i
$634$	0	0
$635$	0	0
$636$	−26.0000	−1.03097
$637$	0	0
$638$	0	0
$639$	13.0000	0.514272
$640$	0	0
$641$	−20.0000	−0.789953	−0.394976	−	0.918691i	$-0.629247\pi$
$641$	−20.0000	−0.789953	−0.394976	+	0.918691i	$0.629247\pi$
$642$	0	0
$643$	− 13.0000i	− 0.512670i	−0.966588	−	0.256335i	$-0.917485\pi$
$643$	− 13.0000i	− 0.512670i	0.966588	−	0.256335i	$-0.0825150\pi$
$644$	−6.00000	−0.236433
$645$	0	0
$646$	0	0
$647$	12.0000i	0.471769i	0.971781	+	0.235884i	$0.0757987\pi$
$647$	12.0000i	0.471769i	−0.971781	+	0.235884i	$0.924201\pi$
$648$	0	0
$649$	−12.0000	−0.471041
$650$	0	0
$651$	15.0000	0.587896
$652$	4.00000i	0.156652i
$653$	− 16.0000i	− 0.626128i	−0.949732	−	0.313064i	$-0.898644\pi$
$653$	− 16.0000i	− 0.626128i	0.949732	−	0.313064i	$-0.101356\pi$
$654$	0	0
$655$	0	0
$656$	28.0000	1.09322
$657$	− 4.00000i	− 0.156055i
$658$	0	0
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	0	0
$661$	−14.0000	−0.544537	−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000	−0.544537	−0.272268	+	0.962221i	$0.587774\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	9.00000i	0.348481i
$668$	− 44.0000i	− 1.70241i
$669$	−10.0000	−0.386622
$670$	0	0
$671$	56.0000	2.16186
$672$	0	0
$673$	34.0000i	1.31060i	0.755367	+	0.655302i	$0.227459\pi$
$673$	34.0000i	1.31060i	−0.755367	+	0.655302i	$0.772541\pi$
$674$	0	0
$675$	0	0
$676$	26.0000	1.00000
$677$	− 45.0000i	− 1.72949i	−0.502211	−	0.864745i	$-0.667480\pi$
$677$	− 45.0000i	− 1.72949i	0.502211	−	0.864745i	$-0.332520\pi$
$678$	0	0
$679$	30.0000	1.15129
$680$	0	0
$681$	−4.00000	−0.153280
$682$	0	0
$683$	− 26.0000i	− 0.994862i	−0.867503	−	0.497431i	$-0.834277\pi$
$683$	− 26.0000i	− 0.994862i	0.867503	−	0.497431i	$-0.165723\pi$
$684$	−16.0000	−0.611775
$685$	0	0
$686$	0	0
$687$	4.00000i	0.152610i
$688$	− 16.0000i	− 0.609994i
$689$	0	0
$690$	0	0
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	48.0000i	1.82469i
$693$	− 12.0000i	− 0.455842i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 21.0000i	− 0.795432i
$698$	0	0
$699$	8.00000	0.302588
$700$	0	0
$701$	−6.00000	−0.226617	−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000	−0.226617	−0.113308	+	0.993560i	$0.536145\pi$
$702$	0	0
$703$	− 72.0000i	− 2.71553i
$704$	−32.0000	−1.20605
$705$	0	0
$706$	0	0
$707$	− 27.0000i	− 1.01544i
$708$	− 6.00000i	− 0.225494i
$709$	−22.0000	−0.826227	−0.413114	−	0.910679i	$-0.635559\pi$
$709$	−22.0000	−0.826227	−0.413114	+	0.910679i	$0.635559\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	5.00000i	0.187251i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	13.0000i	0.485494i
$718$	0	0
$719$	7.00000	0.261056	0.130528	−	0.991445i	$-0.458333\pi$
$719$	7.00000	0.261056	0.130528	+	0.991445i	$0.458333\pi$
$720$	0	0
$721$	−24.0000	−0.893807
$722$	0	0
$723$	− 20.0000i	− 0.743808i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 37.0000i	− 1.37225i	−0.727482	−	0.686127i	$-0.759309\pi$
$727$	− 37.0000i	− 1.37225i	0.727482	−	0.686127i	$-0.240691\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−12.0000	−0.443836
$732$	28.0000i	1.03491i
$733$	− 51.0000i	− 1.88373i	−0.335994	−	0.941864i	$-0.609072\pi$
$733$	− 51.0000i	− 1.88373i	0.335994	−	0.941864i	$-0.390928\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 52.0000i	− 1.91544i
$738$	0	0
$739$	−27.0000	−0.993211	−0.496606	−	0.867976i	$-0.665420\pi$
$739$	−27.0000	−0.993211	−0.496606	+	0.867976i	$0.665420\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	48.0000i	1.76095i	0.474093	+	0.880475i	$0.342776\pi$
$743$	48.0000i	1.76095i	−0.474093	+	0.880475i	$0.657224\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 1.00000i	− 0.0365881i
$748$	24.0000i	0.877527i
$749$	33.0000	1.20579
$750$	0	0
$751$	−16.0000	−0.583848	−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000	−0.583848	−0.291924	+	0.956441i	$0.594295\pi$
$752$	− 8.00000i	− 0.291730i
$753$	− 18.0000i	− 0.655956i
$754$	0	0
$755$	0	0
$756$	6.00000	0.218218
$757$	45.0000i	1.63555i	0.575536	+	0.817776i	$0.304793\pi$
$757$	45.0000i	1.63555i	−0.575536	+	0.817776i	$0.695207\pi$
$758$	0	0
$759$	4.00000	0.145191
$760$	0	0
$761$	−41.0000	−1.48625	−0.743124	−	0.669153i	$-0.766657\pi$
$761$	−41.0000	−1.48625	−0.743124	+	0.669153i	$0.766657\pi$
$762$	0	0
$763$	− 48.0000i	− 1.73772i
$764$	−4.00000	−0.144715
$765$	0	0
$766$	0	0
$767$	0	0
$768$	− 16.0000i	− 0.577350i
$769$	−6.00000	−0.216366	−0.108183	−	0.994131i	$-0.534503\pi$
$769$	−6.00000	−0.216366	−0.108183	+	0.994131i	$0.534503\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	− 8.00000i	− 0.287926i
$773$	46.0000i	1.65451i	0.561830	+	0.827253i	$0.310097\pi$
$773$	46.0000i	1.65451i	−0.561830	+	0.827253i	$0.689903\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	27.0000i	0.968620i
$778$	0	0
$779$	56.0000	2.00641
$780$	0	0
$781$	52.0000	1.86071
$782$	0	0
$783$	− 9.00000i	− 0.321634i
$784$	−8.00000	−0.285714
$785$	0	0
$786$	0	0
$787$	− 27.0000i	− 0.962446i	−0.876598	−	0.481223i	$-0.840193\pi$
$787$	− 27.0000i	− 0.962446i	0.876598	−	0.481223i	$-0.159807\pi$
$788$	4.00000i	0.142494i
$789$	23.0000	0.818822
$790$	0	0
$791$	−3.00000	−0.106668
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	36.0000	1.27599
$797$	27.0000i	0.956389i	0.878254	+	0.478195i	$0.158709\pi$
$797$	27.0000i	0.956389i	−0.878254	+	0.478195i	$0.841291\pi$
$798$	0	0
$799$	−6.00000	−0.212265
$800$	0	0
$801$	−8.00000	−0.282666
$802$	0	0
$803$	− 16.0000i	− 0.564628i
$804$	26.0000	0.916949
$805$	0	0
$806$	0	0
$807$	− 17.0000i	− 0.598428i
$808$	0	0
$809$	39.0000	1.37117	0.685583	−	0.727994i	$-0.259547\pi$
$809$	39.0000	1.37117	0.685583	+	0.727994i	$0.259547\pi$
$810$	0	0
$811$	−25.0000	−0.877869	−0.438934	−	0.898519i	$-0.644644\pi$
$811$	−25.0000	−0.877869	−0.438934	+	0.898519i	$0.644644\pi$
$812$	54.0000i	1.89503i
$813$	23.0000i	0.806645i
$814$	0	0
$815$	0	0
$816$	−12.0000	−0.420084
$817$	− 32.0000i	− 1.11954i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−10.0000	−0.349002	−0.174501	−	0.984657i	$-0.555831\pi$
$821$	−10.0000	−0.349002	−0.174501	+	0.984657i	$0.555831\pi$
$822$	0	0
$823$	16.0000i	0.557725i	0.960331	+	0.278862i	$0.0899574\pi$
$823$	16.0000i	0.557725i	−0.960331	+	0.278862i	$0.910043\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	37.0000i	1.28662i	0.765607	+	0.643308i	$0.222439\pi$
$827$	37.0000i	1.28662i	−0.765607	+	0.643308i	$0.777561\pi$
$828$	2.00000i	0.0695048i
$829$	27.0000	0.937749	0.468874	−	0.883265i	$-0.344660\pi$
$829$	27.0000	0.937749	0.468874	+	0.883265i	$0.344660\pi$
$830$	0	0
$831$	18.0000	0.624413
$832$	0	0
$833$	6.00000i	0.207888i
$834$	0	0
$835$	0	0
$836$	−64.0000	−2.21349
$837$	− 5.00000i	− 0.172825i
$838$	0	0
$839$	6.00000	0.207143	0.103572	−	0.994622i	$-0.466973\pi$
$839$	6.00000	0.207143	0.103572	+	0.994622i	$0.466973\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	0	0
$843$	− 24.0000i	− 0.826604i
$844$	14.0000	0.481900
$845$	0	0
$846$	0	0
$847$	− 15.0000i	− 0.515406i
$848$	− 52.0000i	− 1.78569i
$849$	5.00000	0.171600
$850$	0	0
$851$	−9.00000	−0.308516
$852$	26.0000i	0.890745i
$853$	46.0000i	1.57501i	0.616308	+	0.787505i	$0.288628\pi$
$853$	46.0000i	1.57501i	−0.616308	+	0.787505i	$0.711372\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 18.0000i	− 0.614868i	−0.951569	−	0.307434i	$-0.900530\pi$
$857$	− 18.0000i	− 0.614868i	0.951569	−	0.307434i	$-0.0994704\pi$
$858$	0	0
$859$	−43.0000	−1.46714	−0.733571	−	0.679613i	$-0.762148\pi$
$859$	−43.0000	−1.46714	−0.733571	+	0.679613i	$0.762148\pi$
$860$	0	0
$861$	−21.0000	−0.715678
$862$	0	0
$863$	− 54.0000i	− 1.83818i	−0.394046	−	0.919091i	$-0.628925\pi$
$863$	− 54.0000i	− 1.83818i	0.394046	−	0.919091i	$-0.371075\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 8.00000i	− 0.271694i
$868$	30.0000i	1.01827i
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 10.0000i	− 0.338449i
$874$	0	0
$875$	0	0
$876$	8.00000	0.270295
$877$	− 22.0000i	− 0.742887i	−0.928456	−	0.371444i	$-0.878863\pi$
$877$	− 22.0000i	− 0.742887i	0.928456	−	0.371444i	$-0.121137\pi$
$878$	0	0
$879$	−19.0000	−0.640854
$880$	0	0
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	0	0
$883$	34.0000i	1.14419i	0.820187	+	0.572096i	$0.193869\pi$
$883$	34.0000i	1.14419i	−0.820187	+	0.572096i	$0.806131\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 46.0000i	− 1.54453i	−0.635301	−	0.772264i	$-0.719124\pi$
$887$	− 46.0000i	− 1.54453i	0.635301	−	0.772264i	$-0.280876\pi$
$888$	0	0
$889$	−54.0000	−1.81110
$890$	0	0
$891$	−4.00000	−0.134005
$892$	− 20.0000i	− 0.669650i
$893$	− 16.0000i	− 0.535420i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	45.0000	1.50083
$900$	0	0
$901$	−39.0000	−1.29928
$902$	0	0
$903$	12.0000i	0.399335i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	53.0000i	1.75984i	0.475125	+	0.879918i	$0.342403\pi$
$907$	53.0000i	1.75984i	−0.475125	+	0.879918i	$0.657597\pi$
$908$	− 8.00000i	− 0.265489i
$909$	−9.00000	−0.298511
$910$	0	0
$911$	−40.0000	−1.32526	−0.662630	−	0.748947i	$-0.730560\pi$
$911$	−40.0000	−1.32526	−0.662630	+	0.748947i	$0.730560\pi$
$912$	− 32.0000i	− 1.05963i
$913$	− 4.00000i	− 0.132381i
$914$	0	0
$915$	0	0
$916$	−8.00000	−0.264327
$917$	12.0000i	0.396275i
$918$	0	0
$919$	−24.0000	−0.791687	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000	−0.791687	−0.395843	+	0.918318i	$0.629548\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	24.0000	0.789542
$925$	0	0
$926$	0	0
$927$	8.00000i	0.262754i
$928$	0	0
$929$	−57.0000	−1.87011	−0.935055	−	0.354504i	$-0.884650\pi$
$929$	−57.0000	−1.87011	−0.935055	+	0.354504i	$0.884650\pi$
$930$	0	0
$931$	−16.0000	−0.524379
$932$	16.0000i	0.524097i
$933$	8.00000i	0.261908i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	10.0000i	0.326686i	0.986569	+	0.163343i	$0.0522277\pi$
$937$	10.0000i	0.326686i	−0.986569	+	0.163343i	$0.947772\pi$
$938$	0	0
$939$	−25.0000	−0.815844
$940$	0	0
$941$	10.0000	0.325991	0.162995	−	0.986627i	$-0.447884\pi$
$941$	10.0000	0.325991	0.162995	+	0.986627i	$0.447884\pi$
$942$	0	0
$943$	− 7.00000i	− 0.227951i
$944$	12.0000	0.390567
$945$	0	0
$946$	0	0
$947$	− 8.00000i	− 0.259965i	−0.991516	−	0.129983i	$-0.958508\pi$
$947$	− 8.00000i	− 0.259965i	0.991516	−	0.129983i	$-0.0414921\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	30.0000	0.972817
$952$	0	0
$953$	22.0000i	0.712650i	0.934362	+	0.356325i	$0.115970\pi$
$953$	22.0000i	0.712650i	−0.934362	+	0.356325i	$0.884030\pi$
$954$	0	0
$955$	0	0
$956$	−26.0000	−0.840900
$957$	− 36.0000i	− 1.16371i
$958$	0	0
$959$	6.00000	0.193750
$960$	0	0
$961$	−6.00000	−0.193548
$962$	0	0
$963$	− 11.0000i	− 0.354470i
$964$	40.0000	1.28831
$965$	0	0
$966$	0	0
$967$	10.0000i	0.321578i	0.986989	+	0.160789i	$0.0514039\pi$
$967$	10.0000i	0.321578i	−0.986989	+	0.160789i	$0.948596\pi$
$968$	0	0
$969$	−24.0000	−0.770991
$970$	0	0
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	− 2.00000i	− 0.0641500i
$973$	− 9.00000i	− 0.288527i
$974$	0	0
$975$	0	0
$976$	−56.0000	−1.79252
$977$	− 25.0000i	− 0.799821i	−0.916554	−	0.399910i	$-0.869041\pi$
$977$	− 25.0000i	− 0.799821i	0.916554	−	0.399910i	$-0.130959\pi$
$978$	0	0
$979$	−32.0000	−1.02272
$980$	0	0
$981$	−16.0000	−0.510841
$982$	0	0
$983$	− 15.0000i	− 0.478426i	−0.970967	−	0.239213i	$-0.923111\pi$
$983$	− 15.0000i	− 0.478426i	0.970967	−	0.239213i	$-0.0768894\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	6.00000i	0.190982i
$988$	0	0
$989$	−4.00000	−0.127193
$990$	0	0
$991$	−33.0000	−1.04828	−0.524140	−	0.851632i	$-0.675613\pi$
$991$	−33.0000	−1.04828	−0.524140	+	0.851632i	$0.675613\pi$
$992$	0	0
$993$	− 7.00000i	− 0.222138i
$994$	0	0
$995$	0	0
$996$	2.00000	0.0633724
$997$	8.00000i	0.253363i	0.991943	+	0.126681i	$0.0404325\pi$
$997$	8.00000i	0.253363i	−0.991943	+	0.126681i	$0.959567\pi$
$998$	0	0
$999$	9.00000	0.284747

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1725.2.b.j.1174.1		2
5.2	odd	4		1725.2.a.k.1.1		1
5.3	odd	4		345.2.a.d.1.1	✓	1
5.4	even	2	inner	1725.2.b.j.1174.2		2
15.2	even	4		5175.2.a.o.1.1		1
15.8	even	4		1035.2.a.d.1.1		1
20.3	even	4		5520.2.a.h.1.1		1
115.68	even	4		7935.2.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
345.2.a.d.1.1	✓	1	5.3	odd	4
1035.2.a.d.1.1		1	15.8	even	4
1725.2.a.k.1.1		1	5.2	odd	4
1725.2.b.j.1174.1		2	1.1	even	1	trivial
1725.2.b.j.1174.2		2	5.4	even	2	inner
5175.2.a.o.1.1		1	15.2	even	4
5520.2.a.h.1.1		1	20.3	even	4
7935.2.a.i.1.1		1	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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +2.00000 q^{4} -3.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +2.00000 q^{4} -3.00000i q^{7} -1.00000 q^{9} -4.00000 q^{11} -2.00000i q^{12} +4.00000 q^{16} -3.00000i q^{17} +8.00000 q^{19} -3.00000 q^{21} -1.00000i q^{23} +1.00000i q^{27} -6.00000i q^{28} -9.00000 q^{29} -5.00000 q^{31} +4.00000i q^{33} -2.00000 q^{36} -9.00000i q^{37} +7.00000 q^{41} -4.00000i q^{43} -8.00000 q^{44} -2.00000i q^{47} -4.00000i q^{48} -2.00000 q^{49} -3.00000 q^{51} -13.0000i q^{53} -8.00000i q^{57} +3.00000 q^{59} -14.0000 q^{61} +3.00000i q^{63} +8.00000 q^{64} +13.0000i q^{67} -6.00000i q^{68} -1.00000 q^{69} -13.0000 q^{71} +4.00000i q^{73} +16.0000 q^{76} +12.0000i q^{77} +1.00000 q^{81} +1.00000i q^{83} -6.00000 q^{84} +9.00000i q^{87} +8.00000 q^{89} -2.00000i q^{92} +5.00000i q^{93} +10.0000i q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{4} - 2 q^{9}+O(q^{10}) \) 2 * q + 4 * q^4 - 2 * q^9	\( 2 q + 4 q^{4} - 2 q^{9} - 8 q^{11} + 8 q^{16} + 16 q^{19} - 6 q^{21} - 18 q^{29} - 10 q^{31} - 4 q^{36} + 14 q^{41} - 16 q^{44} - 4 q^{49} - 6 q^{51} + 6 q^{59} - 28 q^{61} + 16 q^{64} - 2 q^{69} - 26 q^{71} + 32 q^{76} + 2 q^{81} - 12 q^{84} + 16 q^{89} + 8 q^{99}+O(q^{100}) \) 2 * q + 4 * q^4 - 2 * q^9 - 8 * q^11 + 8 * q^16 + 16 * q^19 - 6 * q^21 - 18 * q^29 - 10 * q^31 - 4 * q^36 + 14 * q^41 - 16 * q^44 - 4 * q^49 - 6 * q^51 + 6 * q^59 - 28 * q^61 + 16 * q^64 - 2 * q^69 - 26 * q^71 + 32 * q^76 + 2 * q^81 - 12 * q^84 + 16 * q^89 + 8 * q^99

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