LMFDB - Embedded newform 75.2.b.b.49.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,2,Mod(49,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("75.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	75.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:	$0.598878015160$
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 15)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	75.49
Dual form			75.2.b.b.49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} +1.00000 q^{6} +3.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} +1.00000 q^{6} +3.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} -1.00000i q^{12} -2.00000i q^{13} -1.00000 q^{16} -2.00000i q^{17} -1.00000i q^{18} -4.00000 q^{19} -4.00000i q^{22} +3.00000 q^{24} +2.00000 q^{26} +1.00000i q^{27} +2.00000 q^{29} +5.00000i q^{32} +4.00000i q^{33} +2.00000 q^{34} -1.00000 q^{36} +10.0000i q^{37} -4.00000i q^{38} -2.00000 q^{39} +10.0000 q^{41} +4.00000i q^{43} -4.00000 q^{44} -8.00000i q^{47} +1.00000i q^{48} +7.00000 q^{49} -2.00000 q^{51} -2.00000i q^{52} -10.0000i q^{53} -1.00000 q^{54} +4.00000i q^{57} +2.00000i q^{58} +4.00000 q^{59} -2.00000 q^{61} -7.00000 q^{64} -4.00000 q^{66} -12.0000i q^{67} -2.00000i q^{68} -8.00000 q^{71} -3.00000i q^{72} +10.0000i q^{73} -10.0000 q^{74} -4.00000 q^{76} -2.00000i q^{78} +1.00000 q^{81} +10.0000i q^{82} +12.0000i q^{83} -4.00000 q^{86} -2.00000i q^{87} -12.0000i q^{88} +6.00000 q^{89} +8.00000 q^{94} +5.00000 q^{96} -2.00000i q^{97} +7.00000i q^{98} +4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^4 + 2 * q^6 - 2 * q^9	$ 2 q + 2 q^{4} + 2 q^{6} - 2 q^{9} - 8 q^{11} - 2 q^{16} - 8 q^{19} + 6 q^{24} + 4 q^{26} + 4 q^{29} + 4 q^{34} - 2 q^{36} - 4 q^{39} + 20 q^{41} - 8 q^{44} + 14 q^{49} - 4 q^{51} - 2 q^{54} + 8 q^{59} - 4 q^{61} - 14 q^{64} - 8 q^{66} - 16 q^{71} - 20 q^{74} - 8 q^{76} + 2 q^{81} - 8 q^{86} + 12 q^{89} + 16 q^{94} + 10 q^{96} + 8 q^{99}+O(q^{100}) $ 2 * q + 2 * q^4 + 2 * q^6 - 2 * q^9 - 8 * q^11 - 2 * q^16 - 8 * q^19 + 6 * q^24 + 4 * q^26 + 4 * q^29 + 4 * q^34 - 2 * q^36 - 4 * q^39 + 20 * q^41 - 8 * q^44 + 14 * q^49 - 4 * q^51 - 2 * q^54 + 8 * q^59 - 4 * q^61 - 14 * q^64 - 8 * q^66 - 16 * q^71 - 20 * q^74 - 8 * q^76 + 2 * q^81 - 8 * q^86 + 12 * q^89 + 16 * q^94 + 10 * q^96 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$.

$n$	$26$	$52$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000i	0.707107i	0.935414	+	0.353553i	$0.115027\pi$
$2$	1.00000i	0.707107i	−0.935414	+	0.353553i	$0.884973\pi$
$3$	− 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
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	3.00000i	1.06066i
$9$	−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	$0.706037\pi$
$12$	− 1.00000i	− 0.288675i
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	0	0
$15$	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
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	0	0
$22$	− 4.00000i	− 0.852803i
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	3.00000	0.612372
$25$	0	0
$26$	2.00000	0.392232
$27$	1.00000i	0.192450i
$28$	0	0
$29$	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
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	5.00000i	0.883883i
$33$	4.00000i	0.696311i
$34$	2.00000	0.342997
$35$	0	0
$36$	−1.00000	−0.166667
$37$	10.0000i	1.64399i	0.569495	+	0.821995i	$0.307139\pi$
$37$	10.0000i	1.64399i	−0.569495	+	0.821995i	$0.692861\pi$
$38$	− 4.00000i	− 0.648886i
$39$	−2.00000	−0.320256
$40$	0	0
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	4.00000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	0	0
$47$	− 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
$49$	7.00000	1.00000
$50$	0	0
$51$	−2.00000	−0.280056
$52$	− 2.00000i	− 0.277350i
$53$	− 10.0000i	− 1.37361i	−0.726844	−	0.686803i	$-0.759014\pi$
$53$	− 10.0000i	− 1.37361i	0.726844	−	0.686803i	$-0.240986\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	0	0
$57$	4.00000i	0.529813i
$58$	2.00000i	0.262613i
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	0	0
$64$	−7.00000	−0.875000
$65$	0	0
$66$	−4.00000	−0.492366
$67$	− 12.0000i	− 1.46603i	−0.680211	−	0.733017i	$-0.738112\pi$
$67$	− 12.0000i	− 1.46603i	0.680211	−	0.733017i	$-0.261888\pi$
$68$	− 2.00000i	− 0.242536i
$69$	0	0
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	− 3.00000i	− 0.353553i
$73$	10.0000i	1.17041i	0.810885	+	0.585206i	$0.198986\pi$
$73$	10.0000i	1.17041i	−0.810885	+	0.585206i	$0.801014\pi$
$74$	−10.0000	−1.16248
$75$	0	0
$76$	−4.00000	−0.458831
$77$	0	0
$78$	− 2.00000i	− 0.226455i
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	10.0000i	1.10432i
$83$	12.0000i	1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$	12.0000i	1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$	0	0
$85$	0	0
$86$	−4.00000	−0.431331
$87$	− 2.00000i	− 0.214423i
$88$	− 12.0000i	− 1.27920i
$89$	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
$91$	0	0
$92$	0	0
$93$	0	0
$94$	8.00000	0.825137
$95$	0	0
$96$	5.00000	0.510310
$97$	− 2.00000i	− 0.203069i	−0.994832	−	0.101535i	$-0.967625\pi$
$97$	− 2.00000i	− 0.203069i	0.994832	−	0.101535i	$-0.0323753\pi$
$98$	7.00000i	0.707107i
$99$	4.00000	0.402015
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	− 2.00000i	− 0.198030i
$103$	− 16.0000i	− 1.57653i	−0.615338	−	0.788263i	$-0.710980\pi$
$103$	− 16.0000i	− 1.57653i	0.615338	−	0.788263i	$-0.289020\pi$
$104$	6.00000	0.588348
$105$	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
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	0	0
$111$	10.0000	0.949158
$112$	0	0
$113$	2.00000i	0.188144i	0.995565	+	0.0940721i	$0.0299884\pi$
$113$	2.00000i	0.188144i	−0.995565	+	0.0940721i	$0.970012\pi$
$114$	−4.00000	−0.374634
$115$	0	0
$116$	2.00000	0.185695
$117$	2.00000i	0.184900i
$118$	4.00000i	0.368230i
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	− 2.00000i	− 0.181071i
$123$	− 10.0000i	− 0.901670i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	8.00000i	0.709885i	0.934888	+	0.354943i	$0.115500\pi$
$127$	8.00000i	0.709885i	−0.934888	+	0.354943i	$0.884500\pi$
$128$	3.00000i	0.265165i
$129$	4.00000	0.352180
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	4.00000i	0.348155i
$133$	0	0
$134$	12.0000	1.03664
$135$	0	0
$136$	6.00000	0.514496
$137$	6.00000i	0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$	6.00000i	0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	− 8.00000i	− 0.671345i
$143$	8.00000i	0.668994i
$144$	1.00000	0.0833333
$145$	0	0
$146$	−10.0000	−0.827606
$147$	− 7.00000i	− 0.577350i
$148$	10.0000i	0.821995i
$149$	−22.0000	−1.80231	−0.901155	−	0.433497i	$-0.857280\pi$
$149$	−22.0000	−1.80231	−0.901155	+	0.433497i	$0.857280\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	− 12.0000i	− 0.973329i
$153$	2.00000i	0.161690i
$154$	0	0
$155$	0	0
$156$	−2.00000	−0.160128
$157$	− 14.0000i	− 1.11732i	−0.829396	−	0.558661i	$-0.811315\pi$
$157$	− 14.0000i	− 1.11732i	0.829396	−	0.558661i	$-0.188685\pi$
$158$	0	0
$159$	−10.0000	−0.793052
$160$	0	0
$161$	0	0
$162$	1.00000i	0.0785674i
$163$	− 4.00000i	− 0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$	− 4.00000i	− 0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	10.0000	0.780869
$165$	0	0
$166$	−12.0000	−0.931381
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	4.00000	0.305888
$172$	4.00000i	0.304997i
$173$	− 18.0000i	− 1.36851i	−0.729241	−	0.684257i	$-0.760127\pi$
$173$	− 18.0000i	− 1.36851i	0.729241	−	0.684257i	$-0.239873\pi$
$174$	2.00000	0.151620
$175$	0	0
$176$	4.00000	0.301511
$177$	− 4.00000i	− 0.300658i
$178$	6.00000i	0.449719i
$179$	−20.0000	−1.49487	−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000	−1.49487	−0.747435	+	0.664335i	$0.768715\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	2.00000i	0.147844i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	8.00000i	0.585018i
$188$	− 8.00000i	− 0.583460i
$189$	0	0
$190$	0	0
$191$	16.0000	1.15772	0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000	1.15772	0.578860	+	0.815427i	$0.303498\pi$
$192$	7.00000i	0.505181i
$193$	2.00000i	0.143963i	0.997406	+	0.0719816i	$0.0229323\pi$
$193$	2.00000i	0.143963i	−0.997406	+	0.0719816i	$0.977068\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	7.00000	0.500000
$197$	− 6.00000i	− 0.427482i	−0.976890	−	0.213741i	$-0.931435\pi$
$197$	− 6.00000i	− 0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$	4.00000i	0.284268i
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	−12.0000	−0.846415
$202$	6.00000i	0.422159i
$203$	0	0
$204$	−2.00000	−0.140028
$205$	0	0
$206$	16.0000	1.11477
$207$	0	0
$208$	2.00000i	0.138675i
$209$	16.0000	1.10674
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	− 10.0000i	− 0.686803i
$213$	8.00000i	0.548151i
$214$	−12.0000	−0.820303
$215$	0	0
$216$	−3.00000	−0.204124
$217$	0	0
$218$	− 14.0000i	− 0.948200i
$219$	10.0000	0.675737
$220$	0	0
$221$	−4.00000	−0.269069
$222$	10.0000i	0.671156i
$223$	8.00000i	0.535720i	0.963458	+	0.267860i	$0.0863164\pi$
$223$	8.00000i	0.535720i	−0.963458	+	0.267860i	$0.913684\pi$
$224$	0	0
$225$	0	0
$226$	−2.00000	−0.133038
$227$	20.0000i	1.32745i	0.747978	+	0.663723i	$0.231025\pi$
$227$	20.0000i	1.32745i	−0.747978	+	0.663723i	$0.768975\pi$
$228$	4.00000i	0.264906i
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	0	0
$232$	6.00000i	0.393919i
$233$	− 6.00000i	− 0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$	− 6.00000i	− 0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	4.00000	0.260378
$237$	0	0
$238$	0	0
$239$	16.0000	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$240$	0	0
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	5.00000i	0.321412i
$243$	− 1.00000i	− 0.0641500i
$244$	−2.00000	−0.128037
$245$	0	0
$246$	10.0000	0.637577
$247$	8.00000i	0.509028i
$248$	0	0
$249$	12.0000	0.760469
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	0	0
$254$	−8.00000	−0.501965
$255$	0	0
$256$	−17.0000	−1.06250
$257$	− 18.0000i	− 1.12281i	−0.827541	−	0.561405i	$-0.810261\pi$
$257$	− 18.0000i	− 1.12281i	0.827541	−	0.561405i	$-0.189739\pi$
$258$	4.00000i	0.249029i
$259$	0	0
$260$	0	0
$261$	−2.00000	−0.123797
$262$	− 12.0000i	− 0.741362i
$263$	16.0000i	0.986602i	0.869859	+	0.493301i	$0.164210\pi$
$263$	16.0000i	0.986602i	−0.869859	+	0.493301i	$0.835790\pi$
$264$	−12.0000	−0.738549
$265$	0	0
$266$	0	0
$267$	− 6.00000i	− 0.367194i
$268$	− 12.0000i	− 0.733017i
$269$	−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
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	2.00000i	0.121268i
$273$	0	0
$274$	−6.00000	−0.362473
$275$	0	0
$276$	0	0
$277$	− 6.00000i	− 0.360505i	−0.983620	−	0.180253i	$-0.942309\pi$
$277$	− 6.00000i	− 0.360505i	0.983620	−	0.180253i	$-0.0576915\pi$
$278$	4.00000i	0.239904i
$279$	0	0
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	− 8.00000i	− 0.476393i
$283$	− 12.0000i	− 0.713326i	−0.934233	−	0.356663i	$-0.883914\pi$
$283$	− 12.0000i	− 0.713326i	0.934233	−	0.356663i	$-0.116086\pi$
$284$	−8.00000	−0.474713
$285$	0	0
$286$	−8.00000	−0.473050
$287$	0	0
$288$	− 5.00000i	− 0.294628i
$289$	13.0000	0.764706
$290$	0	0
$291$	−2.00000	−0.117242
$292$	10.0000i	0.585206i
$293$	6.00000i	0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$	6.00000i	0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$	7.00000	0.408248
$295$	0	0
$296$	−30.0000	−1.74371
$297$	− 4.00000i	− 0.232104i
$298$	− 22.0000i	− 1.27443i
$299$	0	0
$300$	0	0
$301$	0	0
$302$	− 8.00000i	− 0.460348i
$303$	− 6.00000i	− 0.344691i
$304$	4.00000	0.229416
$305$	0	0
$306$	−2.00000	−0.114332
$307$	− 28.0000i	− 1.59804i	−0.601302	−	0.799022i	$-0.705351\pi$
$307$	− 28.0000i	− 1.59804i	0.601302	−	0.799022i	$-0.294649\pi$
$308$	0	0
$309$	−16.0000	−0.910208
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	− 6.00000i	− 0.339683i
$313$	26.0000i	1.46961i	0.678280	+	0.734803i	$0.262726\pi$
$313$	26.0000i	1.46961i	−0.678280	+	0.734803i	$0.737274\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	0	0
$317$	2.00000i	0.112331i	0.998421	+	0.0561656i	$0.0178875\pi$
$317$	2.00000i	0.112331i	−0.998421	+	0.0561656i	$0.982113\pi$
$318$	− 10.0000i	− 0.560772i
$319$	−8.00000	−0.447914
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	8.00000i	0.445132i
$324$	1.00000	0.0555556
$325$	0	0
$326$	4.00000	0.221540
$327$	14.0000i	0.774202i
$328$	30.0000i	1.65647i
$329$	0	0
$330$	0	0
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	12.0000i	0.658586i
$333$	− 10.0000i	− 0.547997i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	14.0000i	0.762629i	0.924445	+	0.381314i	$0.124528\pi$
$337$	14.0000i	0.762629i	−0.924445	+	0.381314i	$0.875472\pi$
$338$	9.00000i	0.489535i
$339$	2.00000	0.108625
$340$	0	0
$341$	0	0
$342$	4.00000i	0.216295i
$343$	0	0
$344$	−12.0000	−0.646997
$345$	0	0
$346$	18.0000	0.967686
$347$	28.0000i	1.50312i	0.659665	+	0.751559i	$0.270698\pi$
$347$	28.0000i	1.50312i	−0.659665	+	0.751559i	$0.729302\pi$
$348$	− 2.00000i	− 0.107211i
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	− 20.0000i	− 1.06600i
$353$	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
$357$	0	0
$358$	− 20.0000i	− 1.05703i
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	− 10.0000i	− 0.525588i
$363$	− 5.00000i	− 0.262432i
$364$	0	0
$365$	0	0
$366$	−2.00000	−0.104542
$367$	24.0000i	1.25279i	0.779506	+	0.626395i	$0.215470\pi$
$367$	24.0000i	1.25279i	−0.779506	+	0.626395i	$0.784530\pi$
$368$	0	0
$369$	−10.0000	−0.520579
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 26.0000i	− 1.34623i	−0.739538	−	0.673114i	$-0.764956\pi$
$373$	− 26.0000i	− 1.34623i	0.739538	−	0.673114i	$-0.235044\pi$
$374$	−8.00000	−0.413670
$375$	0	0
$376$	24.0000	1.23771
$377$	− 4.00000i	− 0.206010i
$378$	0	0
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	16.0000i	0.818631i
$383$	− 24.0000i	− 1.22634i	−0.789950	−	0.613171i	$-0.789894\pi$
$383$	− 24.0000i	− 1.22634i	0.789950	−	0.613171i	$-0.210106\pi$
$384$	3.00000	0.153093
$385$	0	0
$386$	−2.00000	−0.101797
$387$	− 4.00000i	− 0.203331i
$388$	− 2.00000i	− 0.101535i
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	0	0
$392$	21.0000i	1.06066i
$393$	12.0000i	0.605320i
$394$	6.00000	0.302276
$395$	0	0
$396$	4.00000	0.201008
$397$	2.00000i	0.100377i	0.998740	+	0.0501886i	$0.0159822\pi$
$397$	2.00000i	0.100377i	−0.998740	+	0.0501886i	$0.984018\pi$
$398$	8.00000i	0.401004i
$399$	0	0
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	− 12.0000i	− 0.598506i
$403$	0	0
$404$	6.00000	0.298511
$405$	0	0
$406$	0	0
$407$	− 40.0000i	− 1.98273i
$408$	− 6.00000i	− 0.297044i
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	0	0
$411$	6.00000	0.295958
$412$	− 16.0000i	− 0.788263i
$413$	0	0
$414$	0	0
$415$	0	0
$416$	10.0000	0.490290
$417$	− 4.00000i	− 0.195881i
$418$	16.0000i	0.782586i
$419$	−4.00000	−0.195413	−0.0977064	−	0.995215i	$-0.531151\pi$
$419$	−4.00000	−0.195413	−0.0977064	+	0.995215i	$0.531151\pi$
$420$	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
$426$	−8.00000	−0.387601
$427$	0	0
$428$	12.0000i	0.580042i
$429$	8.00000	0.386244
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	− 14.0000i	− 0.672797i	−0.941720	−	0.336399i	$-0.890791\pi$
$433$	− 14.0000i	− 0.672797i	0.941720	−	0.336399i	$-0.109209\pi$
$434$	0	0
$435$	0	0
$436$	−14.0000	−0.670478
$437$	0	0
$438$	10.0000i	0.477818i
$439$	−40.0000	−1.90910	−0.954548	−	0.298057i	$-0.903661\pi$
$439$	−40.0000	−1.90910	−0.954548	+	0.298057i	$0.903661\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	− 4.00000i	− 0.190261i
$443$	− 12.0000i	− 0.570137i	−0.958507	−	0.285069i	$-0.907984\pi$
$443$	− 12.0000i	− 0.570137i	0.958507	−	0.285069i	$-0.0920164\pi$
$444$	10.0000	0.474579
$445$	0	0
$446$	−8.00000	−0.378811
$447$	22.0000i	1.04056i
$448$	0	0
$449$	−2.00000	−0.0943858	−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000	−0.0943858	−0.0471929	+	0.998886i	$0.515028\pi$
$450$	0	0
$451$	−40.0000	−1.88353
$452$	2.00000i	0.0940721i
$453$	8.00000i	0.375873i
$454$	−20.0000	−0.938647
$455$	0	0
$456$	−12.0000	−0.561951
$457$	− 10.0000i	− 0.467780i	−0.972263	−	0.233890i	$-0.924854\pi$
$457$	− 10.0000i	− 0.467780i	0.972263	−	0.233890i	$-0.0751456\pi$
$458$	− 6.00000i	− 0.280362i
$459$	2.00000	0.0933520
$460$	0	0
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	0	0
$463$	24.0000i	1.11537i	0.830051	+	0.557687i	$0.188311\pi$
$463$	24.0000i	1.11537i	−0.830051	+	0.557687i	$0.811689\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	6.00000	0.277945
$467$	− 28.0000i	− 1.29569i	−0.761774	−	0.647843i	$-0.775671\pi$
$467$	− 28.0000i	− 1.29569i	0.761774	−	0.647843i	$-0.224329\pi$
$468$	2.00000i	0.0924500i
$469$	0	0
$470$	0	0
$471$	−14.0000	−0.645086
$472$	12.0000i	0.552345i
$473$	− 16.0000i	− 0.735681i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	10.0000i	0.457869i
$478$	16.0000i	0.731823i
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	20.0000	0.911922
$482$	− 14.0000i	− 0.637683i
$483$	0	0
$484$	5.00000	0.227273
$485$	0	0
$486$	1.00000	0.0453609
$487$	− 32.0000i	− 1.45006i	−0.688718	−	0.725029i	$-0.741826\pi$
$487$	− 32.0000i	− 1.45006i	0.688718	−	0.725029i	$-0.258174\pi$
$488$	− 6.00000i	− 0.271607i
$489$	−4.00000	−0.180886
$490$	0	0
$491$	28.0000	1.26362	0.631811	−	0.775122i	$-0.282312\pi$
$491$	28.0000	1.26362	0.631811	+	0.775122i	$0.282312\pi$
$492$	− 10.0000i	− 0.450835i
$493$	− 4.00000i	− 0.180151i
$494$	−8.00000	−0.359937
$495$	0	0
$496$	0	0
$497$	0	0
$498$	12.0000i	0.537733i
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	0	0
$501$	0	0
$502$	12.0000i	0.535586i
$503$	− 32.0000i	− 1.42681i	−0.700752	−	0.713405i	$-0.747152\pi$
$503$	− 32.0000i	− 1.42681i	0.700752	−	0.713405i	$-0.252848\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 9.00000i	− 0.399704i
$508$	8.00000i	0.354943i
$509$	34.0000	1.50702	0.753512	−	0.657434i	$-0.228358\pi$
$509$	34.0000	1.50702	0.753512	+	0.657434i	$0.228358\pi$
$510$	0	0
$511$	0	0
$512$	− 11.0000i	− 0.486136i
$513$	− 4.00000i	− 0.176604i
$514$	18.0000	0.793946
$515$	0	0
$516$	4.00000	0.176090
$517$	32.0000i	1.40736i
$518$	0	0
$519$	−18.0000	−0.790112
$520$	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
$523$	4.00000i	0.174908i	0.996169	+	0.0874539i	$0.0278730\pi$
$523$	4.00000i	0.174908i	−0.996169	+	0.0874539i	$0.972127\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	−16.0000	−0.697633
$527$	0	0
$528$	− 4.00000i	− 0.174078i
$529$	23.0000	1.00000
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0	0
$533$	− 20.0000i	− 0.866296i
$534$	6.00000	0.259645
$535$	0	0
$536$	36.0000	1.55496
$537$	20.0000i	0.863064i
$538$	− 14.0000i	− 0.603583i
$539$	−28.0000	−1.20605
$540$	0	0
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	16.0000i	0.687259i
$543$	10.0000i	0.429141i
$544$	10.0000	0.428746
$545$	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	$0.859370\pi$
$548$	6.00000i	0.256307i
$549$	2.00000	0.0853579
$550$	0	0
$551$	−8.00000	−0.340811
$552$	0	0
$553$	0	0
$554$	6.00000	0.254916
$555$	0	0
$556$	4.00000	0.169638
$557$	18.0000i	0.762684i	0.924434	+	0.381342i	$0.124538\pi$
$557$	18.0000i	0.762684i	−0.924434	+	0.381342i	$0.875462\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	0	0
$561$	8.00000	0.337760
$562$	− 6.00000i	− 0.253095i
$563$	12.0000i	0.505740i	0.967500	+	0.252870i	$0.0813744\pi$
$563$	12.0000i	0.505740i	−0.967500	+	0.252870i	$0.918626\pi$
$564$	−8.00000	−0.336861
$565$	0	0
$566$	12.0000	0.504398
$567$	0	0
$568$	− 24.0000i	− 1.00702i
$569$	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
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	8.00000i	0.334497i
$573$	− 16.0000i	− 0.668410i
$574$	0	0
$575$	0	0
$576$	7.00000	0.291667
$577$	− 2.00000i	− 0.0832611i	−0.999133	−	0.0416305i	$-0.986745\pi$
$577$	− 2.00000i	− 0.0832611i	0.999133	−	0.0416305i	$-0.0132552\pi$
$578$	13.0000i	0.540729i
$579$	2.00000	0.0831172
$580$	0	0
$581$	0	0
$582$	− 2.00000i	− 0.0829027i
$583$	40.0000i	1.65663i
$584$	−30.0000	−1.24141
$585$	0	0
$586$	−6.00000	−0.247858
$587$	12.0000i	0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$	12.0000i	0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$	− 7.00000i	− 0.288675i
$589$	0	0
$590$	0	0
$591$	−6.00000	−0.246807
$592$	− 10.0000i	− 0.410997i
$593$	34.0000i	1.39621i	0.715994	+	0.698106i	$0.245974\pi$
$593$	34.0000i	1.39621i	−0.715994	+	0.698106i	$0.754026\pi$
$594$	4.00000	0.164122
$595$	0	0
$596$	−22.0000	−0.901155
$597$	− 8.00000i	− 0.327418i
$598$	0	0
$599$	8.00000	0.326871	0.163436	−	0.986554i	$-0.447742\pi$
$599$	8.00000	0.326871	0.163436	+	0.986554i	$0.447742\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	0	0
$603$	12.0000i	0.488678i
$604$	−8.00000	−0.325515
$605$	0	0
$606$	6.00000	0.243733
$607$	8.00000i	0.324710i	0.986732	+	0.162355i	$0.0519090\pi$
$607$	8.00000i	0.324710i	−0.986732	+	0.162355i	$0.948091\pi$
$608$	− 20.0000i	− 0.811107i
$609$	0	0
$610$	0	0
$611$	−16.0000	−0.647291
$612$	2.00000i	0.0808452i
$613$	22.0000i	0.888572i	0.895885	+	0.444286i	$0.146543\pi$
$613$	22.0000i	0.888572i	−0.895885	+	0.444286i	$0.853457\pi$
$614$	28.0000	1.12999
$615$	0	0
$616$	0	0
$617$	6.00000i	0.241551i	0.992680	+	0.120775i	$0.0385381\pi$
$617$	6.00000i	0.241551i	−0.992680	+	0.120775i	$0.961462\pi$
$618$	− 16.0000i	− 0.643614i
$619$	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
$621$	0	0
$622$	− 24.0000i	− 0.962312i
$623$	0	0
$624$	2.00000	0.0800641
$625$	0	0
$626$	−26.0000	−1.03917
$627$	− 16.0000i	− 0.638978i
$628$	− 14.0000i	− 0.558661i
$629$	20.0000	0.797452
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	0	0
$633$	− 20.0000i	− 0.794929i
$634$	−2.00000	−0.0794301
$635$	0	0
$636$	−10.0000	−0.396526
$637$	− 14.0000i	− 0.554700i
$638$	− 8.00000i	− 0.316723i
$639$	8.00000	0.316475
$640$	0	0
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	12.0000i	0.473602i
$643$	− 36.0000i	− 1.41970i	−0.704352	−	0.709851i	$-0.748762\pi$
$643$	− 36.0000i	− 1.41970i	0.704352	−	0.709851i	$-0.251238\pi$
$644$	0	0
$645$	0	0
$646$	−8.00000	−0.314756
$647$	− 32.0000i	− 1.25805i	−0.777385	−	0.629025i	$-0.783454\pi$
$647$	− 32.0000i	− 1.25805i	0.777385	−	0.629025i	$-0.216546\pi$
$648$	3.00000i	0.117851i
$649$	−16.0000	−0.628055
$650$	0	0
$651$	0	0
$652$	− 4.00000i	− 0.156652i
$653$	46.0000i	1.80012i	0.435767	+	0.900060i	$0.356477\pi$
$653$	46.0000i	1.80012i	−0.435767	+	0.900060i	$0.643523\pi$
$654$	−14.0000	−0.547443
$655$	0	0
$656$	−10.0000	−0.390434
$657$	− 10.0000i	− 0.390137i
$658$	0	0
$659$	−20.0000	−0.779089	−0.389545	−	0.921008i	$-0.627368\pi$
$659$	−20.0000	−0.779089	−0.389545	+	0.921008i	$0.627368\pi$
$660$	0	0
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	12.0000i	0.466393i
$663$	4.00000i	0.155347i
$664$	−36.0000	−1.39707
$665$	0	0
$666$	10.0000	0.387492
$667$	0	0
$668$	0	0
$669$	8.00000	0.309298
$670$	0	0
$671$	8.00000	0.308837
$672$	0	0
$673$	− 30.0000i	− 1.15642i	−0.815890	−	0.578208i	$-0.803752\pi$
$673$	− 30.0000i	− 1.15642i	0.815890	−	0.578208i	$-0.196248\pi$
$674$	−14.0000	−0.539260
$675$	0	0
$676$	9.00000	0.346154
$677$	− 6.00000i	− 0.230599i	−0.993331	−	0.115299i	$-0.963217\pi$
$677$	− 6.00000i	− 0.230599i	0.993331	−	0.115299i	$-0.0367827\pi$
$678$	2.00000i	0.0768095i
$679$	0	0
$680$	0	0
$681$	20.0000	0.766402
$682$	0	0
$683$	36.0000i	1.37750i	0.724998	+	0.688751i	$0.241841\pi$
$683$	36.0000i	1.37750i	−0.724998	+	0.688751i	$0.758159\pi$
$684$	4.00000	0.152944
$685$	0	0
$686$	0	0
$687$	6.00000i	0.228914i
$688$	− 4.00000i	− 0.152499i
$689$	−20.0000	−0.761939
$690$	0	0
$691$	−44.0000	−1.67384	−0.836919	−	0.547326i	$-0.815646\pi$
$691$	−44.0000	−1.67384	−0.836919	+	0.547326i	$0.815646\pi$
$692$	− 18.0000i	− 0.684257i
$693$	0	0
$694$	−28.0000	−1.06287
$695$	0	0
$696$	6.00000	0.227429
$697$	− 20.0000i	− 0.757554i
$698$	2.00000i	0.0757011i
$699$	−6.00000	−0.226941
$700$	0	0
$701$	−2.00000	−0.0755390	−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	−2.00000	−0.0755390	−0.0377695	+	0.999286i	$0.512025\pi$
$702$	2.00000i	0.0754851i
$703$	− 40.0000i	− 1.50863i
$704$	28.0000	1.05529
$705$	0	0
$706$	−18.0000	−0.677439
$707$	0	0
$708$	− 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
$711$	0	0
$712$	18.0000i	0.674579i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−20.0000	−0.747435
$717$	− 16.0000i	− 0.597531i
$718$	24.0000i	0.895672i
$719$	48.0000	1.79010	0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000	1.79010	0.895049	+	0.445968i	$0.147140\pi$
$720$	0	0
$721$	0	0
$722$	− 3.00000i	− 0.111648i
$723$	14.0000i	0.520666i
$724$	−10.0000	−0.371647
$725$	0	0
$726$	5.00000	0.185567
$727$	16.0000i	0.593407i	0.954970	+	0.296704i	$0.0958873\pi$
$727$	16.0000i	0.593407i	−0.954970	+	0.296704i	$0.904113\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	8.00000	0.295891
$732$	2.00000i	0.0739221i
$733$	14.0000i	0.517102i	0.965998	+	0.258551i	$0.0832450\pi$
$733$	14.0000i	0.517102i	−0.965998	+	0.258551i	$0.916755\pi$
$734$	−24.0000	−0.885856
$735$	0	0
$736$	0	0
$737$	48.0000i	1.76810i
$738$	− 10.0000i	− 0.368105i
$739$	44.0000	1.61857	0.809283	−	0.587419i	$-0.199856\pi$
$739$	44.0000	1.61857	0.809283	+	0.587419i	$0.199856\pi$
$740$	0	0
$741$	8.00000	0.293887
$742$	0	0
$743$	− 16.0000i	− 0.586983i	−0.955962	−	0.293492i	$-0.905183\pi$
$743$	− 16.0000i	− 0.586983i	0.955962	−	0.293492i	$-0.0948173\pi$
$744$	0	0
$745$	0	0
$746$	26.0000	0.951928
$747$	− 12.0000i	− 0.439057i
$748$	8.00000i	0.292509i
$749$	0	0
$750$	0	0
$751$	16.0000	0.583848	0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000	0.583848	0.291924	+	0.956441i	$0.405705\pi$
$752$	8.00000i	0.291730i
$753$	− 12.0000i	− 0.437304i
$754$	4.00000	0.145671
$755$	0	0
$756$	0	0
$757$	26.0000i	0.944986i	0.881334	+	0.472493i	$0.156646\pi$
$757$	26.0000i	0.944986i	−0.881334	+	0.472493i	$0.843354\pi$
$758$	20.0000i	0.726433i
$759$	0	0
$760$	0	0
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	8.00000i	0.289809i
$763$	0	0
$764$	16.0000	0.578860
$765$	0	0
$766$	24.0000	0.867155
$767$	− 8.00000i	− 0.288863i
$768$	17.0000i	0.613435i
$769$	−2.00000	−0.0721218	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000	−0.0721218	−0.0360609	+	0.999350i	$0.511481\pi$
$770$	0	0
$771$	−18.0000	−0.648254
$772$	2.00000i	0.0719816i
$773$	6.00000i	0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$	6.00000i	0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$	4.00000	0.143777
$775$	0	0
$776$	6.00000	0.215387
$777$	0	0
$778$	− 6.00000i	− 0.215110i
$779$	−40.0000	−1.43315
$780$	0	0
$781$	32.0000	1.14505
$782$	0	0
$783$	2.00000i	0.0714742i
$784$	−7.00000	−0.250000
$785$	0	0
$786$	−12.0000	−0.428026
$787$	− 28.0000i	− 0.998092i	−0.866575	−	0.499046i	$-0.833684\pi$
$787$	− 28.0000i	− 0.998092i	0.866575	−	0.499046i	$-0.166316\pi$
$788$	− 6.00000i	− 0.213741i
$789$	16.0000	0.569615
$790$	0	0
$791$	0	0
$792$	12.0000i	0.426401i
$793$	4.00000i	0.142044i
$794$	−2.00000	−0.0709773
$795$	0	0
$796$	8.00000	0.283552
$797$	2.00000i	0.0708436i	0.999372	+	0.0354218i	$0.0112775\pi$
$797$	2.00000i	0.0708436i	−0.999372	+	0.0354218i	$0.988723\pi$
$798$	0	0
$799$	−16.0000	−0.566039
$800$	0	0
$801$	−6.00000	−0.212000
$802$	18.0000i	0.635602i
$803$	− 40.0000i	− 1.41157i
$804$	−12.0000	−0.423207
$805$	0	0
$806$	0	0
$807$	14.0000i	0.492823i
$808$	18.0000i	0.633238i
$809$	−10.0000	−0.351581	−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000	−0.351581	−0.175791	+	0.984428i	$0.556248\pi$
$810$	0	0
$811$	12.0000	0.421377	0.210688	−	0.977553i	$-0.432429\pi$
$811$	12.0000	0.421377	0.210688	+	0.977553i	$0.432429\pi$
$812$	0	0
$813$	− 16.0000i	− 0.561144i
$814$	40.0000	1.40200
$815$	0	0
$816$	2.00000	0.0700140
$817$	− 16.0000i	− 0.559769i
$818$	− 26.0000i	− 0.909069i
$819$	0	0
$820$	0	0
$821$	54.0000	1.88461	0.942306	−	0.334751i	$-0.108652\pi$
$821$	54.0000	1.88461	0.942306	+	0.334751i	$0.108652\pi$
$822$	6.00000i	0.209274i
$823$	32.0000i	1.11545i	0.830026	+	0.557725i	$0.188326\pi$
$823$	32.0000i	1.11545i	−0.830026	+	0.557725i	$0.811674\pi$
$824$	48.0000	1.67216
$825$	0	0
$826$	0	0
$827$	28.0000i	0.973655i	0.873498	+	0.486828i	$0.161846\pi$
$827$	28.0000i	0.973655i	−0.873498	+	0.486828i	$0.838154\pi$
$828$	0	0
$829$	−30.0000	−1.04194	−0.520972	−	0.853574i	$-0.674430\pi$
$829$	−30.0000	−1.04194	−0.520972	+	0.853574i	$0.674430\pi$
$830$	0	0
$831$	−6.00000	−0.208138
$832$	14.0000i	0.485363i
$833$	− 14.0000i	− 0.485071i
$834$	4.00000	0.138509
$835$	0	0
$836$	16.0000	0.553372
$837$	0	0
$838$	− 4.00000i	− 0.138178i
$839$	−40.0000	−1.38095	−0.690477	−	0.723355i	$-0.742599\pi$
$839$	−40.0000	−1.38095	−0.690477	+	0.723355i	$0.742599\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	− 26.0000i	− 0.896019i
$843$	6.00000i	0.206651i
$844$	20.0000	0.688428
$845$	0	0
$846$	−8.00000	−0.275046
$847$	0	0
$848$	10.0000i	0.343401i
$849$	−12.0000	−0.411839
$850$	0	0
$851$	0	0
$852$	8.00000i	0.274075i
$853$	6.00000i	0.205436i	0.994711	+	0.102718i	$0.0327539\pi$
$853$	6.00000i	0.205436i	−0.994711	+	0.102718i	$0.967246\pi$
$854$	0	0
$855$	0	0
$856$	−36.0000	−1.23045
$857$	22.0000i	0.751506i	0.926720	+	0.375753i	$0.122616\pi$
$857$	22.0000i	0.751506i	−0.926720	+	0.375753i	$0.877384\pi$
$858$	8.00000i	0.273115i
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 56.0000i	− 1.90626i	−0.302558	−	0.953131i	$-0.597840\pi$
$863$	− 56.0000i	− 1.90626i	0.302558	−	0.953131i	$-0.402160\pi$
$864$	−5.00000	−0.170103
$865$	0	0
$866$	14.0000	0.475739
$867$	− 13.0000i	− 0.441503i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−24.0000	−0.813209
$872$	− 42.0000i	− 1.42230i
$873$	2.00000i	0.0676897i
$874$	0	0
$875$	0	0
$876$	10.0000	0.337869
$877$	− 30.0000i	− 1.01303i	−0.862232	−	0.506514i	$-0.830934\pi$
$877$	− 30.0000i	− 1.01303i	0.862232	−	0.506514i	$-0.169066\pi$
$878$	− 40.0000i	− 1.34993i
$879$	6.00000	0.202375
$880$	0	0
$881$	−46.0000	−1.54978	−0.774890	−	0.632096i	$-0.782195\pi$
$881$	−46.0000	−1.54978	−0.774890	+	0.632096i	$0.782195\pi$
$882$	− 7.00000i	− 0.235702i
$883$	44.0000i	1.48072i	0.672212	+	0.740359i	$0.265344\pi$
$883$	44.0000i	1.48072i	−0.672212	+	0.740359i	$0.734656\pi$
$884$	−4.00000	−0.134535
$885$	0	0
$886$	12.0000	0.403148
$887$	− 48.0000i	− 1.61168i	−0.592132	−	0.805841i	$-0.701714\pi$
$887$	− 48.0000i	− 1.61168i	0.592132	−	0.805841i	$-0.298286\pi$
$888$	30.0000i	1.00673i
$889$	0	0
$890$	0	0
$891$	−4.00000	−0.134005
$892$	8.00000i	0.267860i
$893$	32.0000i	1.07084i
$894$	−22.0000	−0.735790
$895$	0	0
$896$	0	0
$897$	0	0
$898$	− 2.00000i	− 0.0667409i
$899$	0	0
$900$	0	0
$901$	−20.0000	−0.666297
$902$	− 40.0000i	− 1.33185i
$903$	0	0
$904$	−6.00000	−0.199557
$905$	0	0
$906$	−8.00000	−0.265782
$907$	12.0000i	0.398453i	0.979953	+	0.199227i	$0.0638430\pi$
$907$	12.0000i	0.398453i	−0.979953	+	0.199227i	$0.936157\pi$
$908$	20.0000i	0.663723i
$909$	−6.00000	−0.199007
$910$	0	0
$911$	32.0000	1.06021	0.530104	−	0.847933i	$-0.322153\pi$
$911$	32.0000	1.06021	0.530104	+	0.847933i	$0.322153\pi$
$912$	− 4.00000i	− 0.132453i
$913$	− 48.0000i	− 1.58857i
$914$	10.0000	0.330771
$915$	0	0
$916$	−6.00000	−0.198246
$917$	0	0
$918$	2.00000i	0.0660098i
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	0	0
$921$	−28.0000	−0.922631
$922$	− 18.0000i	− 0.592798i
$923$	16.0000i	0.526646i
$924$	0	0
$925$	0	0
$926$	−24.0000	−0.788689
$927$	16.0000i	0.525509i
$928$	10.0000i	0.328266i
$929$	−34.0000	−1.11550	−0.557752	−	0.830008i	$-0.688336\pi$
$929$	−34.0000	−1.11550	−0.557752	+	0.830008i	$0.688336\pi$
$930$	0	0
$931$	−28.0000	−0.917663
$932$	− 6.00000i	− 0.196537i
$933$	24.0000i	0.785725i
$934$	28.0000	0.916188
$935$	0	0
$936$	−6.00000	−0.196116
$937$	54.0000i	1.76410i	0.471153	+	0.882052i	$0.343838\pi$
$937$	54.0000i	1.76410i	−0.471153	+	0.882052i	$0.656162\pi$
$938$	0	0
$939$	26.0000	0.848478
$940$	0	0
$941$	−50.0000	−1.62995	−0.814977	−	0.579494i	$-0.803250\pi$
$941$	−50.0000	−1.62995	−0.814977	+	0.579494i	$0.803250\pi$
$942$	− 14.0000i	− 0.456145i
$943$	0	0
$944$	−4.00000	−0.130189
$945$	0	0
$946$	16.0000	0.520205
$947$	36.0000i	1.16984i	0.811090	+	0.584921i	$0.198875\pi$
$947$	36.0000i	1.16984i	−0.811090	+	0.584921i	$0.801125\pi$
$948$	0	0
$949$	20.0000	0.649227
$950$	0	0
$951$	2.00000	0.0648544
$952$	0	0
$953$	− 22.0000i	− 0.712650i	−0.934362	−	0.356325i	$-0.884030\pi$
$953$	− 22.0000i	− 0.712650i	0.934362	−	0.356325i	$-0.115970\pi$
$954$	−10.0000	−0.323762
$955$	0	0
$956$	16.0000	0.517477
$957$	8.00000i	0.258603i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−31.0000	−1.00000
$962$	20.0000i	0.644826i
$963$	− 12.0000i	− 0.386695i
$964$	−14.0000	−0.450910
$965$	0	0
$966$	0	0
$967$	− 32.0000i	− 1.02905i	−0.857475	−	0.514525i	$-0.827968\pi$
$967$	− 32.0000i	− 1.02905i	0.857475	−	0.514525i	$-0.172032\pi$
$968$	15.0000i	0.482118i
$969$	8.00000	0.256997
$970$	0	0
$971$	60.0000	1.92549	0.962746	−	0.270408i	$-0.0871586\pi$
$971$	60.0000	1.92549	0.962746	+	0.270408i	$0.0871586\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	0	0
$974$	32.0000	1.02535
$975$	0	0
$976$	2.00000	0.0640184
$977$	− 2.00000i	− 0.0639857i	−0.999488	−	0.0319928i	$-0.989815\pi$
$977$	− 2.00000i	− 0.0639857i	0.999488	−	0.0319928i	$-0.0101854\pi$
$978$	− 4.00000i	− 0.127906i
$979$	−24.0000	−0.767043
$980$	0	0
$981$	14.0000	0.446986
$982$	28.0000i	0.893516i
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	30.0000	0.956365
$985$	0	0
$986$	4.00000	0.127386
$987$	0	0
$988$	8.00000i	0.254514i
$989$	0	0
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	0	0
$993$	− 12.0000i	− 0.380808i
$994$	0	0
$995$	0	0
$996$	12.0000	0.380235
$997$	− 54.0000i	− 1.71020i	−0.518465	−	0.855099i	$-0.673497\pi$
$997$	− 54.0000i	− 1.71020i	0.518465	−	0.855099i	$-0.326503\pi$
$998$	− 4.00000i	− 0.126618i
$999$	−10.0000	−0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.2.b.b.49.2		2
3.2	odd	2		225.2.b.b.199.1		2
4.3	odd	2		1200.2.f.h.49.2		2
5.2	odd	4		15.2.a.a.1.1	✓	1
5.3	odd	4		75.2.a.b.1.1		1
5.4	even	2	inner	75.2.b.b.49.1		2
8.3	odd	2		4800.2.f.c.3649.1		2
8.5	even	2		4800.2.f.bf.3649.2		2
12.11	even	2		3600.2.f.e.2449.1		2
15.2	even	4		45.2.a.a.1.1		1
15.8	even	4		225.2.a.b.1.1		1
15.14	odd	2		225.2.b.b.199.2		2
20.3	even	4		1200.2.a.e.1.1		1
20.7	even	4		240.2.a.d.1.1		1
20.19	odd	2		1200.2.f.h.49.1		2
35.2	odd	12		735.2.i.e.361.1		2
35.12	even	12		735.2.i.d.361.1		2
35.13	even	4		3675.2.a.j.1.1		1
35.17	even	12		735.2.i.d.226.1		2
35.27	even	4		735.2.a.c.1.1		1
35.32	odd	12		735.2.i.e.226.1		2
40.3	even	4		4800.2.a.bz.1.1		1
40.13	odd	4		4800.2.a.t.1.1		1
40.19	odd	2		4800.2.f.c.3649.2		2
40.27	even	4		960.2.a.a.1.1		1
40.29	even	2		4800.2.f.bf.3649.1		2
40.37	odd	4		960.2.a.l.1.1		1
45.2	even	12		405.2.e.c.271.1		2
45.7	odd	12		405.2.e.f.271.1		2
45.22	odd	12		405.2.e.f.136.1		2
45.32	even	12		405.2.e.c.136.1		2
55.32	even	4		1815.2.a.d.1.1		1
55.43	even	4		9075.2.a.g.1.1		1
60.23	odd	4		3600.2.a.u.1.1		1
60.47	odd	4		720.2.a.c.1.1		1
60.59	even	2		3600.2.f.e.2449.2		2
65.12	odd	4		2535.2.a.j.1.1		1
80.27	even	4		3840.2.k.r.1921.1		2
80.37	odd	4		3840.2.k.m.1921.2		2
80.67	even	4		3840.2.k.r.1921.2		2
80.77	odd	4		3840.2.k.m.1921.1		2
85.67	odd	4		4335.2.a.c.1.1		1
95.37	even	4		5415.2.a.j.1.1		1
105.62	odd	4		2205.2.a.i.1.1		1
115.22	even	4		7935.2.a.d.1.1		1
120.77	even	4		2880.2.a.y.1.1		1
120.107	odd	4		2880.2.a.bc.1.1		1
165.32	odd	4		5445.2.a.c.1.1		1
195.77	even	4		7605.2.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.2.a.a.1.1	✓	1	5.2	odd	4
45.2.a.a.1.1		1	15.2	even	4
75.2.a.b.1.1		1	5.3	odd	4
75.2.b.b.49.1		2	5.4	even	2	inner
75.2.b.b.49.2		2	1.1	even	1	trivial
225.2.a.b.1.1		1	15.8	even	4
225.2.b.b.199.1		2	3.2	odd	2
225.2.b.b.199.2		2	15.14	odd	2
240.2.a.d.1.1		1	20.7	even	4
405.2.e.c.136.1		2	45.32	even	12
405.2.e.c.271.1		2	45.2	even	12
405.2.e.f.136.1		2	45.22	odd	12
405.2.e.f.271.1		2	45.7	odd	12
720.2.a.c.1.1		1	60.47	odd	4
735.2.a.c.1.1		1	35.27	even	4
735.2.i.d.226.1		2	35.17	even	12
735.2.i.d.361.1		2	35.12	even	12
735.2.i.e.226.1		2	35.32	odd	12
735.2.i.e.361.1		2	35.2	odd	12
960.2.a.a.1.1		1	40.27	even	4
960.2.a.l.1.1		1	40.37	odd	4
1200.2.a.e.1.1		1	20.3	even	4
1200.2.f.h.49.1		2	20.19	odd	2
1200.2.f.h.49.2		2	4.3	odd	2
1815.2.a.d.1.1		1	55.32	even	4
2205.2.a.i.1.1		1	105.62	odd	4
2535.2.a.j.1.1		1	65.12	odd	4
2880.2.a.y.1.1		1	120.77	even	4
2880.2.a.bc.1.1		1	120.107	odd	4
3600.2.a.u.1.1		1	60.23	odd	4
3600.2.f.e.2449.1		2	12.11	even	2
3600.2.f.e.2449.2		2	60.59	even	2
3675.2.a.j.1.1		1	35.13	even	4
3840.2.k.m.1921.1		2	80.77	odd	4
3840.2.k.m.1921.2		2	80.37	odd	4
3840.2.k.r.1921.1		2	80.27	even	4
3840.2.k.r.1921.2		2	80.67	even	4
4335.2.a.c.1.1		1	85.67	odd	4
4800.2.a.t.1.1		1	40.13	odd	4
4800.2.a.bz.1.1		1	40.3	even	4
4800.2.f.c.3649.1		2	8.3	odd	2
4800.2.f.c.3649.2		2	40.19	odd	2
4800.2.f.bf.3649.1		2	40.29	even	2
4800.2.f.bf.3649.2		2	8.5	even	2
5415.2.a.j.1.1		1	95.37	even	4
5445.2.a.c.1.1		1	165.32	odd	4
7605.2.a.g.1.1		1	195.77	even	4
7935.2.a.d.1.1		1	115.22	even	4
9075.2.a.g.1.1		1	55.43	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:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	75.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} +1.00000 q^{6} +3.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} +1.00000 q^{6} +3.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} -1.00000i q^{12} -2.00000i q^{13} -1.00000 q^{16} -2.00000i q^{17} -1.00000i q^{18} -4.00000 q^{19} -4.00000i q^{22} +3.00000 q^{24} +2.00000 q^{26} +1.00000i q^{27} +2.00000 q^{29} +5.00000i q^{32} +4.00000i q^{33} +2.00000 q^{34} -1.00000 q^{36} +10.0000i q^{37} -4.00000i q^{38} -2.00000 q^{39} +10.0000 q^{41} +4.00000i q^{43} -4.00000 q^{44} -8.00000i q^{47} +1.00000i q^{48} +7.00000 q^{49} -2.00000 q^{51} -2.00000i q^{52} -10.0000i q^{53} -1.00000 q^{54} +4.00000i q^{57} +2.00000i q^{58} +4.00000 q^{59} -2.00000 q^{61} -7.00000 q^{64} -4.00000 q^{66} -12.0000i q^{67} -2.00000i q^{68} -8.00000 q^{71} -3.00000i q^{72} +10.0000i q^{73} -10.0000 q^{74} -4.00000 q^{76} -2.00000i q^{78} +1.00000 q^{81} +10.0000i q^{82} +12.0000i q^{83} -4.00000 q^{86} -2.00000i q^{87} -12.0000i q^{88} +6.00000 q^{89} +8.00000 q^{94} +5.00000 q^{96} -2.00000i q^{97} +7.00000i q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^4 + 2 * q^6 - 2 * q^9	\( 2 q + 2 q^{4} + 2 q^{6} - 2 q^{9} - 8 q^{11} - 2 q^{16} - 8 q^{19} + 6 q^{24} + 4 q^{26} + 4 q^{29} + 4 q^{34} - 2 q^{36} - 4 q^{39} + 20 q^{41} - 8 q^{44} + 14 q^{49} - 4 q^{51} - 2 q^{54} + 8 q^{59} - 4 q^{61} - 14 q^{64} - 8 q^{66} - 16 q^{71} - 20 q^{74} - 8 q^{76} + 2 q^{81} - 8 q^{86} + 12 q^{89} + 16 q^{94} + 10 q^{96} + 8 q^{99}+O(q^{100}) \) 2 * q + 2 * q^4 + 2 * q^6 - 2 * q^9 - 8 * q^11 - 2 * q^16 - 8 * q^19 + 6 * q^24 + 4 * q^26 + 4 * q^29 + 4 * q^34 - 2 * q^36 - 4 * q^39 + 20 * q^41 - 8 * q^44 + 14 * q^49 - 4 * q^51 - 2 * q^54 + 8 * q^59 - 4 * q^61 - 14 * q^64 - 8 * q^66 - 16 * q^71 - 20 * q^74 - 8 * q^76 + 2 * q^81 - 8 * q^86 + 12 * q^89 + 16 * q^94 + 10 * q^96 + 8 * q^99

Introduction

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