LMFDB - Embedded newform 525.2.d.b.274.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [525,2,Mod(274,525)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("525.274"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(525, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 525 = 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	525.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,2,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$4.19214610612$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 105)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	525.274
Dual form			525.2.d.b.274.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} +1.00000i q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000i q^{7} -3.00000i q^{8} -1.00000 q^{9} +1.00000i q^{12} -6.00000i q^{13} -1.00000 q^{14} -1.00000 q^{16} -2.00000i q^{17} +1.00000i q^{18} +8.00000 q^{19} +1.00000 q^{21} +8.00000i q^{23} +3.00000 q^{24} -6.00000 q^{26} -1.00000i q^{27} -1.00000i q^{28} +2.00000 q^{29} +4.00000 q^{31} -5.00000i q^{32} -2.00000 q^{34} -1.00000 q^{36} +2.00000i q^{37} -8.00000i q^{38} +6.00000 q^{39} -6.00000 q^{41} -1.00000i q^{42} +4.00000i q^{43} +8.00000 q^{46} -8.00000i q^{47} -1.00000i q^{48} -1.00000 q^{49} +2.00000 q^{51} -6.00000i q^{52} +10.0000i q^{53} -1.00000 q^{54} -3.00000 q^{56} +8.00000i q^{57} -2.00000i q^{58} -4.00000 q^{59} -2.00000 q^{61} -4.00000i q^{62} +1.00000i q^{63} -7.00000 q^{64} -4.00000i q^{67} -2.00000i q^{68} -8.00000 q^{69} -12.0000 q^{71} +3.00000i q^{72} -2.00000i q^{73} +2.00000 q^{74} +8.00000 q^{76} -6.00000i q^{78} -8.00000 q^{79} +1.00000 q^{81} +6.00000i q^{82} -4.00000i q^{83} +1.00000 q^{84} +4.00000 q^{86} +2.00000i q^{87} +6.00000 q^{89} -6.00000 q^{91} +8.00000i q^{92} +4.00000i q^{93} -8.00000 q^{94} +5.00000 q^{96} +18.0000i q^{97} +1.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} + 2 q^{6} - 2 q^{9} - 2 q^{14} - 2 q^{16} + 16 q^{19} + 2 q^{21} + 6 q^{24} - 12 q^{26} + 4 q^{29} + 8 q^{31} - 4 q^{34} - 2 q^{36} + 12 q^{39} - 12 q^{41} + 16 q^{46} - 2 q^{49} + 4 q^{51}+ \cdots + 10 q^{96}+O(q^{100}) $ 2 * q + 2 * q^4 + 2 * q^6 - 2 * q^9 - 2 * q^14 - 2 * q^16 + 16 * q^19 + 2 * q^21 + 6 * q^24 - 12 * q^26 + 4 * q^29 + 8 * q^31 - 4 * q^34 - 2 * q^36 + 12 * q^39 - 12 * q^41 + 16 * q^46 - 2 * q^49 + 4 * q^51 - 2 * q^54 - 6 * q^56 - 8 * q^59 - 4 * q^61 - 14 * q^64 - 16 * q^69 - 24 * q^71 + 4 * q^74 + 16 * q^76 - 16 * q^79 + 2 * q^81 + 2 * q^84 + 8 * q^86 + 12 * q^89 - 12 * q^91 - 16 * q^94 + 10 * q^96

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/525\mathbb{Z}\right)^\times$.

$n$	$127$	$176$	$451$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	− 1.00000i	− 0.707107i	−0.935414	−	0.353553i	$-0.884973\pi$
$2$	− 1.00000i	− 0.707107i	0.935414	−	0.353553i	$-0.115027\pi$
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	− 3.00000i	− 1.06066i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	1.00000i	0.288675i
$13$	− 6.00000i	− 1.66410i	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	− 6.00000i	− 1.66410i	0.554700	−	0.832050i	$-0.312833\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	−1.00000	−0.250000
$17$	− 2.00000i	− 0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	− 2.00000i	− 0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$	1.00000i	0.235702i
$19$	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
$21$	1.00000	0.218218
$22$	0	0
$23$	8.00000i	1.66812i	0.551677	+	0.834058i	$0.313988\pi$
$23$	8.00000i	1.66812i	−0.551677	+	0.834058i	$0.686012\pi$
$24$	3.00000	0.612372
$25$	0	0
$26$	−6.00000	−1.17670
$27$	− 1.00000i	− 0.192450i
$28$	− 1.00000i	− 0.188982i
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	− 5.00000i	− 0.883883i
$33$	0	0
$34$	−2.00000	−0.342997
$35$	0	0
$36$	−1.00000	−0.166667
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	− 8.00000i	− 1.29777i
$39$	6.00000	0.960769
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	− 1.00000i	− 0.154303i
$43$	4.00000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	0	0
$45$	0	0
$46$	8.00000	1.17954
$47$	− 8.00000i	− 1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$	− 8.00000i	− 1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$	− 1.00000i	− 0.144338i
$49$	−1.00000	−0.142857
$50$	0	0
$51$	2.00000	0.280056
$52$	− 6.00000i	− 0.832050i
$53$	10.0000i	1.37361i	0.726844	+	0.686803i	$0.240986\pi$
$53$	10.0000i	1.37361i	−0.726844	+	0.686803i	$0.759014\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	−3.00000	−0.400892
$57$	8.00000i	1.05963i
$58$	− 2.00000i	− 0.262613i
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	− 4.00000i	− 0.508001i
$63$	1.00000i	0.125988i
$64$	−7.00000	−0.875000
$65$	0	0
$66$	0	0
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	− 2.00000i	− 0.242536i
$69$	−8.00000	−0.963087
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	3.00000i	0.353553i
$73$	− 2.00000i	− 0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$	− 2.00000i	− 0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	8.00000	0.917663
$77$	0	0
$78$	− 6.00000i	− 0.679366i
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	6.00000i	0.662589i
$83$	− 4.00000i	− 0.439057i	−0.975606	−	0.219529i	$-0.929548\pi$
$83$	− 4.00000i	− 0.439057i	0.975606	−	0.219529i	$-0.0704519\pi$
$84$	1.00000	0.109109
$85$	0	0
$86$	4.00000	0.431331
$87$	2.00000i	0.214423i
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−6.00000	−0.628971
$92$	8.00000i	0.834058i
$93$	4.00000i	0.414781i
$94$	−8.00000	−0.825137
$95$	0	0
$96$	5.00000	0.510310
$97$	18.0000i	1.82762i	0.406138	+	0.913812i	$0.366875\pi$
$97$	18.0000i	1.82762i	−0.406138	+	0.913812i	$0.633125\pi$
$98$	1.00000i	0.101015i
$99$	0	0
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	− 2.00000i	− 0.198030i
$103$	8.00000i	0.788263i	0.919054	+	0.394132i	$0.128955\pi$
$103$	8.00000i	0.788263i	−0.919054	+	0.394132i	$0.871045\pi$
$104$	−18.0000	−1.76505
$105$	0	0
$106$	10.0000	0.971286
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	18.0000	1.72409	0.862044	−	0.506834i	$-0.169184\pi$
$109$	18.0000	1.72409	0.862044	+	0.506834i	$0.169184\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	1.00000i	0.0944911i
$113$	6.00000i	0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$	6.00000i	0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$	8.00000	0.749269
$115$	0	0
$116$	2.00000	0.185695
$117$	6.00000i	0.554700i
$118$	4.00000i	0.368230i
$119$	−2.00000	−0.183340
$120$	0	0
$121$	−11.0000	−1.00000
$122$	2.00000i	0.181071i
$123$	− 6.00000i	− 0.541002i
$124$	4.00000	0.359211
$125$	0	0
$126$	1.00000	0.0890871
$127$	− 8.00000i	− 0.709885i	−0.934888	−	0.354943i	$-0.884500\pi$
$127$	− 8.00000i	− 0.709885i	0.934888	−	0.354943i	$-0.115500\pi$
$128$	− 3.00000i	− 0.265165i
$129$	−4.00000	−0.352180
$130$	0	0
$131$	20.0000	1.74741	0.873704	−	0.486458i	$-0.161711\pi$
$131$	20.0000	1.74741	0.873704	+	0.486458i	$0.161711\pi$
$132$	0	0
$133$	− 8.00000i	− 0.693688i
$134$	−4.00000	−0.345547
$135$	0	0
$136$	−6.00000	−0.514496
$137$	10.0000i	0.854358i	0.904167	+	0.427179i	$0.140493\pi$
$137$	10.0000i	0.854358i	−0.904167	+	0.427179i	$0.859507\pi$
$138$	8.00000i	0.681005i
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	12.0000i	1.00702i
$143$	0	0
$144$	1.00000	0.0833333
$145$	0	0
$146$	−2.00000	−0.165521
$147$	− 1.00000i	− 0.0824786i
$148$	2.00000i	0.164399i
$149$	−14.0000	−1.14692	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000	−1.14692	−0.573462	+	0.819232i	$0.694400\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	− 24.0000i	− 1.94666i
$153$	2.00000i	0.161690i
$154$	0	0
$155$	0	0
$156$	6.00000	0.480384
$157$	14.0000i	1.11732i	0.829396	+	0.558661i	$0.188685\pi$
$157$	14.0000i	1.11732i	−0.829396	+	0.558661i	$0.811315\pi$
$158$	8.00000i	0.636446i
$159$	−10.0000	−0.793052
$160$	0	0
$161$	8.00000	0.630488
$162$	− 1.00000i	− 0.0785674i
$163$	12.0000i	0.939913i	0.882690	+	0.469956i	$0.155730\pi$
$163$	12.0000i	0.939913i	−0.882690	+	0.469956i	$0.844270\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	−4.00000	−0.310460
$167$	− 8.00000i	− 0.619059i	−0.950890	−	0.309529i	$-0.899829\pi$
$167$	− 8.00000i	− 0.619059i	0.950890	−	0.309529i	$-0.100171\pi$
$168$	− 3.00000i	− 0.231455i
$169$	−23.0000	−1.76923
$170$	0	0
$171$	−8.00000	−0.611775
$172$	4.00000i	0.304997i
$173$	6.00000i	0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$	6.00000i	0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	2.00000	0.151620
$175$	0	0
$176$	0	0
$177$	− 4.00000i	− 0.300658i
$178$	− 6.00000i	− 0.449719i
$179$	24.0000	1.79384	0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000	1.79384	0.896922	+	0.442189i	$0.145798\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	6.00000i	0.444750i
$183$	− 2.00000i	− 0.147844i
$184$	24.0000	1.76930
$185$	0	0
$186$	4.00000	0.293294
$187$	0	0
$188$	− 8.00000i	− 0.583460i
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	4.00000	0.289430	0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000	0.289430	0.144715	+	0.989473i	$0.453773\pi$
$192$	− 7.00000i	− 0.505181i
$193$	18.0000i	1.29567i	0.761781	+	0.647834i	$0.224325\pi$
$193$	18.0000i	1.29567i	−0.761781	+	0.647834i	$0.775675\pi$
$194$	18.0000	1.29232
$195$	0	0
$196$	−1.00000	−0.0714286
$197$	− 18.0000i	− 1.28245i	−0.767354	−	0.641223i	$-0.778427\pi$
$197$	− 18.0000i	− 1.28245i	0.767354	−	0.641223i	$-0.221573\pi$
$198$	0	0
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	10.0000i	0.703598i
$203$	− 2.00000i	− 0.140372i
$204$	2.00000	0.140028
$205$	0	0
$206$	8.00000	0.557386
$207$	− 8.00000i	− 0.556038i
$208$	6.00000i	0.416025i
$209$	0	0
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	10.0000i	0.686803i
$213$	− 12.0000i	− 0.822226i
$214$	12.0000	0.820303
$215$	0	0
$216$	−3.00000	−0.204124
$217$	− 4.00000i	− 0.271538i
$218$	− 18.0000i	− 1.21911i
$219$	2.00000	0.135147
$220$	0	0
$221$	−12.0000	−0.807207
$222$	2.00000i	0.134231i
$223$	− 24.0000i	− 1.60716i	−0.595198	−	0.803579i	$-0.702926\pi$
$223$	− 24.0000i	− 1.60716i	0.595198	−	0.803579i	$-0.297074\pi$
$224$	−5.00000	−0.334077
$225$	0	0
$226$	6.00000	0.399114
$227$	4.00000i	0.265489i	0.991150	+	0.132745i	$0.0423790\pi$
$227$	4.00000i	0.265489i	−0.991150	+	0.132745i	$0.957621\pi$
$228$	8.00000i	0.529813i
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	0	0
$231$	0	0
$232$	− 6.00000i	− 0.393919i
$233$	− 18.0000i	− 1.17922i	−0.807688	−	0.589610i	$-0.799282\pi$
$233$	− 18.0000i	− 1.17922i	0.807688	−	0.589610i	$-0.200718\pi$
$234$	6.00000	0.392232
$235$	0	0
$236$	−4.00000	−0.260378
$237$	− 8.00000i	− 0.519656i
$238$	2.00000i	0.129641i
$239$	4.00000	0.258738	0.129369	−	0.991596i	$-0.458705\pi$
$239$	4.00000	0.258738	0.129369	+	0.991596i	$0.458705\pi$
$240$	0	0
$241$	−6.00000	−0.386494	−0.193247	−	0.981150i	$-0.561902\pi$
$241$	−6.00000	−0.386494	−0.193247	+	0.981150i	$0.561902\pi$
$242$	11.0000i	0.707107i
$243$	1.00000i	0.0641500i
$244$	−2.00000	−0.128037
$245$	0	0
$246$	−6.00000	−0.382546
$247$	− 48.0000i	− 3.05417i
$248$	− 12.0000i	− 0.762001i
$249$	4.00000	0.253490
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	1.00000i	0.0629941i
$253$	0	0
$254$	−8.00000	−0.501965
$255$	0	0
$256$	−17.0000	−1.06250
$257$	6.00000i	0.374270i	0.982334	+	0.187135i	$0.0599201\pi$
$257$	6.00000i	0.374270i	−0.982334	+	0.187135i	$0.940080\pi$
$258$	4.00000i	0.249029i
$259$	2.00000	0.124274
$260$	0	0
$261$	−2.00000	−0.123797
$262$	− 20.0000i	− 1.23560i
$263$	16.0000i	0.986602i	0.869859	+	0.493301i	$0.164210\pi$
$263$	16.0000i	0.986602i	−0.869859	+	0.493301i	$0.835790\pi$
$264$	0	0
$265$	0	0
$266$	−8.00000	−0.490511
$267$	6.00000i	0.367194i
$268$	− 4.00000i	− 0.244339i
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	0	0
$271$	12.0000	0.728948	0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000	0.728948	0.364474	+	0.931214i	$0.381249\pi$
$272$	2.00000i	0.121268i
$273$	− 6.00000i	− 0.363137i
$274$	10.0000	0.604122
$275$	0	0
$276$	−8.00000	−0.481543
$277$	− 14.0000i	− 0.841178i	−0.907251	−	0.420589i	$-0.861823\pi$
$277$	− 14.0000i	− 0.841178i	0.907251	−	0.420589i	$-0.138177\pi$
$278$	0	0
$279$	−4.00000	−0.239474
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	− 8.00000i	− 0.476393i
$283$	− 4.00000i	− 0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$	− 4.00000i	− 0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$	−12.0000	−0.712069
$285$	0	0
$286$	0	0
$287$	6.00000i	0.354169i
$288$	5.00000i	0.294628i
$289$	13.0000	0.764706
$290$	0	0
$291$	−18.0000	−1.05518
$292$	− 2.00000i	− 0.117041i
$293$	14.0000i	0.817889i	0.912559	+	0.408944i	$0.134103\pi$
$293$	14.0000i	0.817889i	−0.912559	+	0.408944i	$0.865897\pi$
$294$	−1.00000	−0.0583212
$295$	0	0
$296$	6.00000	0.348743
$297$	0	0
$298$	14.0000i	0.810998i
$299$	48.0000	2.77591
$300$	0	0
$301$	4.00000	0.230556
$302$	− 8.00000i	− 0.460348i
$303$	− 10.0000i	− 0.574485i
$304$	−8.00000	−0.458831
$305$	0	0
$306$	2.00000	0.114332
$307$	− 12.0000i	− 0.684876i	−0.939540	−	0.342438i	$-0.888747\pi$
$307$	− 12.0000i	− 0.684876i	0.939540	−	0.342438i	$-0.111253\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	− 18.0000i	− 1.01905i
$313$	− 10.0000i	− 0.565233i	−0.959233	−	0.282617i	$-0.908798\pi$
$313$	− 10.0000i	− 0.565233i	0.959233	−	0.282617i	$-0.0912024\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	−8.00000	−0.450035
$317$	− 2.00000i	− 0.112331i	−0.998421	−	0.0561656i	$-0.982113\pi$
$317$	− 2.00000i	− 0.112331i	0.998421	−	0.0561656i	$-0.0178875\pi$
$318$	10.0000i	0.560772i
$319$	0	0
$320$	0	0
$321$	−12.0000	−0.669775
$322$	− 8.00000i	− 0.445823i
$323$	− 16.0000i	− 0.890264i
$324$	1.00000	0.0555556
$325$	0	0
$326$	12.0000	0.664619
$327$	18.0000i	0.995402i
$328$	18.0000i	0.993884i
$329$	−8.00000	−0.441054
$330$	0	0
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	− 4.00000i	− 0.219529i
$333$	− 2.00000i	− 0.109599i
$334$	−8.00000	−0.437741
$335$	0	0
$336$	−1.00000	−0.0545545
$337$	14.0000i	0.762629i	0.924445	+	0.381314i	$0.124528\pi$
$337$	14.0000i	0.762629i	−0.924445	+	0.381314i	$0.875472\pi$
$338$	23.0000i	1.25104i
$339$	−6.00000	−0.325875
$340$	0	0
$341$	0	0
$342$	8.00000i	0.432590i
$343$	1.00000i	0.0539949i
$344$	12.0000	0.646997
$345$	0	0
$346$	6.00000	0.322562
$347$	20.0000i	1.07366i	0.843692	+	0.536828i	$0.180378\pi$
$347$	20.0000i	1.07366i	−0.843692	+	0.536828i	$0.819622\pi$
$348$	2.00000i	0.107211i
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	0	0
$351$	−6.00000	−0.320256
$352$	0	0
$353$	18.0000i	0.958043i	0.877803	+	0.479022i	$0.159008\pi$
$353$	18.0000i	0.958043i	−0.877803	+	0.479022i	$0.840992\pi$
$354$	−4.00000	−0.212598
$355$	0	0
$356$	6.00000	0.317999
$357$	− 2.00000i	− 0.105851i
$358$	− 24.0000i	− 1.26844i
$359$	36.0000	1.90001	0.950004	−	0.312239i	$-0.101079\pi$
$359$	36.0000	1.90001	0.950004	+	0.312239i	$0.101079\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	2.00000i	0.105118i
$363$	− 11.0000i	− 0.577350i
$364$	−6.00000	−0.314485
$365$	0	0
$366$	−2.00000	−0.104542
$367$	8.00000i	0.417597i	0.977959	+	0.208798i	$0.0669552\pi$
$367$	8.00000i	0.417597i	−0.977959	+	0.208798i	$0.933045\pi$
$368$	− 8.00000i	− 0.417029i
$369$	6.00000	0.312348
$370$	0	0
$371$	10.0000	0.519174
$372$	4.00000i	0.207390i
$373$	− 10.0000i	− 0.517780i	−0.965907	−	0.258890i	$-0.916643\pi$
$373$	− 10.0000i	− 0.517780i	0.965907	−	0.258890i	$-0.0833568\pi$
$374$	0	0
$375$	0	0
$376$	−24.0000	−1.23771
$377$	− 12.0000i	− 0.618031i
$378$	1.00000i	0.0514344i
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	− 4.00000i	− 0.204658i
$383$	− 32.0000i	− 1.63512i	−0.575841	−	0.817562i	$-0.695325\pi$
$383$	− 32.0000i	− 1.63512i	0.575841	−	0.817562i	$-0.304675\pi$
$384$	3.00000	0.153093
$385$	0	0
$386$	18.0000	0.916176
$387$	− 4.00000i	− 0.203331i
$388$	18.0000i	0.913812i
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	16.0000	0.809155
$392$	3.00000i	0.151523i
$393$	20.0000i	1.00887i
$394$	−18.0000	−0.906827
$395$	0	0
$396$	0	0
$397$	22.0000i	1.10415i	0.833795	+	0.552074i	$0.186163\pi$
$397$	22.0000i	1.10415i	−0.833795	+	0.552074i	$0.813837\pi$
$398$	− 4.00000i	− 0.200502i
$399$	8.00000	0.400501
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	− 4.00000i	− 0.199502i
$403$	− 24.0000i	− 1.19553i
$404$	−10.0000	−0.497519
$405$	0	0
$406$	−2.00000	−0.0992583
$407$	0	0
$408$	− 6.00000i	− 0.297044i
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	0	0
$411$	−10.0000	−0.493264
$412$	8.00000i	0.394132i
$413$	4.00000i	0.196827i
$414$	−8.00000	−0.393179
$415$	0	0
$416$	−30.0000	−1.47087
$417$	0	0
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	20.0000i	0.973585i
$423$	8.00000i	0.388973i
$424$	30.0000	1.45693
$425$	0	0
$426$	−12.0000	−0.581402
$427$	2.00000i	0.0967868i
$428$	12.0000i	0.580042i
$429$	0	0
$430$	0	0
$431$	28.0000	1.34871	0.674356	−	0.738406i	$-0.264421\pi$
$431$	28.0000	1.34871	0.674356	+	0.738406i	$0.264421\pi$
$432$	1.00000i	0.0481125i
$433$	− 2.00000i	− 0.0961139i	−0.998845	−	0.0480569i	$-0.984697\pi$
$433$	− 2.00000i	− 0.0961139i	0.998845	−	0.0480569i	$-0.0153029\pi$
$434$	−4.00000	−0.192006
$435$	0	0
$436$	18.0000	0.862044
$437$	64.0000i	3.06154i
$438$	− 2.00000i	− 0.0955637i
$439$	−28.0000	−1.33637	−0.668184	−	0.743996i	$-0.732928\pi$
$439$	−28.0000	−1.33637	−0.668184	+	0.743996i	$0.732928\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	12.0000i	0.570782i
$443$	12.0000i	0.570137i	0.958507	+	0.285069i	$0.0920164\pi$
$443$	12.0000i	0.570137i	−0.958507	+	0.285069i	$0.907984\pi$
$444$	−2.00000	−0.0949158
$445$	0	0
$446$	−24.0000	−1.13643
$447$	− 14.0000i	− 0.662177i
$448$	7.00000i	0.330719i
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	0	0
$451$	0	0
$452$	6.00000i	0.282216i
$453$	8.00000i	0.375873i
$454$	4.00000	0.187729
$455$	0	0
$456$	24.0000	1.12390
$457$	− 18.0000i	− 0.842004i	−0.907060	−	0.421002i	$-0.861678\pi$
$457$	− 18.0000i	− 0.842004i	0.907060	−	0.421002i	$-0.138322\pi$
$458$	22.0000i	1.02799i
$459$	−2.00000	−0.0933520
$460$	0	0
$461$	−2.00000	−0.0931493	−0.0465746	−	0.998915i	$-0.514831\pi$
$461$	−2.00000	−0.0931493	−0.0465746	+	0.998915i	$0.514831\pi$
$462$	0	0
$463$	− 24.0000i	− 1.11537i	−0.830051	−	0.557687i	$-0.811689\pi$
$463$	− 24.0000i	− 1.11537i	0.830051	−	0.557687i	$-0.188311\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	−18.0000	−0.833834
$467$	− 28.0000i	− 1.29569i	−0.761774	−	0.647843i	$-0.775671\pi$
$467$	− 28.0000i	− 1.29569i	0.761774	−	0.647843i	$-0.224329\pi$
$468$	6.00000i	0.277350i
$469$	−4.00000	−0.184703
$470$	0	0
$471$	−14.0000	−0.645086
$472$	12.0000i	0.552345i
$473$	0	0
$474$	−8.00000	−0.367452
$475$	0	0
$476$	−2.00000	−0.0916698
$477$	− 10.0000i	− 0.457869i
$478$	− 4.00000i	− 0.182956i
$479$	−32.0000	−1.46212	−0.731059	−	0.682315i	$-0.760973\pi$
$479$	−32.0000	−1.46212	−0.731059	+	0.682315i	$0.760973\pi$
$480$	0	0
$481$	12.0000	0.547153
$482$	6.00000i	0.273293i
$483$	8.00000i	0.364013i
$484$	−11.0000	−0.500000
$485$	0	0
$486$	1.00000	0.0453609
$487$	16.0000i	0.725029i	0.931978	+	0.362515i	$0.118082\pi$
$487$	16.0000i	0.725029i	−0.931978	+	0.362515i	$0.881918\pi$
$488$	6.00000i	0.271607i
$489$	−12.0000	−0.542659
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	− 6.00000i	− 0.270501i
$493$	− 4.00000i	− 0.180151i
$494$	−48.0000	−2.15962
$495$	0	0
$496$	−4.00000	−0.179605
$497$	12.0000i	0.538274i
$498$	− 4.00000i	− 0.179244i
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	0	0
$501$	8.00000	0.357414
$502$	12.0000i	0.535586i
$503$	− 8.00000i	− 0.356702i	−0.983967	−	0.178351i	$-0.942924\pi$
$503$	− 8.00000i	− 0.356702i	0.983967	−	0.178351i	$-0.0570763\pi$
$504$	3.00000	0.133631
$505$	0	0
$506$	0	0
$507$	− 23.0000i	− 1.02147i
$508$	− 8.00000i	− 0.354943i
$509$	2.00000	0.0886484	0.0443242	−	0.999017i	$-0.485887\pi$
$509$	2.00000	0.0886484	0.0443242	+	0.999017i	$0.485887\pi$
$510$	0	0
$511$	−2.00000	−0.0884748
$512$	11.0000i	0.486136i
$513$	− 8.00000i	− 0.353209i
$514$	6.00000	0.264649
$515$	0	0
$516$	−4.00000	−0.176090
$517$	0	0
$518$	− 2.00000i	− 0.0878750i
$519$	−6.00000	−0.263371
$520$	0	0
$521$	10.0000	0.438108	0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000	0.438108	0.219054	+	0.975713i	$0.429703\pi$
$522$	2.00000i	0.0875376i
$523$	20.0000i	0.874539i	0.899331	+	0.437269i	$0.144054\pi$
$523$	20.0000i	0.874539i	−0.899331	+	0.437269i	$0.855946\pi$
$524$	20.0000	0.873704
$525$	0	0
$526$	16.0000	0.697633
$527$	− 8.00000i	− 0.348485i
$528$	0	0
$529$	−41.0000	−1.78261
$530$	0	0
$531$	4.00000	0.173585
$532$	− 8.00000i	− 0.346844i
$533$	36.0000i	1.55933i
$534$	6.00000	0.259645
$535$	0	0
$536$	−12.0000	−0.518321
$537$	24.0000i	1.03568i
$538$	14.0000i	0.603583i
$539$	0	0
$540$	0	0
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	− 12.0000i	− 0.515444i
$543$	− 2.00000i	− 0.0858282i
$544$	−10.0000	−0.428746
$545$	0	0
$546$	−6.00000	−0.256776
$547$	12.0000i	0.513083i	0.966533	+	0.256541i	$0.0825830\pi$
$547$	12.0000i	0.513083i	−0.966533	+	0.256541i	$0.917417\pi$
$548$	10.0000i	0.427179i
$549$	2.00000	0.0853579
$550$	0	0
$551$	16.0000	0.681623
$552$	24.0000i	1.02151i
$553$	8.00000i	0.340195i
$554$	−14.0000	−0.594803
$555$	0	0
$556$	0	0
$557$	− 2.00000i	− 0.0847427i	−0.999102	−	0.0423714i	$-0.986509\pi$
$557$	− 2.00000i	− 0.0847427i	0.999102	−	0.0423714i	$-0.0134913\pi$
$558$	4.00000i	0.169334i
$559$	24.0000	1.01509
$560$	0	0
$561$	0	0
$562$	− 18.0000i	− 0.759284i
$563$	4.00000i	0.168580i	0.996441	+	0.0842900i	$0.0268622\pi$
$563$	4.00000i	0.168580i	−0.996441	+	0.0842900i	$0.973138\pi$
$564$	8.00000	0.336861
$565$	0	0
$566$	−4.00000	−0.168133
$567$	− 1.00000i	− 0.0419961i
$568$	36.0000i	1.51053i
$569$	−42.0000	−1.76073	−0.880366	−	0.474295i	$-0.842703\pi$
$569$	−42.0000	−1.76073	−0.880366	+	0.474295i	$0.842703\pi$
$570$	0	0
$571$	−36.0000	−1.50655	−0.753277	−	0.657704i	$-0.771528\pi$
$571$	−36.0000	−1.50655	−0.753277	+	0.657704i	$0.771528\pi$
$572$	0	0
$573$	4.00000i	0.167102i
$574$	6.00000	0.250435
$575$	0	0
$576$	7.00000	0.291667
$577$	2.00000i	0.0832611i	0.999133	+	0.0416305i	$0.0132552\pi$
$577$	2.00000i	0.0832611i	−0.999133	+	0.0416305i	$0.986745\pi$
$578$	− 13.0000i	− 0.540729i
$579$	−18.0000	−0.748054
$580$	0	0
$581$	−4.00000	−0.165948
$582$	18.0000i	0.746124i
$583$	0	0
$584$	−6.00000	−0.248282
$585$	0	0
$586$	14.0000	0.578335
$587$	− 12.0000i	− 0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$	− 12.0000i	− 0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$	− 1.00000i	− 0.0412393i
$589$	32.0000	1.31854
$590$	0	0
$591$	18.0000	0.740421
$592$	− 2.00000i	− 0.0821995i
$593$	− 30.0000i	− 1.23195i	−0.787765	−	0.615976i	$-0.788762\pi$
$593$	− 30.0000i	− 1.23195i	0.787765	−	0.615976i	$-0.211238\pi$
$594$	0	0
$595$	0	0
$596$	−14.0000	−0.573462
$597$	4.00000i	0.163709i
$598$	− 48.0000i	− 1.96287i
$599$	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
$601$	18.0000	0.734235	0.367118	−	0.930175i	$-0.380345\pi$
$601$	18.0000	0.734235	0.367118	+	0.930175i	$0.380345\pi$
$602$	− 4.00000i	− 0.163028i
$603$	4.00000i	0.162893i
$604$	8.00000	0.325515
$605$	0	0
$606$	−10.0000	−0.406222
$607$	− 8.00000i	− 0.324710i	−0.986732	−	0.162355i	$-0.948091\pi$
$607$	− 8.00000i	− 0.324710i	0.986732	−	0.162355i	$-0.0519090\pi$
$608$	− 40.0000i	− 1.62221i
$609$	2.00000	0.0810441
$610$	0	0
$611$	−48.0000	−1.94187
$612$	2.00000i	0.0808452i
$613$	− 18.0000i	− 0.727013i	−0.931592	−	0.363507i	$-0.881579\pi$
$613$	− 18.0000i	− 0.727013i	0.931592	−	0.363507i	$-0.118421\pi$
$614$	−12.0000	−0.484281
$615$	0	0
$616$	0	0
$617$	− 30.0000i	− 1.20775i	−0.797077	−	0.603877i	$-0.793622\pi$
$617$	− 30.0000i	− 1.20775i	0.797077	−	0.603877i	$-0.206378\pi$
$618$	8.00000i	0.321807i
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	8.00000	0.321029
$622$	− 24.0000i	− 0.962312i
$623$	− 6.00000i	− 0.240385i
$624$	−6.00000	−0.240192
$625$	0	0
$626$	−10.0000	−0.399680
$627$	0	0
$628$	14.0000i	0.558661i
$629$	4.00000	0.159490
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	24.0000i	0.954669i
$633$	− 20.0000i	− 0.794929i
$634$	−2.00000	−0.0794301
$635$	0	0
$636$	−10.0000	−0.396526
$637$	6.00000i	0.237729i
$638$	0	0
$639$	12.0000	0.474713
$640$	0	0
$641$	−6.00000	−0.236986	−0.118493	−	0.992955i	$-0.537806\pi$
$641$	−6.00000	−0.236986	−0.118493	+	0.992955i	$0.537806\pi$
$642$	12.0000i	0.473602i
$643$	28.0000i	1.10421i	0.833774	+	0.552106i	$0.186176\pi$
$643$	28.0000i	1.10421i	−0.833774	+	0.552106i	$0.813824\pi$
$644$	8.00000	0.315244
$645$	0	0
$646$	−16.0000	−0.629512
$647$	− 24.0000i	− 0.943537i	−0.881722	−	0.471769i	$-0.843616\pi$
$647$	− 24.0000i	− 0.943537i	0.881722	−	0.471769i	$-0.156384\pi$
$648$	− 3.00000i	− 0.117851i
$649$	0	0
$650$	0	0
$651$	4.00000	0.156772
$652$	12.0000i	0.469956i
$653$	− 22.0000i	− 0.860927i	−0.902608	−	0.430463i	$-0.858350\pi$
$653$	− 22.0000i	− 0.860927i	0.902608	−	0.430463i	$-0.141650\pi$
$654$	18.0000	0.703856
$655$	0	0
$656$	6.00000	0.234261
$657$	2.00000i	0.0780274i
$658$	8.00000i	0.311872i
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	38.0000	1.47803	0.739014	−	0.673690i	$-0.235292\pi$
$661$	38.0000	1.47803	0.739014	+	0.673690i	$0.235292\pi$
$662$	12.0000i	0.466393i
$663$	− 12.0000i	− 0.466041i
$664$	−12.0000	−0.465690
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	16.0000i	0.619522i
$668$	− 8.00000i	− 0.309529i
$669$	24.0000	0.927894
$670$	0	0
$671$	0	0
$672$	− 5.00000i	− 0.192879i
$673$	26.0000i	1.00223i	0.865382	+	0.501113i	$0.167076\pi$
$673$	26.0000i	1.00223i	−0.865382	+	0.501113i	$0.832924\pi$
$674$	14.0000	0.539260
$675$	0	0
$676$	−23.0000	−0.884615
$677$	− 46.0000i	− 1.76792i	−0.467559	−	0.883962i	$-0.654866\pi$
$677$	− 46.0000i	− 1.76792i	0.467559	−	0.883962i	$-0.345134\pi$
$678$	6.00000i	0.230429i
$679$	18.0000	0.690777
$680$	0	0
$681$	−4.00000	−0.153280
$682$	0	0
$683$	12.0000i	0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$	12.0000i	0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$	−8.00000	−0.305888
$685$	0	0
$686$	1.00000	0.0381802
$687$	− 22.0000i	− 0.839352i
$688$	− 4.00000i	− 0.152499i
$689$	60.0000	2.28582
$690$	0	0
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	6.00000i	0.228086i
$693$	0	0
$694$	20.0000	0.759190
$695$	0	0
$696$	6.00000	0.227429
$697$	12.0000i	0.454532i
$698$	14.0000i	0.529908i
$699$	18.0000	0.680823
$700$	0	0
$701$	−2.00000	−0.0755390	−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	−2.00000	−0.0755390	−0.0377695	+	0.999286i	$0.512025\pi$
$702$	6.00000i	0.226455i
$703$	16.0000i	0.603451i
$704$	0	0
$705$	0	0
$706$	18.0000	0.677439
$707$	10.0000i	0.376089i
$708$	− 4.00000i	− 0.150329i
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	− 18.0000i	− 0.674579i
$713$	32.0000i	1.19841i
$714$	−2.00000	−0.0748481
$715$	0	0
$716$	24.0000	0.896922
$717$	4.00000i	0.149383i
$718$	− 36.0000i	− 1.34351i
$719$	−8.00000	−0.298350	−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000	−0.298350	−0.149175	+	0.988811i	$0.547662\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	− 45.0000i	− 1.67473i
$723$	− 6.00000i	− 0.223142i
$724$	−2.00000	−0.0743294
$725$	0	0
$726$	−11.0000	−0.408248
$727$	− 16.0000i	− 0.593407i	−0.954970	−	0.296704i	$-0.904113\pi$
$727$	− 16.0000i	− 0.593407i	0.954970	−	0.296704i	$-0.0958873\pi$
$728$	18.0000i	0.667124i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	8.00000	0.295891
$732$	− 2.00000i	− 0.0739221i
$733$	26.0000i	0.960332i	0.877178	+	0.480166i	$0.159424\pi$
$733$	26.0000i	0.960332i	−0.877178	+	0.480166i	$0.840576\pi$
$734$	8.00000	0.295285
$735$	0	0
$736$	40.0000	1.47442
$737$	0	0
$738$	− 6.00000i	− 0.220863i
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	48.0000	1.76332
$742$	− 10.0000i	− 0.367112i
$743$	− 48.0000i	− 1.76095i	−0.474093	−	0.880475i	$-0.657224\pi$
$743$	− 48.0000i	− 1.76095i	0.474093	−	0.880475i	$-0.342776\pi$
$744$	12.0000	0.439941
$745$	0	0
$746$	−10.0000	−0.366126
$747$	4.00000i	0.146352i
$748$	0	0
$749$	12.0000	0.438470
$750$	0	0
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	8.00000i	0.291730i
$753$	− 12.0000i	− 0.437304i
$754$	−12.0000	−0.437014
$755$	0	0
$756$	−1.00000	−0.0363696
$757$	10.0000i	0.363456i	0.983349	+	0.181728i	$0.0581691\pi$
$757$	10.0000i	0.363456i	−0.983349	+	0.181728i	$0.941831\pi$
$758$	− 4.00000i	− 0.145287i
$759$	0	0
$760$	0	0
$761$	−54.0000	−1.95750	−0.978749	−	0.205061i	$-0.934261\pi$
$761$	−54.0000	−1.95750	−0.978749	+	0.205061i	$0.934261\pi$
$762$	− 8.00000i	− 0.289809i
$763$	− 18.0000i	− 0.651644i
$764$	4.00000	0.144715
$765$	0	0
$766$	−32.0000	−1.15621
$767$	24.0000i	0.866590i
$768$	− 17.0000i	− 0.613435i
$769$	−26.0000	−0.937584	−0.468792	−	0.883309i	$-0.655311\pi$
$769$	−26.0000	−0.937584	−0.468792	+	0.883309i	$0.655311\pi$
$770$	0	0
$771$	−6.00000	−0.216085
$772$	18.0000i	0.647834i
$773$	30.0000i	1.07903i	0.841978	+	0.539513i	$0.181391\pi$
$773$	30.0000i	1.07903i	−0.841978	+	0.539513i	$0.818609\pi$
$774$	−4.00000	−0.143777
$775$	0	0
$776$	54.0000	1.93849
$777$	2.00000i	0.0717496i
$778$	30.0000i	1.07555i
$779$	−48.0000	−1.71978
$780$	0	0
$781$	0	0
$782$	− 16.0000i	− 0.572159i
$783$	− 2.00000i	− 0.0714742i
$784$	1.00000	0.0357143
$785$	0	0
$786$	20.0000	0.713376
$787$	− 28.0000i	− 0.998092i	−0.866575	−	0.499046i	$-0.833684\pi$
$787$	− 28.0000i	− 0.998092i	0.866575	−	0.499046i	$-0.166316\pi$
$788$	− 18.0000i	− 0.641223i
$789$	−16.0000	−0.569615
$790$	0	0
$791$	6.00000	0.213335
$792$	0	0
$793$	12.0000i	0.426132i
$794$	22.0000	0.780751
$795$	0	0
$796$	4.00000	0.141776
$797$	− 54.0000i	− 1.91278i	−0.292096	−	0.956389i	$-0.594353\pi$
$797$	− 54.0000i	− 1.91278i	0.292096	−	0.956389i	$-0.405647\pi$
$798$	− 8.00000i	− 0.283197i
$799$	−16.0000	−0.566039
$800$	0	0
$801$	−6.00000	−0.212000
$802$	− 18.0000i	− 0.635602i
$803$	0	0
$804$	4.00000	0.141069
$805$	0	0
$806$	−24.0000	−0.845364
$807$	− 14.0000i	− 0.492823i
$808$	30.0000i	1.05540i
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	− 2.00000i	− 0.0701862i
$813$	12.0000i	0.420858i
$814$	0	0
$815$	0	0
$816$	−2.00000	−0.0700140
$817$	32.0000i	1.11954i
$818$	− 22.0000i	− 0.769212i
$819$	6.00000	0.209657
$820$	0	0
$821$	−42.0000	−1.46581	−0.732905	−	0.680331i	$-0.761836\pi$
$821$	−42.0000	−1.46581	−0.732905	+	0.680331i	$0.761836\pi$
$822$	10.0000i	0.348790i
$823$	− 16.0000i	− 0.557725i	−0.960331	−	0.278862i	$-0.910043\pi$
$823$	− 16.0000i	− 0.557725i	0.960331	−	0.278862i	$-0.0899574\pi$
$824$	24.0000	0.836080
$825$	0	0
$826$	4.00000	0.139178
$827$	12.0000i	0.417281i	0.977992	+	0.208640i	$0.0669038\pi$
$827$	12.0000i	0.417281i	−0.977992	+	0.208640i	$0.933096\pi$
$828$	− 8.00000i	− 0.278019i
$829$	42.0000	1.45872	0.729360	−	0.684130i	$-0.239818\pi$
$829$	42.0000	1.45872	0.729360	+	0.684130i	$0.239818\pi$
$830$	0	0
$831$	14.0000	0.485655
$832$	42.0000i	1.45609i
$833$	2.00000i	0.0692959i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 4.00000i	− 0.138260i
$838$	− 12.0000i	− 0.414533i
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	26.0000i	0.896019i
$843$	18.0000i	0.619953i
$844$	−20.0000	−0.688428
$845$	0	0
$846$	8.00000	0.275046
$847$	11.0000i	0.377964i
$848$	− 10.0000i	− 0.343401i
$849$	4.00000	0.137280
$850$	0	0
$851$	−16.0000	−0.548473
$852$	− 12.0000i	− 0.411113i
$853$	− 30.0000i	− 1.02718i	−0.858036	−	0.513590i	$-0.828315\pi$
$853$	− 30.0000i	− 1.02718i	0.858036	−	0.513590i	$-0.171685\pi$
$854$	2.00000	0.0684386
$855$	0	0
$856$	36.0000	1.23045
$857$	− 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
$859$	40.0000	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	0	0
$861$	−6.00000	−0.204479
$862$	− 28.0000i	− 0.953684i
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	−5.00000	−0.170103
$865$	0	0
$866$	−2.00000	−0.0679628
$867$	13.0000i	0.441503i
$868$	− 4.00000i	− 0.135769i
$869$	0	0
$870$	0	0
$871$	−24.0000	−0.813209
$872$	− 54.0000i	− 1.82867i
$873$	− 18.0000i	− 0.609208i
$874$	64.0000	2.16483
$875$	0	0
$876$	2.00000	0.0675737
$877$	58.0000i	1.95852i	0.202606	+	0.979260i	$0.435059\pi$
$877$	58.0000i	1.95852i	−0.202606	+	0.979260i	$0.564941\pi$
$878$	28.0000i	0.944954i
$879$	−14.0000	−0.472208
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	− 1.00000i	− 0.0336718i
$883$	4.00000i	0.134611i	0.997732	+	0.0673054i	$0.0214402\pi$
$883$	4.00000i	0.134611i	−0.997732	+	0.0673054i	$0.978560\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	12.0000	0.403148
$887$	− 16.0000i	− 0.537227i	−0.963248	−	0.268614i	$-0.913434\pi$
$887$	− 16.0000i	− 0.537227i	0.963248	−	0.268614i	$-0.0865655\pi$
$888$	6.00000i	0.201347i
$889$	−8.00000	−0.268311
$890$	0	0
$891$	0	0
$892$	− 24.0000i	− 0.803579i
$893$	− 64.0000i	− 2.14168i
$894$	−14.0000	−0.468230
$895$	0	0
$896$	−3.00000	−0.100223
$897$	48.0000i	1.60267i
$898$	− 30.0000i	− 1.00111i
$899$	8.00000	0.266815
$900$	0	0
$901$	20.0000	0.666297
$902$	0	0
$903$	4.00000i	0.133112i
$904$	18.0000	0.598671
$905$	0	0
$906$	8.00000	0.265782
$907$	− 28.0000i	− 0.929725i	−0.885383	−	0.464862i	$-0.846104\pi$
$907$	− 28.0000i	− 0.929725i	0.885383	−	0.464862i	$-0.153896\pi$
$908$	4.00000i	0.132745i
$909$	10.0000	0.331679
$910$	0	0
$911$	36.0000	1.19273	0.596367	−	0.802712i	$-0.296610\pi$
$911$	36.0000	1.19273	0.596367	+	0.802712i	$0.296610\pi$
$912$	− 8.00000i	− 0.264906i
$913$	0	0
$914$	−18.0000	−0.595387
$915$	0	0
$916$	−22.0000	−0.726900
$917$	− 20.0000i	− 0.660458i
$918$	2.00000i	0.0660098i
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	0	0
$921$	12.0000	0.395413
$922$	2.00000i	0.0658665i
$923$	72.0000i	2.36991i
$924$	0	0
$925$	0	0
$926$	−24.0000	−0.788689
$927$	− 8.00000i	− 0.262754i
$928$	− 10.0000i	− 0.328266i
$929$	14.0000	0.459325	0.229663	−	0.973270i	$-0.426238\pi$
$929$	14.0000	0.459325	0.229663	+	0.973270i	$0.426238\pi$
$930$	0	0
$931$	−8.00000	−0.262189
$932$	− 18.0000i	− 0.589610i
$933$	24.0000i	0.785725i
$934$	−28.0000	−0.916188
$935$	0	0
$936$	18.0000	0.588348
$937$	2.00000i	0.0653372i	0.999466	+	0.0326686i	$0.0104006\pi$
$937$	2.00000i	0.0653372i	−0.999466	+	0.0326686i	$0.989599\pi$
$938$	4.00000i	0.130605i
$939$	10.0000	0.326338
$940$	0	0
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	14.0000i	0.456145i
$943$	− 48.0000i	− 1.56310i
$944$	4.00000	0.130189
$945$	0	0
$946$	0	0
$947$	− 20.0000i	− 0.649913i	−0.945729	−	0.324956i	$-0.894650\pi$
$947$	− 20.0000i	− 0.649913i	0.945729	−	0.324956i	$-0.105350\pi$
$948$	− 8.00000i	− 0.259828i
$949$	−12.0000	−0.389536
$950$	0	0
$951$	2.00000	0.0648544
$952$	6.00000i	0.194461i
$953$	− 2.00000i	− 0.0647864i	−0.999475	−	0.0323932i	$-0.989687\pi$
$953$	− 2.00000i	− 0.0647864i	0.999475	−	0.0323932i	$-0.0103129\pi$
$954$	−10.0000	−0.323762
$955$	0	0
$956$	4.00000	0.129369
$957$	0	0
$958$	32.0000i	1.03387i
$959$	10.0000	0.322917
$960$	0	0
$961$	−15.0000	−0.483871
$962$	− 12.0000i	− 0.386896i
$963$	− 12.0000i	− 0.386695i
$964$	−6.00000	−0.193247
$965$	0	0
$966$	8.00000	0.257396
$967$	8.00000i	0.257263i	0.991692	+	0.128631i	$0.0410584\pi$
$967$	8.00000i	0.257263i	−0.991692	+	0.128631i	$0.958942\pi$
$968$	33.0000i	1.06066i
$969$	16.0000	0.513994
$970$	0	0
$971$	20.0000	0.641831	0.320915	−	0.947108i	$-0.396010\pi$
$971$	20.0000	0.641831	0.320915	+	0.947108i	$0.396010\pi$
$972$	1.00000i	0.0320750i
$973$	0	0
$974$	16.0000	0.512673
$975$	0	0
$976$	2.00000	0.0640184
$977$	10.0000i	0.319928i	0.987123	+	0.159964i	$0.0511379\pi$
$977$	10.0000i	0.319928i	−0.987123	+	0.159964i	$0.948862\pi$
$978$	12.0000i	0.383718i
$979$	0	0
$980$	0	0
$981$	−18.0000	−0.574696
$982$	0	0
$983$	− 16.0000i	− 0.510321i	−0.966899	−	0.255160i	$-0.917872\pi$
$983$	− 16.0000i	− 0.510321i	0.966899	−	0.255160i	$-0.0821283\pi$
$984$	−18.0000	−0.573819
$985$	0	0
$986$	−4.00000	−0.127386
$987$	− 8.00000i	− 0.254643i
$988$	− 48.0000i	− 1.52708i
$989$	−32.0000	−1.01754
$990$	0	0
$991$	−24.0000	−0.762385	−0.381193	−	0.924496i	$-0.624487\pi$
$991$	−24.0000	−0.762385	−0.381193	+	0.924496i	$0.624487\pi$
$992$	− 20.0000i	− 0.635001i
$993$	− 12.0000i	− 0.380808i
$994$	12.0000	0.380617
$995$	0	0
$996$	4.00000	0.126745
$997$	− 2.00000i	− 0.0633406i	−0.999498	−	0.0316703i	$-0.989917\pi$
$997$	− 2.00000i	− 0.0633406i	0.999498	−	0.0316703i	$-0.0100827\pi$
$998$	20.0000i	0.633089i
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.d.b.274.1		2
3.2	odd	2		1575.2.d.b.1324.2		2
5.2	odd	4		105.2.a.a.1.1	✓	1
5.3	odd	4		525.2.a.a.1.1		1
5.4	even	2	inner	525.2.d.b.274.2		2
15.2	even	4		315.2.a.a.1.1		1
15.8	even	4		1575.2.a.h.1.1		1
15.14	odd	2		1575.2.d.b.1324.1		2
20.3	even	4		8400.2.a.co.1.1		1
20.7	even	4		1680.2.a.f.1.1		1
35.2	odd	12		735.2.i.a.361.1		2
35.12	even	12		735.2.i.b.361.1		2
35.13	even	4		3675.2.a.f.1.1		1
35.17	even	12		735.2.i.b.226.1		2
35.27	even	4		735.2.a.f.1.1		1
35.32	odd	12		735.2.i.a.226.1		2
40.27	even	4		6720.2.a.bk.1.1		1
40.37	odd	4		6720.2.a.p.1.1		1
60.47	odd	4		5040.2.a.d.1.1		1
105.62	odd	4		2205.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.a.a.1.1	✓	1	5.2	odd	4
315.2.a.a.1.1		1	15.2	even	4
525.2.a.a.1.1		1	5.3	odd	4
525.2.d.b.274.1		2	1.1	even	1	trivial
525.2.d.b.274.2		2	5.4	even	2	inner
735.2.a.f.1.1		1	35.27	even	4
735.2.i.a.226.1		2	35.32	odd	12
735.2.i.a.361.1		2	35.2	odd	12
735.2.i.b.226.1		2	35.17	even	12
735.2.i.b.361.1		2	35.12	even	12
1575.2.a.h.1.1		1	15.8	even	4
1575.2.d.b.1324.1		2	15.14	odd	2
1575.2.d.b.1324.2		2	3.2	odd	2
1680.2.a.f.1.1		1	20.7	even	4
2205.2.a.b.1.1		1	105.62	odd	4
3675.2.a.f.1.1		1	35.13	even	4
5040.2.a.d.1.1		1	60.47	odd	4
6720.2.a.p.1.1		1	40.37	odd	4
6720.2.a.bk.1.1		1	40.27	even	4
8400.2.a.co.1.1		1	20.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	525.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000i q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000i q^{7} -3.00000i q^{8} -1.00000 q^{9} +1.00000i q^{12} -6.00000i q^{13} -1.00000 q^{14} -1.00000 q^{16} -2.00000i q^{17} +1.00000i q^{18} +8.00000 q^{19} +1.00000 q^{21} +8.00000i q^{23} +3.00000 q^{24} -6.00000 q^{26} -1.00000i q^{27} -1.00000i q^{28} +2.00000 q^{29} +4.00000 q^{31} -5.00000i q^{32} -2.00000 q^{34} -1.00000 q^{36} +2.00000i q^{37} -8.00000i q^{38} +6.00000 q^{39} -6.00000 q^{41} -1.00000i q^{42} +4.00000i q^{43} +8.00000 q^{46} -8.00000i q^{47} -1.00000i q^{48} -1.00000 q^{49} +2.00000 q^{51} -6.00000i q^{52} +10.0000i q^{53} -1.00000 q^{54} -3.00000 q^{56} +8.00000i q^{57} -2.00000i q^{58} -4.00000 q^{59} -2.00000 q^{61} -4.00000i q^{62} +1.00000i q^{63} -7.00000 q^{64} -4.00000i q^{67} -2.00000i q^{68} -8.00000 q^{69} -12.0000 q^{71} +3.00000i q^{72} -2.00000i q^{73} +2.00000 q^{74} +8.00000 q^{76} -6.00000i q^{78} -8.00000 q^{79} +1.00000 q^{81} +6.00000i q^{82} -4.00000i q^{83} +1.00000 q^{84} +4.00000 q^{86} +2.00000i q^{87} +6.00000 q^{89} -6.00000 q^{91} +8.00000i q^{92} +4.00000i q^{93} -8.00000 q^{94} +5.00000 q^{96} +18.0000i q^{97} +1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} + 2 q^{6} - 2 q^{9} - 2 q^{14} - 2 q^{16} + 16 q^{19} + 2 q^{21} + 6 q^{24} - 12 q^{26} + 4 q^{29} + 8 q^{31} - 4 q^{34} - 2 q^{36} + 12 q^{39} - 12 q^{41} + 16 q^{46} - 2 q^{49} + 4 q^{51}+ \cdots + 10 q^{96}+O(q^{100}) \) 2 * q + 2 * q^4 + 2 * q^6 - 2 * q^9 - 2 * q^14 - 2 * q^16 + 16 * q^19 + 2 * q^21 + 6 * q^24 - 12 * q^26 + 4 * q^29 + 8 * q^31 - 4 * q^34 - 2 * q^36 + 12 * q^39 - 12 * q^41 + 16 * q^46 - 2 * q^49 + 4 * q^51 - 2 * q^54 - 6 * q^56 - 8 * q^59 - 4 * q^61 - 14 * q^64 - 16 * q^69 - 24 * q^71 + 4 * q^74 + 16 * q^76 - 16 * q^79 + 2 * q^81 + 2 * q^84 + 8 * q^86 + 12 * q^89 - 12 * q^91 - 16 * q^94 + 10 * q^96

Introduction

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