LMFDB - Embedded newform 975.2.c.c.274.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [975,2,Mod(274,975)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(975, 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("975.274");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 975 = 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	975.c (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:	$7.78541419707$
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 195)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	975.274
Dual form			975.2.c.c.274.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{2} +1.00000i q^{3} -2.00000 q^{4} +2.00000 q^{6} +1.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	\(q-2.00000i q^{2} +1.00000i q^{3} -2.00000 q^{4} +2.00000 q^{6} +1.00000i q^{7} -1.00000 q^{9} +5.00000 q^{11} -2.00000i q^{12} -1.00000i q^{13} +2.00000 q^{14} -4.00000 q^{16} +7.00000i q^{17} +2.00000i q^{18} +6.00000 q^{19} -1.00000 q^{21} -10.0000i q^{22} +3.00000i q^{23} -2.00000 q^{26} -1.00000i q^{27} -2.00000i q^{28} -2.00000 q^{29} +2.00000 q^{31} +8.00000i q^{32} +5.00000i q^{33} +14.0000 q^{34} +2.00000 q^{36} -7.00000i q^{37} -12.0000i q^{38} +1.00000 q^{39} +9.00000 q^{41} +2.00000i q^{42} -8.00000i q^{43} -10.0000 q^{44} +6.00000 q^{46} -10.0000i q^{47} -4.00000i q^{48} +6.00000 q^{49} -7.00000 q^{51} +2.00000i q^{52} +5.00000i q^{53} -2.00000 q^{54} +6.00000i q^{57} +4.00000i q^{58} +5.00000 q^{61} -4.00000i q^{62} -1.00000i q^{63} +8.00000 q^{64} +10.0000 q^{66} +4.00000i q^{67} -14.0000i q^{68} -3.00000 q^{69} +9.00000 q^{71} -6.00000i q^{73} -14.0000 q^{74} -12.0000 q^{76} +5.00000i q^{77} -2.00000i q^{78} +3.00000 q^{79} +1.00000 q^{81} -18.0000i q^{82} -4.00000i q^{83} +2.00000 q^{84} -16.0000 q^{86} -2.00000i q^{87} -11.0000 q^{89} +1.00000 q^{91} -6.00000i q^{92} +2.00000i q^{93} -20.0000 q^{94} -8.00000 q^{96} +11.0000i q^{97} -12.0000i q^{98} -5.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{4} + 4 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q - 4 * q^4 + 4 * q^6 - 2 * q^9	$ 2 q - 4 q^{4} + 4 q^{6} - 2 q^{9} + 10 q^{11} + 4 q^{14} - 8 q^{16} + 12 q^{19} - 2 q^{21} - 4 q^{26} - 4 q^{29} + 4 q^{31} + 28 q^{34} + 4 q^{36} + 2 q^{39} + 18 q^{41} - 20 q^{44} + 12 q^{46} + 12 q^{49} - 14 q^{51} - 4 q^{54} + 10 q^{61} + 16 q^{64} + 20 q^{66} - 6 q^{69} + 18 q^{71} - 28 q^{74} - 24 q^{76} + 6 q^{79} + 2 q^{81} + 4 q^{84} - 32 q^{86} - 22 q^{89} + 2 q^{91} - 40 q^{94} - 16 q^{96} - 10 q^{99}+O(q^{100}) $ 2 * q - 4 * q^4 + 4 * q^6 - 2 * q^9 + 10 * q^11 + 4 * q^14 - 8 * q^16 + 12 * q^19 - 2 * q^21 - 4 * q^26 - 4 * q^29 + 4 * q^31 + 28 * q^34 + 4 * q^36 + 2 * q^39 + 18 * q^41 - 20 * q^44 + 12 * q^46 + 12 * q^49 - 14 * q^51 - 4 * q^54 + 10 * q^61 + 16 * q^64 + 20 * q^66 - 6 * q^69 + 18 * q^71 - 28 * q^74 - 24 * q^76 + 6 * q^79 + 2 * q^81 + 4 * q^84 - 32 * q^86 - 22 * q^89 + 2 * q^91 - 40 * q^94 - 16 * q^96 - 10 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/975\mathbb{Z}\right)^\times$.

$n$	$301$	$326$	$352$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$	− 2.00000i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	− 2.00000i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−2.00000	−1.00000
$5$	0	0
$5$	0	0
$6$	2.00000	0.816497
$7$	1.00000i	0.377964i	0.981981	+	0.188982i	$0.0605189\pi$
$7$	1.00000i	0.377964i	−0.981981	+	0.188982i	$0.939481\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	5.00000	1.50756	0.753778	−	0.657129i	$-0.228229\pi$
$11$	5.00000	1.50756	0.753778	+	0.657129i	$0.228229\pi$
$12$	− 2.00000i	− 0.577350i
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	2.00000	0.534522
$15$	0	0
$16$	−4.00000	−1.00000
$17$	7.00000i	1.69775i	0.528594	+	0.848875i	$0.322719\pi$
$17$	7.00000i	1.69775i	−0.528594	+	0.848875i	$0.677281\pi$
$18$	2.00000i	0.471405i
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	− 10.0000i	− 2.13201i
$23$	3.00000i	0.625543i	0.949828	+	0.312772i	$0.101257\pi$
$23$	3.00000i	0.625543i	−0.949828	+	0.312772i	$0.898743\pi$
$24$	0	0
$25$	0	0
$26$	−2.00000	−0.392232
$27$	− 1.00000i	− 0.192450i
$28$	− 2.00000i	− 0.377964i
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	8.00000i	1.41421i
$33$	5.00000i	0.870388i
$34$	14.0000	2.40098
$35$	0	0
$36$	2.00000	0.333333
$37$	− 7.00000i	− 1.15079i	−0.817875	−	0.575396i	$-0.804848\pi$
$37$	− 7.00000i	− 1.15079i	0.817875	−	0.575396i	$-0.195152\pi$
$38$	− 12.0000i	− 1.94666i
$39$	1.00000	0.160128
$40$	0	0
$41$	9.00000	1.40556	0.702782	−	0.711405i	$-0.251941\pi$
$41$	9.00000	1.40556	0.702782	+	0.711405i	$0.251941\pi$
$42$	2.00000i	0.308607i
$43$	− 8.00000i	− 1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$	− 8.00000i	− 1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$	−10.0000	−1.50756
$45$	0	0
$46$	6.00000	0.884652
$47$	− 10.0000i	− 1.45865i	−0.684167	−	0.729325i	$-0.739834\pi$
$47$	− 10.0000i	− 1.45865i	0.684167	−	0.729325i	$-0.260166\pi$
$48$	− 4.00000i	− 0.577350i
$49$	6.00000	0.857143
$50$	0	0
$51$	−7.00000	−0.980196
$52$	2.00000i	0.277350i
$53$	5.00000i	0.686803i	0.939189	+	0.343401i	$0.111579\pi$
$53$	5.00000i	0.686803i	−0.939189	+	0.343401i	$0.888421\pi$
$54$	−2.00000	−0.272166
$55$	0	0
$56$	0	0
$57$	6.00000i	0.794719i
$58$	4.00000i	0.525226i
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	5.00000	0.640184	0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000	0.640184	0.320092	+	0.947386i	$0.396286\pi$
$62$	− 4.00000i	− 0.508001i
$63$	− 1.00000i	− 0.125988i
$64$	8.00000	1.00000
$65$	0	0
$66$	10.0000	1.23091
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	− 14.0000i	− 1.69775i
$69$	−3.00000	−0.361158
$70$	0	0
$71$	9.00000	1.06810	0.534052	−	0.845452i	$-0.320669\pi$
$71$	9.00000	1.06810	0.534052	+	0.845452i	$0.320669\pi$
$72$	0	0
$73$	− 6.00000i	− 0.702247i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$	− 6.00000i	− 0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$	−14.0000	−1.62747
$75$	0	0
$76$	−12.0000	−1.37649
$77$	5.00000i	0.569803i
$78$	− 2.00000i	− 0.226455i
$79$	3.00000	0.337526	0.168763	−	0.985657i	$-0.446023\pi$
$79$	3.00000	0.337526	0.168763	+	0.985657i	$0.446023\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 18.0000i	− 1.98777i
$83$	− 4.00000i	− 0.439057i	−0.975606	−	0.219529i	$-0.929548\pi$
$83$	− 4.00000i	− 0.439057i	0.975606	−	0.219529i	$-0.0704519\pi$
$84$	2.00000	0.218218
$85$	0	0
$86$	−16.0000	−1.72532
$87$	− 2.00000i	− 0.214423i
$88$	0	0
$89$	−11.0000	−1.16600	−0.582999	−	0.812473i	$-0.698121\pi$
$89$	−11.0000	−1.16600	−0.582999	+	0.812473i	$0.698121\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	− 6.00000i	− 0.625543i
$93$	2.00000i	0.207390i
$94$	−20.0000	−2.06284
$95$	0	0
$96$	−8.00000	−0.816497
$97$	11.0000i	1.11688i	0.829545	+	0.558440i	$0.188600\pi$
$97$	11.0000i	1.11688i	−0.829545	+	0.558440i	$0.811400\pi$
$98$	− 12.0000i	− 1.21218i
$99$	−5.00000	−0.502519
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	14.0000i	1.38621i
$103$	− 4.00000i	− 0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$	− 4.00000i	− 0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$	0	0
$105$	0	0
$106$	10.0000	0.971286
$107$	17.0000i	1.64345i	0.569883	+	0.821726i	$0.306989\pi$
$107$	17.0000i	1.64345i	−0.569883	+	0.821726i	$0.693011\pi$
$108$	2.00000i	0.192450i
$109$	−4.00000	−0.383131	−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000	−0.383131	−0.191565	+	0.981480i	$0.561356\pi$
$110$	0	0
$111$	7.00000	0.664411
$112$	− 4.00000i	− 0.377964i
$113$	10.0000i	0.940721i	0.882474	+	0.470360i	$0.155876\pi$
$113$	10.0000i	0.940721i	−0.882474	+	0.470360i	$0.844124\pi$
$114$	12.0000	1.12390
$115$	0	0
$116$	4.00000	0.371391
$117$	1.00000i	0.0924500i
$118$	0	0
$119$	−7.00000	−0.641689
$120$	0	0
$121$	14.0000	1.27273
$122$	− 10.0000i	− 0.905357i
$123$	9.00000i	0.811503i
$124$	−4.00000	−0.359211
$125$	0	0
$126$	−2.00000	−0.178174
$127$	2.00000i	0.177471i	0.996055	+	0.0887357i	$0.0282826\pi$
$127$	2.00000i	0.177471i	−0.996055	+	0.0887357i	$0.971717\pi$
$128$	0	0
$129$	8.00000	0.704361
$130$	0	0
$131$	−22.0000	−1.92215	−0.961074	−	0.276289i	$-0.910895\pi$
$131$	−22.0000	−1.92215	−0.961074	+	0.276289i	$0.910895\pi$
$132$	− 10.0000i	− 0.870388i
$133$	6.00000i	0.520266i
$134$	8.00000	0.691095
$135$	0	0
$136$	0	0
$137$	14.0000i	1.19610i	0.801459	+	0.598050i	$0.204058\pi$
$137$	14.0000i	1.19610i	−0.801459	+	0.598050i	$0.795942\pi$
$138$	6.00000i	0.510754i
$139$	−15.0000	−1.27228	−0.636142	−	0.771572i	$-0.719471\pi$
$139$	−15.0000	−1.27228	−0.636142	+	0.771572i	$0.719471\pi$
$140$	0	0
$141$	10.0000	0.842152
$142$	− 18.0000i	− 1.51053i
$143$	− 5.00000i	− 0.418121i
$144$	4.00000	0.333333
$145$	0	0
$146$	−12.0000	−0.993127
$147$	6.00000i	0.494872i
$148$	14.0000i	1.15079i
$149$	−15.0000	−1.22885	−0.614424	−	0.788976i	$-0.710612\pi$
$149$	−15.0000	−1.22885	−0.614424	+	0.788976i	$0.710612\pi$
$150$	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
$153$	− 7.00000i	− 0.565916i
$154$	10.0000	0.805823
$155$	0	0
$156$	−2.00000	−0.160128
$157$	− 18.0000i	− 1.43656i	−0.695756	−	0.718278i	$-0.744931\pi$
$157$	− 18.0000i	− 1.43656i	0.695756	−	0.718278i	$-0.255069\pi$
$158$	− 6.00000i	− 0.477334i
$159$	−5.00000	−0.396526
$160$	0	0
$161$	−3.00000	−0.236433
$162$	− 2.00000i	− 0.157135i
$163$	15.0000i	1.17489i	0.809264	+	0.587445i	$0.199866\pi$
$163$	15.0000i	1.17489i	−0.809264	+	0.587445i	$0.800134\pi$
$164$	−18.0000	−1.40556
$165$	0	0
$166$	−8.00000	−0.620920
$167$	24.0000i	1.85718i	0.371113	+	0.928588i	$0.378976\pi$
$167$	24.0000i	1.85718i	−0.371113	+	0.928588i	$0.621024\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	−6.00000	−0.458831
$172$	16.0000i	1.21999i
$173$	− 18.0000i	− 1.36851i	−0.729241	−	0.684257i	$-0.760127\pi$
$173$	− 18.0000i	− 1.36851i	0.729241	−	0.684257i	$-0.239873\pi$
$174$	−4.00000	−0.303239
$175$	0	0
$176$	−20.0000	−1.50756
$177$	0	0
$178$	22.0000i	1.64897i
$179$	−6.00000	−0.448461	−0.224231	−	0.974536i	$-0.571987\pi$
$179$	−6.00000	−0.448461	−0.224231	+	0.974536i	$0.571987\pi$
$180$	0	0
$181$	−7.00000	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000	−0.520306	−0.260153	+	0.965567i	$0.583773\pi$
$182$	− 2.00000i	− 0.148250i
$183$	5.00000i	0.369611i
$184$	0	0
$185$	0	0
$186$	4.00000	0.293294
$187$	35.0000i	2.55945i
$188$	20.0000i	1.45865i
$189$	1.00000	0.0727393
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	8.00000i	0.577350i
$193$	− 17.0000i	− 1.22369i	−0.790979	−	0.611843i	$-0.790428\pi$
$193$	− 17.0000i	− 1.22369i	0.790979	−	0.611843i	$-0.209572\pi$
$194$	22.0000	1.57951
$195$	0	0
$196$	−12.0000	−0.857143
$197$	− 24.0000i	− 1.70993i	−0.518686	−	0.854965i	$-0.673579\pi$
$197$	− 24.0000i	− 1.70993i	0.518686	−	0.854965i	$-0.326421\pi$
$198$	10.0000i	0.710669i
$199$	28.0000	1.98487	0.992434	−	0.122782i	$-0.0391815\pi$
$199$	28.0000	1.98487	0.992434	+	0.122782i	$0.0391815\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	0	0
$203$	− 2.00000i	− 0.140372i
$204$	14.0000	0.980196
$205$	0	0
$206$	−8.00000	−0.557386
$207$	− 3.00000i	− 0.208514i
$208$	4.00000i	0.277350i
$209$	30.0000	2.07514
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	− 10.0000i	− 0.686803i
$213$	9.00000i	0.616670i
$214$	34.0000	2.32419
$215$	0	0
$216$	0	0
$217$	2.00000i	0.135769i
$218$	8.00000i	0.541828i
$219$	6.00000	0.405442
$220$	0	0
$221$	7.00000	0.470871
$222$	− 14.0000i	− 0.939618i
$223$	− 8.00000i	− 0.535720i	−0.963458	−	0.267860i	$-0.913684\pi$
$223$	− 8.00000i	− 0.535720i	0.963458	−	0.267860i	$-0.0863164\pi$
$224$	−8.00000	−0.534522
$225$	0	0
$226$	20.0000	1.33038
$227$	2.00000i	0.132745i	0.997795	+	0.0663723i	$0.0211425\pi$
$227$	2.00000i	0.132745i	−0.997795	+	0.0663723i	$0.978857\pi$
$228$	− 12.0000i	− 0.794719i
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	−5.00000	−0.328976
$232$	0	0
$233$	19.0000i	1.24473i	0.782727	+	0.622366i	$0.213828\pi$
$233$	19.0000i	1.24473i	−0.782727	+	0.622366i	$0.786172\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	0	0
$237$	3.00000i	0.194871i
$238$	14.0000i	0.907485i
$239$	−9.00000	−0.582162	−0.291081	−	0.956698i	$-0.594015\pi$
$239$	−9.00000	−0.582162	−0.291081	+	0.956698i	$0.594015\pi$
$240$	0	0
$241$	22.0000	1.41714	0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000	1.41714	0.708572	+	0.705638i	$0.249340\pi$
$242$	− 28.0000i	− 1.79991i
$243$	1.00000i	0.0641500i
$244$	−10.0000	−0.640184
$245$	0	0
$246$	18.0000	1.14764
$247$	− 6.00000i	− 0.381771i
$248$	0	0
$249$	4.00000	0.253490
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	2.00000i	0.125988i
$253$	15.0000i	0.943042i
$254$	4.00000	0.250982
$255$	0	0
$256$	16.0000	1.00000
$257$	− 2.00000i	− 0.124757i	−0.998053	−	0.0623783i	$-0.980131\pi$
$257$	− 2.00000i	− 0.124757i	0.998053	−	0.0623783i	$-0.0198685\pi$
$258$	− 16.0000i	− 0.996116i
$259$	7.00000	0.434959
$260$	0	0
$261$	2.00000	0.123797
$262$	44.0000i	2.71833i
$263$	− 16.0000i	− 0.986602i	−0.869859	−	0.493301i	$-0.835790\pi$
$263$	− 16.0000i	− 0.986602i	0.869859	−	0.493301i	$-0.164210\pi$
$264$	0	0
$265$	0	0
$266$	12.0000	0.735767
$267$	− 11.0000i	− 0.673189i
$268$	− 8.00000i	− 0.488678i
$269$	24.0000	1.46331	0.731653	−	0.681677i	$-0.238749\pi$
$269$	24.0000	1.46331	0.731653	+	0.681677i	$0.238749\pi$
$270$	0	0
$271$	−22.0000	−1.33640	−0.668202	−	0.743980i	$-0.732936\pi$
$271$	−22.0000	−1.33640	−0.668202	+	0.743980i	$0.732936\pi$
$272$	− 28.0000i	− 1.69775i
$273$	1.00000i	0.0605228i
$274$	28.0000	1.69154
$275$	0	0
$276$	6.00000	0.361158
$277$	2.00000i	0.120168i	0.998193	+	0.0600842i	$0.0191369\pi$
$277$	2.00000i	0.120168i	−0.998193	+	0.0600842i	$0.980863\pi$
$278$	30.0000i	1.79928i
$279$	−2.00000	−0.119737
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	− 20.0000i	− 1.19098i
$283$	20.0000i	1.18888i	0.804141	+	0.594438i	$0.202626\pi$
$283$	20.0000i	1.18888i	−0.804141	+	0.594438i	$0.797374\pi$
$284$	−18.0000	−1.06810
$285$	0	0
$286$	−10.0000	−0.591312
$287$	9.00000i	0.531253i
$288$	− 8.00000i	− 0.471405i
$289$	−32.0000	−1.88235
$290$	0	0
$291$	−11.0000	−0.644831
$292$	12.0000i	0.702247i
$293$	4.00000i	0.233682i	0.993151	+	0.116841i	$0.0372769\pi$
$293$	4.00000i	0.233682i	−0.993151	+	0.116841i	$0.962723\pi$
$294$	12.0000	0.699854
$295$	0	0
$296$	0	0
$297$	− 5.00000i	− 0.290129i
$298$	30.0000i	1.73785i
$299$	3.00000	0.173494
$300$	0	0
$301$	8.00000	0.461112
$302$	16.0000i	0.920697i
$303$	0	0
$304$	−24.0000	−1.37649
$305$	0	0
$306$	−14.0000	−0.800327
$307$	− 23.0000i	− 1.31268i	−0.754466	−	0.656340i	$-0.772104\pi$
$307$	− 23.0000i	− 1.31268i	0.754466	−	0.656340i	$-0.227896\pi$
$308$	− 10.0000i	− 0.569803i
$309$	4.00000	0.227552
$310$	0	0
$311$	20.0000	1.13410	0.567048	−	0.823685i	$-0.308085\pi$
$311$	20.0000	1.13410	0.567048	+	0.823685i	$0.308085\pi$
$312$	0	0
$313$	− 22.0000i	− 1.24351i	−0.783210	−	0.621757i	$-0.786419\pi$
$313$	− 22.0000i	− 1.24351i	0.783210	−	0.621757i	$-0.213581\pi$
$314$	−36.0000	−2.03160
$315$	0	0
$316$	−6.00000	−0.337526
$317$	24.0000i	1.34797i	0.738743	+	0.673987i	$0.235420\pi$
$317$	24.0000i	1.34797i	−0.738743	+	0.673987i	$0.764580\pi$
$318$	10.0000i	0.560772i
$319$	−10.0000	−0.559893
$320$	0	0
$321$	−17.0000	−0.948847
$322$	6.00000i	0.334367i
$323$	42.0000i	2.33694i
$324$	−2.00000	−0.111111
$325$	0	0
$326$	30.0000	1.66155
$327$	− 4.00000i	− 0.221201i
$328$	0	0
$329$	10.0000	0.551318
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	8.00000i	0.439057i
$333$	7.00000i	0.383598i
$334$	48.0000	2.62644
$335$	0	0
$336$	4.00000	0.218218
$337$	− 8.00000i	− 0.435788i	−0.975972	−	0.217894i	$-0.930081\pi$
$337$	− 8.00000i	− 0.435788i	0.975972	−	0.217894i	$-0.0699187\pi$
$338$	2.00000i	0.108786i
$339$	−10.0000	−0.543125
$340$	0	0
$341$	10.0000	0.541530
$342$	12.0000i	0.648886i
$343$	13.0000i	0.701934i
$344$	0	0
$345$	0	0
$346$	−36.0000	−1.93537
$347$	− 11.0000i	− 0.590511i	−0.955418	−	0.295255i	$-0.904595\pi$
$347$	− 11.0000i	− 0.590511i	0.955418	−	0.295255i	$-0.0954048\pi$
$348$	4.00000i	0.214423i
$349$	−24.0000	−1.28469	−0.642345	−	0.766415i	$-0.722038\pi$
$349$	−24.0000	−1.28469	−0.642345	+	0.766415i	$0.722038\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	40.0000i	2.13201i
$353$	− 6.00000i	− 0.319348i	−0.987170	−	0.159674i	$-0.948956\pi$
$353$	− 6.00000i	− 0.319348i	0.987170	−	0.159674i	$-0.0510443\pi$
$354$	0	0
$355$	0	0
$356$	22.0000	1.16600
$357$	− 7.00000i	− 0.370479i
$358$	12.0000i	0.634220i
$359$	16.0000	0.844448	0.422224	−	0.906492i	$-0.361250\pi$
$359$	16.0000	0.844448	0.422224	+	0.906492i	$0.361250\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	14.0000i	0.735824i
$363$	14.0000i	0.734809i
$364$	−2.00000	−0.104828
$365$	0	0
$366$	10.0000	0.522708
$367$	− 4.00000i	− 0.208798i	−0.994535	−	0.104399i	$-0.966708\pi$
$367$	− 4.00000i	− 0.208798i	0.994535	−	0.104399i	$-0.0332919\pi$
$368$	− 12.0000i	− 0.625543i
$369$	−9.00000	−0.468521
$370$	0	0
$371$	−5.00000	−0.259587
$372$	− 4.00000i	− 0.207390i
$373$	− 32.0000i	− 1.65690i	−0.560065	−	0.828449i	$-0.689224\pi$
$373$	− 32.0000i	− 1.65690i	0.560065	−	0.828449i	$-0.310776\pi$
$374$	70.0000	3.61961
$375$	0	0
$376$	0	0
$377$	2.00000i	0.103005i
$378$	− 2.00000i	− 0.102869i
$379$	−10.0000	−0.513665	−0.256833	−	0.966456i	$-0.582679\pi$
$379$	−10.0000	−0.513665	−0.256833	+	0.966456i	$0.582679\pi$
$380$	0	0
$381$	−2.00000	−0.102463
$382$	24.0000i	1.22795i
$383$	− 6.00000i	− 0.306586i	−0.988181	−	0.153293i	$-0.951012\pi$
$383$	− 6.00000i	− 0.306586i	0.988181	−	0.153293i	$-0.0489878\pi$
$384$	0	0
$385$	0	0
$386$	−34.0000	−1.73055
$387$	8.00000i	0.406663i
$388$	− 22.0000i	− 1.11688i
$389$	4.00000	0.202808	0.101404	−	0.994845i	$-0.467667\pi$
$389$	4.00000	0.202808	0.101404	+	0.994845i	$0.467667\pi$
$390$	0	0
$391$	−21.0000	−1.06202
$392$	0	0
$393$	− 22.0000i	− 1.10975i
$394$	−48.0000	−2.41821
$395$	0	0
$396$	10.0000	0.502519
$397$	− 15.0000i	− 0.752828i	−0.926451	−	0.376414i	$-0.877157\pi$
$397$	− 15.0000i	− 0.752828i	0.926451	−	0.376414i	$-0.122843\pi$
$398$	− 56.0000i	− 2.80703i
$399$	−6.00000	−0.300376
$400$	0	0
$401$	−14.0000	−0.699127	−0.349563	−	0.936913i	$-0.613670\pi$
$401$	−14.0000	−0.699127	−0.349563	+	0.936913i	$0.613670\pi$
$402$	8.00000i	0.399004i
$403$	− 2.00000i	− 0.0996271i
$404$	0	0
$405$	0	0
$406$	−4.00000	−0.198517
$407$	− 35.0000i	− 1.73489i
$408$	0	0
$409$	−30.0000	−1.48340	−0.741702	−	0.670729i	$-0.765981\pi$
$409$	−30.0000	−1.48340	−0.741702	+	0.670729i	$0.765981\pi$
$410$	0	0
$411$	−14.0000	−0.690569
$412$	8.00000i	0.394132i
$413$	0	0
$414$	−6.00000	−0.294884
$415$	0	0
$416$	8.00000	0.392232
$417$	− 15.0000i	− 0.734553i
$418$	− 60.0000i	− 2.93470i
$419$	−26.0000	−1.27018	−0.635092	−	0.772437i	$-0.719038\pi$
$419$	−26.0000	−1.27018	−0.635092	+	0.772437i	$0.719038\pi$
$420$	0	0
$421$	−20.0000	−0.974740	−0.487370	−	0.873195i	$-0.662044\pi$
$421$	−20.0000	−0.974740	−0.487370	+	0.873195i	$0.662044\pi$
$422$	24.0000i	1.16830i
$423$	10.0000i	0.486217i
$424$	0	0
$425$	0	0
$426$	18.0000	0.872103
$427$	5.00000i	0.241967i
$428$	− 34.0000i	− 1.64345i
$429$	5.00000	0.241402
$430$	0	0
$431$	−40.0000	−1.92673	−0.963366	−	0.268190i	$-0.913575\pi$
$431$	−40.0000	−1.92673	−0.963366	+	0.268190i	$0.913575\pi$
$432$	4.00000i	0.192450i
$433$	− 20.0000i	− 0.961139i	−0.876957	−	0.480569i	$-0.840430\pi$
$433$	− 20.0000i	− 0.961139i	0.876957	−	0.480569i	$-0.159570\pi$
$434$	4.00000	0.192006
$435$	0	0
$436$	8.00000	0.383131
$437$	18.0000i	0.861057i
$438$	− 12.0000i	− 0.573382i
$439$	−15.0000	−0.715911	−0.357955	−	0.933739i	$-0.616526\pi$
$439$	−15.0000	−0.715911	−0.357955	+	0.933739i	$0.616526\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	− 14.0000i	− 0.665912i
$443$	− 1.00000i	− 0.0475114i	−0.999718	−	0.0237557i	$-0.992438\pi$
$443$	− 1.00000i	− 0.0475114i	0.999718	−	0.0237557i	$-0.00756239\pi$
$444$	−14.0000	−0.664411
$445$	0	0
$446$	−16.0000	−0.757622
$447$	− 15.0000i	− 0.709476i
$448$	8.00000i	0.377964i
$449$	−9.00000	−0.424736	−0.212368	−	0.977190i	$-0.568118\pi$
$449$	−9.00000	−0.424736	−0.212368	+	0.977190i	$0.568118\pi$
$450$	0	0
$451$	45.0000	2.11897
$452$	− 20.0000i	− 0.940721i
$453$	− 8.00000i	− 0.375873i
$454$	4.00000	0.187729
$455$	0	0
$456$	0	0
$457$	7.00000i	0.327446i	0.986506	+	0.163723i	$0.0523504\pi$
$457$	7.00000i	0.327446i	−0.986506	+	0.163723i	$0.947650\pi$
$458$	28.0000i	1.30835i
$459$	7.00000	0.326732
$460$	0	0
$461$	37.0000	1.72326	0.861631	−	0.507535i	$-0.169443\pi$
$461$	37.0000	1.72326	0.861631	+	0.507535i	$0.169443\pi$
$462$	10.0000i	0.465242i
$463$	15.0000i	0.697109i	0.937288	+	0.348555i	$0.113327\pi$
$463$	15.0000i	0.697109i	−0.937288	+	0.348555i	$0.886673\pi$
$464$	8.00000	0.371391
$465$	0	0
$466$	38.0000	1.76032
$467$	1.00000i	0.0462745i	0.999732	+	0.0231372i	$0.00736547\pi$
$467$	1.00000i	0.0462745i	−0.999732	+	0.0231372i	$0.992635\pi$
$468$	− 2.00000i	− 0.0924500i
$469$	−4.00000	−0.184703
$470$	0	0
$471$	18.0000	0.829396
$472$	0	0
$473$	− 40.0000i	− 1.83920i
$474$	6.00000	0.275589
$475$	0	0
$476$	14.0000	0.641689
$477$	− 5.00000i	− 0.228934i
$478$	18.0000i	0.823301i
$479$	−3.00000	−0.137073	−0.0685367	−	0.997649i	$-0.521833\pi$
$479$	−3.00000	−0.137073	−0.0685367	+	0.997649i	$0.521833\pi$
$480$	0	0
$481$	−7.00000	−0.319173
$482$	− 44.0000i	− 2.00415i
$483$	− 3.00000i	− 0.136505i
$484$	−28.0000	−1.27273
$485$	0	0
$486$	2.00000	0.0907218
$487$	− 5.00000i	− 0.226572i	−0.993562	−	0.113286i	$-0.963862\pi$
$487$	− 5.00000i	− 0.226572i	0.993562	−	0.113286i	$-0.0361376\pi$
$488$	0	0
$489$	−15.0000	−0.678323
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	− 18.0000i	− 0.811503i
$493$	− 14.0000i	− 0.630528i
$494$	−12.0000	−0.539906
$495$	0	0
$496$	−8.00000	−0.359211
$497$	9.00000i	0.403705i
$498$	− 8.00000i	− 0.358489i
$499$	14.0000	0.626726	0.313363	−	0.949633i	$-0.398544\pi$
$499$	14.0000	0.626726	0.313363	+	0.949633i	$0.398544\pi$
$500$	0	0
$501$	−24.0000	−1.07224
$502$	0	0
$503$	36.0000i	1.60516i	0.596544	+	0.802580i	$0.296540\pi$
$503$	36.0000i	1.60516i	−0.596544	+	0.802580i	$0.703460\pi$
$504$	0	0
$505$	0	0
$506$	30.0000	1.33366
$507$	− 1.00000i	− 0.0444116i
$508$	− 4.00000i	− 0.177471i
$509$	−27.0000	−1.19675	−0.598377	−	0.801215i	$-0.704187\pi$
$509$	−27.0000	−1.19675	−0.598377	+	0.801215i	$0.704187\pi$
$510$	0	0
$511$	6.00000	0.265424
$512$	− 32.0000i	− 1.41421i
$513$	− 6.00000i	− 0.264906i
$514$	−4.00000	−0.176432
$515$	0	0
$516$	−16.0000	−0.704361
$517$	− 50.0000i	− 2.19900i
$518$	− 14.0000i	− 0.615125i
$519$	18.0000	0.790112
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	− 4.00000i	− 0.175075i
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	44.0000	1.92215
$525$	0	0
$526$	−32.0000	−1.39527
$527$	14.0000i	0.609850i
$528$	− 20.0000i	− 0.870388i
$529$	14.0000	0.608696
$530$	0	0
$531$	0	0
$532$	− 12.0000i	− 0.520266i
$533$	− 9.00000i	− 0.389833i
$534$	−22.0000	−0.952033
$535$	0	0
$536$	0	0
$537$	− 6.00000i	− 0.258919i
$538$	− 48.0000i	− 2.06943i
$539$	30.0000	1.29219
$540$	0	0
$541$	38.0000	1.63375	0.816874	−	0.576816i	$-0.195705\pi$
$541$	38.0000	1.63375	0.816874	+	0.576816i	$0.195705\pi$
$542$	44.0000i	1.88996i
$543$	− 7.00000i	− 0.300399i
$544$	−56.0000	−2.40098
$545$	0	0
$546$	2.00000	0.0855921
$547$	− 36.0000i	− 1.53925i	−0.638497	−	0.769624i	$-0.720443\pi$
$547$	− 36.0000i	− 1.53925i	0.638497	−	0.769624i	$-0.279557\pi$
$548$	− 28.0000i	− 1.19610i
$549$	−5.00000	−0.213395
$550$	0	0
$551$	−12.0000	−0.511217
$552$	0	0
$553$	3.00000i	0.127573i
$554$	4.00000	0.169944
$555$	0	0
$556$	30.0000	1.27228
$557$	2.00000i	0.0847427i	0.999102	+	0.0423714i	$0.0134913\pi$
$557$	2.00000i	0.0847427i	−0.999102	+	0.0423714i	$0.986509\pi$
$558$	4.00000i	0.169334i
$559$	−8.00000	−0.338364
$560$	0	0
$561$	−35.0000	−1.47770
$562$	− 36.0000i	− 1.51857i
$563$	11.0000i	0.463595i	0.972764	+	0.231797i	$0.0744606\pi$
$563$	11.0000i	0.463595i	−0.972764	+	0.231797i	$0.925539\pi$
$564$	−20.0000	−0.842152
$565$	0	0
$566$	40.0000	1.68133
$567$	1.00000i	0.0419961i
$568$	0	0
$569$	20.0000	0.838444	0.419222	−	0.907884i	$-0.362303\pi$
$569$	20.0000	0.838444	0.419222	+	0.907884i	$0.362303\pi$
$570$	0	0
$571$	−39.0000	−1.63210	−0.816050	−	0.577982i	$-0.803840\pi$
$571$	−39.0000	−1.63210	−0.816050	+	0.577982i	$0.803840\pi$
$572$	10.0000i	0.418121i
$573$	− 12.0000i	− 0.501307i
$574$	18.0000	0.751305
$575$	0	0
$576$	−8.00000	−0.333333
$577$	7.00000i	0.291414i	0.989328	+	0.145707i	$0.0465456\pi$
$577$	7.00000i	0.291414i	−0.989328	+	0.145707i	$0.953454\pi$
$578$	64.0000i	2.66205i
$579$	17.0000	0.706496
$580$	0	0
$581$	4.00000	0.165948
$582$	22.0000i	0.911929i
$583$	25.0000i	1.03539i
$584$	0	0
$585$	0	0
$586$	8.00000	0.330477
$587$	2.00000i	0.0825488i	0.999148	+	0.0412744i	$0.0131418\pi$
$587$	2.00000i	0.0825488i	−0.999148	+	0.0412744i	$0.986858\pi$
$588$	− 12.0000i	− 0.494872i
$589$	12.0000	0.494451
$590$	0	0
$591$	24.0000	0.987228
$592$	28.0000i	1.15079i
$593$	− 16.0000i	− 0.657041i	−0.944497	−	0.328521i	$-0.893450\pi$
$593$	− 16.0000i	− 0.657041i	0.944497	−	0.328521i	$-0.106550\pi$
$594$	−10.0000	−0.410305
$595$	0	0
$596$	30.0000	1.22885
$597$	28.0000i	1.14596i
$598$	− 6.00000i	− 0.245358i
$599$	−20.0000	−0.817178	−0.408589	−	0.912719i	$-0.633979\pi$
$599$	−20.0000	−0.817178	−0.408589	+	0.912719i	$0.633979\pi$
$600$	0	0
$601$	−5.00000	−0.203954	−0.101977	−	0.994787i	$-0.532517\pi$
$601$	−5.00000	−0.203954	−0.101977	+	0.994787i	$0.532517\pi$
$602$	− 16.0000i	− 0.652111i
$603$	− 4.00000i	− 0.162893i
$604$	16.0000	0.651031
$605$	0	0
$606$	0	0
$607$	32.0000i	1.29884i	0.760430	+	0.649420i	$0.224988\pi$
$607$	32.0000i	1.29884i	−0.760430	+	0.649420i	$0.775012\pi$
$608$	48.0000i	1.94666i
$609$	2.00000	0.0810441
$610$	0	0
$611$	−10.0000	−0.404557
$612$	14.0000i	0.565916i
$613$	9.00000i	0.363507i	0.983344	+	0.181753i	$0.0581772\pi$
$613$	9.00000i	0.363507i	−0.983344	+	0.181753i	$0.941823\pi$
$614$	−46.0000	−1.85641
$615$	0	0
$616$	0	0
$617$	− 18.0000i	− 0.724653i	−0.932051	−	0.362326i	$-0.881983\pi$
$617$	− 18.0000i	− 0.724653i	0.932051	−	0.362326i	$-0.118017\pi$
$618$	− 8.00000i	− 0.321807i
$619$	38.0000	1.52735	0.763674	−	0.645601i	$-0.223393\pi$
$619$	38.0000	1.52735	0.763674	+	0.645601i	$0.223393\pi$
$620$	0	0
$621$	3.00000	0.120386
$622$	− 40.0000i	− 1.60385i
$623$	− 11.0000i	− 0.440706i
$624$	−4.00000	−0.160128
$625$	0	0
$626$	−44.0000	−1.75859
$627$	30.0000i	1.19808i
$628$	36.0000i	1.43656i
$629$	49.0000	1.95376
$630$	0	0
$631$	−20.0000	−0.796187	−0.398094	−	0.917345i	$-0.630328\pi$
$631$	−20.0000	−0.796187	−0.398094	+	0.917345i	$0.630328\pi$
$632$	0	0
$633$	− 12.0000i	− 0.476957i
$634$	48.0000	1.90632
$635$	0	0
$636$	10.0000	0.396526
$637$	− 6.00000i	− 0.237729i
$638$	20.0000i	0.791808i
$639$	−9.00000	−0.356034
$640$	0	0
$641$	20.0000	0.789953	0.394976	−	0.918691i	$-0.370753\pi$
$641$	20.0000	0.789953	0.394976	+	0.918691i	$0.370753\pi$
$642$	34.0000i	1.34187i
$643$	− 37.0000i	− 1.45914i	−0.683907	−	0.729569i	$-0.739721\pi$
$643$	− 37.0000i	− 1.45914i	0.683907	−	0.729569i	$-0.260279\pi$
$644$	6.00000	0.236433
$645$	0	0
$646$	84.0000	3.30494
$647$	− 17.0000i	− 0.668339i	−0.942513	−	0.334169i	$-0.891544\pi$
$647$	− 17.0000i	− 0.668339i	0.942513	−	0.334169i	$-0.108456\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−2.00000	−0.0783862
$652$	− 30.0000i	− 1.17489i
$653$	18.0000i	0.704394i	0.935926	+	0.352197i	$0.114565\pi$
$653$	18.0000i	0.704394i	−0.935926	+	0.352197i	$0.885435\pi$
$654$	−8.00000	−0.312825
$655$	0	0
$656$	−36.0000	−1.40556
$657$	6.00000i	0.234082i
$658$	− 20.0000i	− 0.779681i
$659$	−24.0000	−0.934907	−0.467454	−	0.884018i	$-0.654829\pi$
$659$	−24.0000	−0.934907	−0.467454	+	0.884018i	$0.654829\pi$
$660$	0	0
$661$	−40.0000	−1.55582	−0.777910	−	0.628376i	$-0.783720\pi$
$661$	−40.0000	−1.55582	−0.777910	+	0.628376i	$0.783720\pi$
$662$	0	0
$663$	7.00000i	0.271857i
$664$	0	0
$665$	0	0
$666$	14.0000	0.542489
$667$	− 6.00000i	− 0.232321i
$668$	− 48.0000i	− 1.85718i
$669$	8.00000	0.309298
$670$	0	0
$671$	25.0000	0.965114
$672$	− 8.00000i	− 0.308607i
$673$	42.0000i	1.61898i	0.587133	+	0.809491i	$0.300257\pi$
$673$	42.0000i	1.61898i	−0.587133	+	0.809491i	$0.699743\pi$
$674$	−16.0000	−0.616297
$675$	0	0
$676$	2.00000	0.0769231
$677$	− 21.0000i	− 0.807096i	−0.914959	−	0.403548i	$-0.867777\pi$
$677$	− 21.0000i	− 0.807096i	0.914959	−	0.403548i	$-0.132223\pi$
$678$	20.0000i	0.768095i
$679$	−11.0000	−0.422141
$680$	0	0
$681$	−2.00000	−0.0766402
$682$	− 20.0000i	− 0.765840i
$683$	16.0000i	0.612223i	0.951996	+	0.306111i	$0.0990280\pi$
$683$	16.0000i	0.612223i	−0.951996	+	0.306111i	$0.900972\pi$
$684$	12.0000	0.458831
$685$	0	0
$686$	26.0000	0.992685
$687$	− 14.0000i	− 0.534133i
$688$	32.0000i	1.21999i
$689$	5.00000	0.190485
$690$	0	0
$691$	10.0000	0.380418	0.190209	−	0.981744i	$-0.439083\pi$
$691$	10.0000	0.380418	0.190209	+	0.981744i	$0.439083\pi$
$692$	36.0000i	1.36851i
$693$	− 5.00000i	− 0.189934i
$694$	−22.0000	−0.835109
$695$	0	0
$696$	0	0
$697$	63.0000i	2.38630i
$698$	48.0000i	1.81683i
$699$	−19.0000	−0.718646
$700$	0	0
$701$	28.0000	1.05755	0.528773	−	0.848763i	$-0.322652\pi$
$701$	28.0000	1.05755	0.528773	+	0.848763i	$0.322652\pi$
$702$	2.00000i	0.0754851i
$703$	− 42.0000i	− 1.58406i
$704$	40.0000	1.50756
$705$	0	0
$706$	−12.0000	−0.451626
$707$	0	0
$708$	0	0
$709$	44.0000	1.65245	0.826227	−	0.563337i	$-0.190483\pi$
$709$	44.0000	1.65245	0.826227	+	0.563337i	$0.190483\pi$
$710$	0	0
$711$	−3.00000	−0.112509
$712$	0	0
$713$	6.00000i	0.224702i
$714$	−14.0000	−0.523937
$715$	0	0
$716$	12.0000	0.448461
$717$	− 9.00000i	− 0.336111i
$718$	− 32.0000i	− 1.19423i
$719$	−16.0000	−0.596699	−0.298350	−	0.954457i	$-0.596436\pi$
$719$	−16.0000	−0.596699	−0.298350	+	0.954457i	$0.596436\pi$
$720$	0	0
$721$	4.00000	0.148968
$722$	− 34.0000i	− 1.26535i
$723$	22.0000i	0.818189i
$724$	14.0000	0.520306
$725$	0	0
$726$	28.0000	1.03918
$727$	18.0000i	0.667583i	0.942647	+	0.333792i	$0.108328\pi$
$727$	18.0000i	0.667583i	−0.942647	+	0.333792i	$0.891672\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	56.0000	2.07123
$732$	− 10.0000i	− 0.369611i
$733$	43.0000i	1.58824i	0.607760	+	0.794121i	$0.292068\pi$
$733$	43.0000i	1.58824i	−0.607760	+	0.794121i	$0.707932\pi$
$734$	−8.00000	−0.295285
$735$	0	0
$736$	−24.0000	−0.884652
$737$	20.0000i	0.736709i
$738$	18.0000i	0.662589i
$739$	6.00000	0.220714	0.110357	−	0.993892i	$-0.464801\pi$
$739$	6.00000	0.220714	0.110357	+	0.993892i	$0.464801\pi$
$740$	0	0
$741$	6.00000	0.220416
$742$	10.0000i	0.367112i
$743$	30.0000i	1.10059i	0.834969	+	0.550297i	$0.185485\pi$
$743$	30.0000i	1.10059i	−0.834969	+	0.550297i	$0.814515\pi$
$744$	0	0
$745$	0	0
$746$	−64.0000	−2.34321
$747$	4.00000i	0.146352i
$748$	− 70.0000i	− 2.55945i
$749$	−17.0000	−0.621166
$750$	0	0
$751$	−13.0000	−0.474377	−0.237188	−	0.971464i	$-0.576226\pi$
$751$	−13.0000	−0.474377	−0.237188	+	0.971464i	$0.576226\pi$
$752$	40.0000i	1.45865i
$753$	0	0
$754$	4.00000	0.145671
$755$	0	0
$756$	−2.00000	−0.0727393
$757$	− 20.0000i	− 0.726912i	−0.931611	−	0.363456i	$-0.881597\pi$
$757$	− 20.0000i	− 0.726912i	0.931611	−	0.363456i	$-0.118403\pi$
$758$	20.0000i	0.726433i
$759$	−15.0000	−0.544466
$760$	0	0
$761$	−42.0000	−1.52250	−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000	−1.52250	−0.761249	+	0.648459i	$0.775414\pi$
$762$	4.00000i	0.144905i
$763$	− 4.00000i	− 0.144810i
$764$	24.0000	0.868290
$765$	0	0
$766$	−12.0000	−0.433578
$767$	0	0
$768$	16.0000i	0.577350i
$769$	28.0000	1.00971	0.504853	−	0.863205i	$-0.331547\pi$
$769$	28.0000	1.00971	0.504853	+	0.863205i	$0.331547\pi$
$770$	0	0
$771$	2.00000	0.0720282
$772$	34.0000i	1.22369i
$773$	− 20.0000i	− 0.719350i	−0.933078	−	0.359675i	$-0.882888\pi$
$773$	− 20.0000i	− 0.719350i	0.933078	−	0.359675i	$-0.117112\pi$
$774$	16.0000	0.575108
$775$	0	0
$776$	0	0
$777$	7.00000i	0.251124i
$778$	− 8.00000i	− 0.286814i
$779$	54.0000	1.93475
$780$	0	0
$781$	45.0000	1.61023
$782$	42.0000i	1.50192i
$783$	2.00000i	0.0714742i
$784$	−24.0000	−0.857143
$785$	0	0
$786$	−44.0000	−1.56943
$787$	− 44.0000i	− 1.56843i	−0.620489	−	0.784215i	$-0.713066\pi$
$787$	− 44.0000i	− 1.56843i	0.620489	−	0.784215i	$-0.286934\pi$
$788$	48.0000i	1.70993i
$789$	16.0000	0.569615
$790$	0	0
$791$	−10.0000	−0.355559
$792$	0	0
$793$	− 5.00000i	− 0.177555i
$794$	−30.0000	−1.06466
$795$	0	0
$796$	−56.0000	−1.98487
$797$	− 39.0000i	− 1.38145i	−0.723117	−	0.690725i	$-0.757291\pi$
$797$	− 39.0000i	− 1.38145i	0.723117	−	0.690725i	$-0.242709\pi$
$798$	12.0000i	0.424795i
$799$	70.0000	2.47642
$800$	0	0
$801$	11.0000	0.388666
$802$	28.0000i	0.988714i
$803$	− 30.0000i	− 1.05868i
$804$	8.00000	0.282138
$805$	0	0
$806$	−4.00000	−0.140894
$807$	24.0000i	0.844840i
$808$	0	0
$809$	50.0000	1.75791	0.878953	−	0.476908i	$-0.158243\pi$
$809$	50.0000	1.75791	0.878953	+	0.476908i	$0.158243\pi$
$810$	0	0
$811$	24.0000	0.842754	0.421377	−	0.906886i	$-0.361547\pi$
$811$	24.0000	0.842754	0.421377	+	0.906886i	$0.361547\pi$
$812$	4.00000i	0.140372i
$813$	− 22.0000i	− 0.771574i
$814$	−70.0000	−2.45350
$815$	0	0
$816$	28.0000	0.980196
$817$	− 48.0000i	− 1.67931i
$818$	60.0000i	2.09785i
$819$	−1.00000	−0.0349428
$820$	0	0
$821$	17.0000	0.593304	0.296652	−	0.954986i	$-0.404130\pi$
$821$	17.0000	0.593304	0.296652	+	0.954986i	$0.404130\pi$
$822$	28.0000i	0.976612i
$823$	− 4.00000i	− 0.139431i	−0.997567	−	0.0697156i	$-0.977791\pi$
$823$	− 4.00000i	− 0.139431i	0.997567	−	0.0697156i	$-0.0222092\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	26.0000i	0.904109i	0.891990	+	0.452054i	$0.149309\pi$
$827$	26.0000i	0.904109i	−0.891990	+	0.452054i	$0.850691\pi$
$828$	6.00000i	0.208514i
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	0	0
$831$	−2.00000	−0.0693792
$832$	− 8.00000i	− 0.277350i
$833$	42.0000i	1.45521i
$834$	−30.0000	−1.03882
$835$	0	0
$836$	−60.0000	−2.07514
$837$	− 2.00000i	− 0.0691301i
$838$	52.0000i	1.79631i
$839$	−17.0000	−0.586905	−0.293453	−	0.955974i	$-0.594804\pi$
$839$	−17.0000	−0.586905	−0.293453	+	0.955974i	$0.594804\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	40.0000i	1.37849i
$843$	18.0000i	0.619953i
$844$	24.0000	0.826114
$845$	0	0
$846$	20.0000	0.687614
$847$	14.0000i	0.481046i
$848$	− 20.0000i	− 0.686803i
$849$	−20.0000	−0.686398
$850$	0	0
$851$	21.0000	0.719871
$852$	− 18.0000i	− 0.616670i
$853$	− 39.0000i	− 1.33533i	−0.744460	−	0.667667i	$-0.767293\pi$
$853$	− 39.0000i	− 1.33533i	0.744460	−	0.667667i	$-0.232707\pi$
$854$	10.0000	0.342193
$855$	0	0
$856$	0	0
$857$	45.0000i	1.53717i	0.639747	+	0.768585i	$0.279039\pi$
$857$	45.0000i	1.53717i	−0.639747	+	0.768585i	$0.720961\pi$
$858$	− 10.0000i	− 0.341394i
$859$	−19.0000	−0.648272	−0.324136	−	0.946011i	$-0.605073\pi$
$859$	−19.0000	−0.648272	−0.324136	+	0.946011i	$0.605073\pi$
$860$	0	0
$861$	−9.00000	−0.306719
$862$	80.0000i	2.72481i
$863$	6.00000i	0.204242i	0.994772	+	0.102121i	$0.0325630\pi$
$863$	6.00000i	0.204242i	−0.994772	+	0.102121i	$0.967437\pi$
$864$	8.00000	0.272166
$865$	0	0
$866$	−40.0000	−1.35926
$867$	− 32.0000i	− 1.08678i
$868$	− 4.00000i	− 0.135769i
$869$	15.0000	0.508840
$870$	0	0
$871$	4.00000	0.135535
$872$	0	0
$873$	− 11.0000i	− 0.372294i
$874$	36.0000	1.21772
$875$	0	0
$876$	−12.0000	−0.405442
$877$	− 18.0000i	− 0.607817i	−0.952701	−	0.303908i	$-0.901708\pi$
$877$	− 18.0000i	− 0.607817i	0.952701	−	0.303908i	$-0.0982917\pi$
$878$	30.0000i	1.01245i
$879$	−4.00000	−0.134917
$880$	0	0
$881$	14.0000	0.471672	0.235836	−	0.971793i	$-0.424217\pi$
$881$	14.0000	0.471672	0.235836	+	0.971793i	$0.424217\pi$
$882$	12.0000i	0.404061i
$883$	44.0000i	1.48072i	0.672212	+	0.740359i	$0.265344\pi$
$883$	44.0000i	1.48072i	−0.672212	+	0.740359i	$0.734656\pi$
$884$	−14.0000	−0.470871
$885$	0	0
$886$	−2.00000	−0.0671913
$887$	− 13.0000i	− 0.436497i	−0.975893	−	0.218249i	$-0.929966\pi$
$887$	− 13.0000i	− 0.436497i	0.975893	−	0.218249i	$-0.0700344\pi$
$888$	0	0
$889$	−2.00000	−0.0670778
$890$	0	0
$891$	5.00000	0.167506
$892$	16.0000i	0.535720i
$893$	− 60.0000i	− 2.00782i
$894$	−30.0000	−1.00335
$895$	0	0
$896$	0	0
$897$	3.00000i	0.100167i
$898$	18.0000i	0.600668i
$899$	−4.00000	−0.133407
$900$	0	0
$901$	−35.0000	−1.16602
$902$	− 90.0000i	− 2.99667i
$903$	8.00000i	0.266223i
$904$	0	0
$905$	0	0
$906$	−16.0000	−0.531564
$907$	− 38.0000i	− 1.26177i	−0.775877	−	0.630885i	$-0.782692\pi$
$907$	− 38.0000i	− 1.26177i	0.775877	−	0.630885i	$-0.217308\pi$
$908$	− 4.00000i	− 0.132745i
$909$	0	0
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	− 24.0000i	− 0.794719i
$913$	− 20.0000i	− 0.661903i
$914$	14.0000	0.463079
$915$	0	0
$916$	28.0000	0.925146
$917$	− 22.0000i	− 0.726504i
$918$	− 14.0000i	− 0.462069i
$919$	19.0000	0.626752	0.313376	−	0.949629i	$-0.398540\pi$
$919$	19.0000	0.626752	0.313376	+	0.949629i	$0.398540\pi$
$920$	0	0
$921$	23.0000	0.757876
$922$	− 74.0000i	− 2.43706i
$923$	− 9.00000i	− 0.296239i
$924$	10.0000	0.328976
$925$	0	0
$926$	30.0000	0.985861
$927$	4.00000i	0.131377i
$928$	− 16.0000i	− 0.525226i
$929$	29.0000	0.951459	0.475730	−	0.879592i	$-0.342184\pi$
$929$	29.0000	0.951459	0.475730	+	0.879592i	$0.342184\pi$
$930$	0	0
$931$	36.0000	1.17985
$932$	− 38.0000i	− 1.24473i
$933$	20.0000i	0.654771i
$934$	2.00000	0.0654420
$935$	0	0
$936$	0	0
$937$	6.00000i	0.196011i	0.995186	+	0.0980057i	$0.0312463\pi$
$937$	6.00000i	0.196011i	−0.995186	+	0.0980057i	$0.968754\pi$
$938$	8.00000i	0.261209i
$939$	22.0000	0.717943
$940$	0	0
$941$	15.0000	0.488986	0.244493	−	0.969651i	$-0.421378\pi$
$941$	15.0000	0.488986	0.244493	+	0.969651i	$0.421378\pi$
$942$	− 36.0000i	− 1.17294i
$943$	27.0000i	0.879241i
$944$	0	0
$945$	0	0
$946$	−80.0000	−2.60102
$947$	− 12.0000i	− 0.389948i	−0.980808	−	0.194974i	$-0.937538\pi$
$947$	− 12.0000i	− 0.389948i	0.980808	−	0.194974i	$-0.0624622\pi$
$948$	− 6.00000i	− 0.194871i
$949$	−6.00000	−0.194768
$950$	0	0
$951$	−24.0000	−0.778253
$952$	0	0
$953$	− 15.0000i	− 0.485898i	−0.970039	−	0.242949i	$-0.921885\pi$
$953$	− 15.0000i	− 0.485898i	0.970039	−	0.242949i	$-0.0781147\pi$
$954$	−10.0000	−0.323762
$955$	0	0
$956$	18.0000	0.582162
$957$	− 10.0000i	− 0.323254i
$958$	6.00000i	0.193851i
$959$	−14.0000	−0.452084
$960$	0	0
$961$	−27.0000	−0.870968
$962$	14.0000i	0.451378i
$963$	− 17.0000i	− 0.547817i
$964$	−44.0000	−1.41714
$965$	0	0
$966$	−6.00000	−0.193047
$967$	8.00000i	0.257263i	0.991692	+	0.128631i	$0.0410584\pi$
$967$	8.00000i	0.257263i	−0.991692	+	0.128631i	$0.958942\pi$
$968$	0	0
$969$	−42.0000	−1.34923
$970$	0	0
$971$	−28.0000	−0.898563	−0.449281	−	0.893390i	$-0.648320\pi$
$971$	−28.0000	−0.898563	−0.449281	+	0.893390i	$0.648320\pi$
$972$	− 2.00000i	− 0.0641500i
$973$	− 15.0000i	− 0.480878i
$974$	−10.0000	−0.320421
$975$	0	0
$976$	−20.0000	−0.640184
$977$	12.0000i	0.383914i	0.981403	+	0.191957i	$0.0614834\pi$
$977$	12.0000i	0.383914i	−0.981403	+	0.191957i	$0.938517\pi$
$978$	30.0000i	0.959294i
$979$	−55.0000	−1.75781
$980$	0	0
$981$	4.00000	0.127710
$982$	0	0
$983$	− 4.00000i	− 0.127580i	−0.997963	−	0.0637901i	$-0.979681\pi$
$983$	− 4.00000i	− 0.127580i	0.997963	−	0.0637901i	$-0.0203188\pi$
$984$	0	0
$985$	0	0
$986$	−28.0000	−0.891702
$987$	10.0000i	0.318304i
$988$	12.0000i	0.381771i
$989$	24.0000	0.763156
$990$	0	0
$991$	−57.0000	−1.81066	−0.905332	−	0.424704i	$-0.860378\pi$
$991$	−57.0000	−1.81066	−0.905332	+	0.424704i	$0.860378\pi$
$992$	16.0000i	0.508001i
$993$	0	0
$994$	18.0000	0.570925
$995$	0	0
$996$	−8.00000	−0.253490
$997$	− 8.00000i	− 0.253363i	−0.991943	−	0.126681i	$-0.959567\pi$
$997$	− 8.00000i	− 0.253363i	0.991943	−	0.126681i	$-0.0404325\pi$
$998$	− 28.0000i	− 0.886325i
$999$	−7.00000	−0.221470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	975.2.c.c.274.1		2
3.2	odd	2		2925.2.c.a.2224.2		2
5.2	odd	4		195.2.a.c.1.1	✓	1
5.3	odd	4		975.2.a.a.1.1		1
5.4	even	2	inner	975.2.c.c.274.2		2
15.2	even	4		585.2.a.c.1.1		1
15.8	even	4		2925.2.a.s.1.1		1
15.14	odd	2		2925.2.c.a.2224.1		2
20.7	even	4		3120.2.a.d.1.1		1
35.27	even	4		9555.2.a.u.1.1		1
60.47	odd	4		9360.2.a.bv.1.1		1
65.12	odd	4		2535.2.a.d.1.1		1
195.77	even	4		7605.2.a.t.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
195.2.a.c.1.1	✓	1	5.2	odd	4
585.2.a.c.1.1		1	15.2	even	4
975.2.a.a.1.1		1	5.3	odd	4
975.2.c.c.274.1		2	1.1	even	1	trivial
975.2.c.c.274.2		2	5.4	even	2	inner
2535.2.a.d.1.1		1	65.12	odd	4
2925.2.a.s.1.1		1	15.8	even	4
2925.2.c.a.2224.1		2	15.14	odd	2
2925.2.c.a.2224.2		2	3.2	odd	2
3120.2.a.d.1.1		1	20.7	even	4
7605.2.a.t.1.1		1	195.77	even	4
9360.2.a.bv.1.1		1	60.47	odd	4
9555.2.a.u.1.1		1	35.27	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:	\( N \)	\(=\)	\( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	975.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} +1.00000i q^{3} -2.00000 q^{4} +2.00000 q^{6} +1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-2.00000i q^{2} +1.00000i q^{3} -2.00000 q^{4} +2.00000 q^{6} +1.00000i q^{7} -1.00000 q^{9} +5.00000 q^{11} -2.00000i q^{12} -1.00000i q^{13} +2.00000 q^{14} -4.00000 q^{16} +7.00000i q^{17} +2.00000i q^{18} +6.00000 q^{19} -1.00000 q^{21} -10.0000i q^{22} +3.00000i q^{23} -2.00000 q^{26} -1.00000i q^{27} -2.00000i q^{28} -2.00000 q^{29} +2.00000 q^{31} +8.00000i q^{32} +5.00000i q^{33} +14.0000 q^{34} +2.00000 q^{36} -7.00000i q^{37} -12.0000i q^{38} +1.00000 q^{39} +9.00000 q^{41} +2.00000i q^{42} -8.00000i q^{43} -10.0000 q^{44} +6.00000 q^{46} -10.0000i q^{47} -4.00000i q^{48} +6.00000 q^{49} -7.00000 q^{51} +2.00000i q^{52} +5.00000i q^{53} -2.00000 q^{54} +6.00000i q^{57} +4.00000i q^{58} +5.00000 q^{61} -4.00000i q^{62} -1.00000i q^{63} +8.00000 q^{64} +10.0000 q^{66} +4.00000i q^{67} -14.0000i q^{68} -3.00000 q^{69} +9.00000 q^{71} -6.00000i q^{73} -14.0000 q^{74} -12.0000 q^{76} +5.00000i q^{77} -2.00000i q^{78} +3.00000 q^{79} +1.00000 q^{81} -18.0000i q^{82} -4.00000i q^{83} +2.00000 q^{84} -16.0000 q^{86} -2.00000i q^{87} -11.0000 q^{89} +1.00000 q^{91} -6.00000i q^{92} +2.00000i q^{93} -20.0000 q^{94} -8.00000 q^{96} +11.0000i q^{97} -12.0000i q^{98} -5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4} + 4 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 4 * q^4 + 4 * q^6 - 2 * q^9	\( 2 q - 4 q^{4} + 4 q^{6} - 2 q^{9} + 10 q^{11} + 4 q^{14} - 8 q^{16} + 12 q^{19} - 2 q^{21} - 4 q^{26} - 4 q^{29} + 4 q^{31} + 28 q^{34} + 4 q^{36} + 2 q^{39} + 18 q^{41} - 20 q^{44} + 12 q^{46} + 12 q^{49} - 14 q^{51} - 4 q^{54} + 10 q^{61} + 16 q^{64} + 20 q^{66} - 6 q^{69} + 18 q^{71} - 28 q^{74} - 24 q^{76} + 6 q^{79} + 2 q^{81} + 4 q^{84} - 32 q^{86} - 22 q^{89} + 2 q^{91} - 40 q^{94} - 16 q^{96} - 10 q^{99}+O(q^{100}) \) 2 * q - 4 * q^4 + 4 * q^6 - 2 * q^9 + 10 * q^11 + 4 * q^14 - 8 * q^16 + 12 * q^19 - 2 * q^21 - 4 * q^26 - 4 * q^29 + 4 * q^31 + 28 * q^34 + 4 * q^36 + 2 * q^39 + 18 * q^41 - 20 * q^44 + 12 * q^46 + 12 * q^49 - 14 * q^51 - 4 * q^54 + 10 * q^61 + 16 * q^64 + 20 * q^66 - 6 * q^69 + 18 * q^71 - 28 * q^74 - 24 * q^76 + 6 * q^79 + 2 * q^81 + 4 * q^84 - 32 * q^86 - 22 * q^89 + 2 * q^91 - 40 * q^94 - 16 * q^96 - 10 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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