LMFDB - Embedded newform 1850.2.b.c.149.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1850,2,Mod(149,1850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1850, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1850.149");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1850 = 2 \cdot 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1850.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:	$14.7723243739$
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 370)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			149.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1850.149
Dual form			1850.2.b.c.149.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} +1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} +1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +3.00000 q^{11} +2.00000i q^{12} +1.00000 q^{14} +1.00000 q^{16} +3.00000i q^{17} +1.00000i q^{18} +6.00000 q^{19} +2.00000 q^{21} -3.00000i q^{22} -2.00000i q^{23} +2.00000 q^{24} -4.00000i q^{27} -1.00000i q^{28} +3.00000 q^{29} +3.00000 q^{31} -1.00000i q^{32} -6.00000i q^{33} +3.00000 q^{34} +1.00000 q^{36} -1.00000i q^{37} -6.00000i q^{38} +3.00000 q^{41} -2.00000i q^{42} +1.00000i q^{43} -3.00000 q^{44} -2.00000 q^{46} +4.00000i q^{47} -2.00000i q^{48} +6.00000 q^{49} +6.00000 q^{51} -13.0000i q^{53} -4.00000 q^{54} -1.00000 q^{56} -12.0000i q^{57} -3.00000i q^{58} -15.0000 q^{61} -3.00000i q^{62} -1.00000i q^{63} -1.00000 q^{64} -6.00000 q^{66} -3.00000i q^{68} -4.00000 q^{69} -2.00000 q^{71} -1.00000i q^{72} -1.00000 q^{74} -6.00000 q^{76} +3.00000i q^{77} +8.00000 q^{79} -11.0000 q^{81} -3.00000i q^{82} +4.00000i q^{83} -2.00000 q^{84} +1.00000 q^{86} -6.00000i q^{87} +3.00000i q^{88} +18.0000 q^{89} +2.00000i q^{92} -6.00000i q^{93} +4.00000 q^{94} -2.00000 q^{96} -7.00000i q^{97} -6.00000i q^{98} -3.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 4 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 - 4 * q^6 - 2 * q^9	$ 2 q - 2 q^{4} - 4 q^{6} - 2 q^{9} + 6 q^{11} + 2 q^{14} + 2 q^{16} + 12 q^{19} + 4 q^{21} + 4 q^{24} + 6 q^{29} + 6 q^{31} + 6 q^{34} + 2 q^{36} + 6 q^{41} - 6 q^{44} - 4 q^{46} + 12 q^{49} + 12 q^{51} - 8 q^{54} - 2 q^{56} - 30 q^{61} - 2 q^{64} - 12 q^{66} - 8 q^{69} - 4 q^{71} - 2 q^{74} - 12 q^{76} + 16 q^{79} - 22 q^{81} - 4 q^{84} + 2 q^{86} + 36 q^{89} + 8 q^{94} - 4 q^{96} - 6 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 - 4 * q^6 - 2 * q^9 + 6 * q^11 + 2 * q^14 + 2 * q^16 + 12 * q^19 + 4 * q^21 + 4 * q^24 + 6 * q^29 + 6 * q^31 + 6 * q^34 + 2 * q^36 + 6 * q^41 - 6 * q^44 - 4 * q^46 + 12 * q^49 + 12 * q^51 - 8 * q^54 - 2 * q^56 - 30 * q^61 - 2 * q^64 - 12 * q^66 - 8 * q^69 - 4 * q^71 - 2 * q^74 - 12 * q^76 + 16 * q^79 - 22 * q^81 - 4 * q^84 + 2 * q^86 + 36 * q^89 + 8 * q^94 - 4 * q^96 - 6 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1850\mathbb{Z}\right)^\times$.

$n$	$1001$	$1777$
$\chi(n)$	$1$	$-1$

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
$2$	− 1.00000i	− 0.707107i
$3$	− 2.00000i	− 1.15470i	−0.816497	−	0.577350i	$-0.804087\pi$
$3$	− 2.00000i	− 1.15470i	0.816497	−	0.577350i	$-0.195913\pi$
$4$	−1.00000	−0.500000
$5$	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	$0.939481\pi$
$8$	1.00000i	0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	2.00000i	0.577350i
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	3.00000i	0.727607i	0.931476	+	0.363803i	$0.118522\pi$
$17$	3.00000i	0.727607i	−0.931476	+	0.363803i	$0.881478\pi$
$18$	1.00000i	0.235702i
$19$	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
$21$	2.00000	0.436436
$22$	− 3.00000i	− 0.639602i
$23$	− 2.00000i	− 0.417029i	−0.978019	−	0.208514i	$-0.933137\pi$
$23$	− 2.00000i	− 0.417029i	0.978019	−	0.208514i	$-0.0668628\pi$
$24$	2.00000	0.408248
$25$	0	0
$26$	0	0
$27$	− 4.00000i	− 0.769800i
$28$	− 1.00000i	− 0.188982i
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	− 1.00000i	− 0.176777i
$33$	− 6.00000i	− 1.04447i
$34$	3.00000	0.514496
$35$	0	0
$36$	1.00000	0.166667
$37$	− 1.00000i	− 0.164399i
$37$	− 1.00000i	− 0.164399i
$38$	− 6.00000i	− 0.973329i
$39$	0	0
$40$	0	0
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	− 2.00000i	− 0.308607i
$43$	1.00000i	0.152499i	0.997089	+	0.0762493i	$0.0242945\pi$
$43$	1.00000i	0.152499i	−0.997089	+	0.0762493i	$0.975706\pi$
$44$	−3.00000	−0.452267
$45$	0	0
$46$	−2.00000	−0.294884
$47$	4.00000i	0.583460i	0.956501	+	0.291730i	$0.0942309\pi$
$47$	4.00000i	0.583460i	−0.956501	+	0.291730i	$0.905769\pi$
$48$	− 2.00000i	− 0.288675i
$49$	6.00000	0.857143
$50$	0	0
$51$	6.00000	0.840168
$52$	0	0
$53$	− 13.0000i	− 1.78569i	−0.450367	−	0.892844i	$-0.648707\pi$
$53$	− 13.0000i	− 1.78569i	0.450367	−	0.892844i	$-0.351293\pi$
$54$	−4.00000	−0.544331
$55$	0	0
$56$	−1.00000	−0.133631
$57$	− 12.0000i	− 1.58944i
$58$	− 3.00000i	− 0.393919i
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−15.0000	−1.92055	−0.960277	−	0.279050i	$-0.909981\pi$
$61$	−15.0000	−1.92055	−0.960277	+	0.279050i	$0.909981\pi$
$62$	− 3.00000i	− 0.381000i
$63$	− 1.00000i	− 0.125988i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−6.00000	−0.738549
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	− 3.00000i	− 0.363803i
$69$	−4.00000	−0.481543
$70$	0	0
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	− 1.00000i	− 0.117851i
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	−1.00000	−0.116248
$75$	0	0
$76$	−6.00000	−0.688247
$77$	3.00000i	0.341882i
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	− 3.00000i	− 0.331295i
$83$	4.00000i	0.439057i	0.975606	+	0.219529i	$0.0704519\pi$
$83$	4.00000i	0.439057i	−0.975606	+	0.219529i	$0.929548\pi$
$84$	−2.00000	−0.218218
$85$	0	0
$86$	1.00000	0.107833
$87$	− 6.00000i	− 0.643268i
$88$	3.00000i	0.319801i
$89$	18.0000	1.90800	0.953998	−	0.299813i	$-0.0969242\pi$
$89$	18.0000	1.90800	0.953998	+	0.299813i	$0.0969242\pi$
$90$	0	0
$91$	0	0
$92$	2.00000i	0.208514i
$93$	− 6.00000i	− 0.622171i
$94$	4.00000	0.412568
$95$	0	0
$96$	−2.00000	−0.204124
$97$	− 7.00000i	− 0.710742i	−0.934725	−	0.355371i	$-0.884354\pi$
$97$	− 7.00000i	− 0.710742i	0.934725	−	0.355371i	$-0.115646\pi$
$98$	− 6.00000i	− 0.606092i
$99$	−3.00000	−0.301511
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	− 6.00000i	− 0.594089i
$103$	− 8.00000i	− 0.788263i	−0.919054	−	0.394132i	$-0.871045\pi$
$103$	− 8.00000i	− 0.788263i	0.919054	−	0.394132i	$-0.128955\pi$
$104$	0	0
$105$	0	0
$106$	−13.0000	−1.26267
$107$	2.00000i	0.193347i	0.995316	+	0.0966736i	$0.0308203\pi$
$107$	2.00000i	0.193347i	−0.995316	+	0.0966736i	$0.969180\pi$
$108$	4.00000i	0.384900i
$109$	3.00000	0.287348	0.143674	−	0.989625i	$-0.454108\pi$
$109$	3.00000	0.287348	0.143674	+	0.989625i	$0.454108\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	1.00000i	0.0944911i
$113$	7.00000i	0.658505i	0.944242	+	0.329252i	$0.106797\pi$
$113$	7.00000i	0.658505i	−0.944242	+	0.329252i	$0.893203\pi$
$114$	−12.0000	−1.12390
$115$	0	0
$116$	−3.00000	−0.278543
$117$	0	0
$118$	0	0
$119$	−3.00000	−0.275010
$120$	0	0
$121$	−2.00000	−0.181818
$122$	15.0000i	1.35804i
$123$	− 6.00000i	− 0.541002i
$124$	−3.00000	−0.269408
$125$	0	0
$126$	−1.00000	−0.0890871
$127$	− 4.00000i	− 0.354943i	−0.984126	−	0.177471i	$-0.943208\pi$
$127$	− 4.00000i	− 0.354943i	0.984126	−	0.177471i	$-0.0567917\pi$
$128$	1.00000i	0.0883883i
$129$	2.00000	0.176090
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	6.00000i	0.522233i
$133$	6.00000i	0.520266i
$134$	0	0
$135$	0	0
$136$	−3.00000	−0.257248
$137$	8.00000i	0.683486i	0.939793	+	0.341743i	$0.111017\pi$
$137$	8.00000i	0.683486i	−0.939793	+	0.341743i	$0.888983\pi$
$138$	4.00000i	0.340503i
$139$	−3.00000	−0.254457	−0.127228	−	0.991873i	$-0.540608\pi$
$139$	−3.00000	−0.254457	−0.127228	+	0.991873i	$0.540608\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	2.00000i	0.167836i
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	0	0
$147$	− 12.0000i	− 0.989743i
$148$	1.00000i	0.0821995i
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	0	0
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	6.00000i	0.486664i
$153$	− 3.00000i	− 0.242536i
$154$	3.00000	0.241747
$155$	0	0
$156$	0	0
$157$	− 3.00000i	− 0.239426i	−0.992809	−	0.119713i	$-0.961803\pi$
$157$	− 3.00000i	− 0.239426i	0.992809	−	0.119713i	$-0.0381975\pi$
$158$	− 8.00000i	− 0.636446i
$159$	−26.0000	−2.06193
$160$	0	0
$161$	2.00000	0.157622
$162$	11.0000i	0.864242i
$163$	5.00000i	0.391630i	0.980641	+	0.195815i	$0.0627352\pi$
$163$	5.00000i	0.391630i	−0.980641	+	0.195815i	$0.937265\pi$
$164$	−3.00000	−0.234261
$165$	0	0
$166$	4.00000	0.310460
$167$	12.0000i	0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$	12.0000i	0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$	2.00000i	0.154303i
$169$	13.0000	1.00000
$170$	0	0
$171$	−6.00000	−0.458831
$172$	− 1.00000i	− 0.0762493i
$173$	− 9.00000i	− 0.684257i	−0.939653	−	0.342129i	$-0.888852\pi$
$173$	− 9.00000i	− 0.684257i	0.939653	−	0.342129i	$-0.111148\pi$
$174$	−6.00000	−0.454859
$175$	0	0
$176$	3.00000	0.226134
$177$	0	0
$178$	− 18.0000i	− 1.34916i
$179$	−24.0000	−1.79384	−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000	−1.79384	−0.896922	+	0.442189i	$0.854202\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	30.0000i	2.21766i
$184$	2.00000	0.147442
$185$	0	0
$186$	−6.00000	−0.439941
$187$	9.00000i	0.658145i
$188$	− 4.00000i	− 0.291730i
$189$	4.00000	0.290957
$190$	0	0
$191$	21.0000	1.51951	0.759753	−	0.650211i	$-0.225320\pi$
$191$	21.0000	1.51951	0.759753	+	0.650211i	$0.225320\pi$
$192$	2.00000i	0.144338i
$193$	10.0000i	0.719816i	0.932988	+	0.359908i	$0.117192\pi$
$193$	10.0000i	0.719816i	−0.932988	+	0.359908i	$0.882808\pi$
$194$	−7.00000	−0.502571
$195$	0	0
$196$	−6.00000	−0.428571
$197$	− 18.0000i	− 1.28245i	−0.767354	−	0.641223i	$-0.778427\pi$
$197$	− 18.0000i	− 1.28245i	0.767354	−	0.641223i	$-0.221573\pi$
$198$	3.00000i	0.213201i
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	0	0
$202$	10.0000i	0.703598i
$203$	3.00000i	0.210559i
$204$	−6.00000	−0.420084
$205$	0	0
$206$	−8.00000	−0.557386
$207$	2.00000i	0.139010i
$208$	0	0
$209$	18.0000	1.24509
$210$	0	0
$211$	−3.00000	−0.206529	−0.103264	−	0.994654i	$-0.532929\pi$
$211$	−3.00000	−0.206529	−0.103264	+	0.994654i	$0.532929\pi$
$212$	13.0000i	0.892844i
$213$	4.00000i	0.274075i
$214$	2.00000	0.136717
$215$	0	0
$216$	4.00000	0.272166
$217$	3.00000i	0.203653i
$218$	− 3.00000i	− 0.203186i
$219$	0	0
$220$	0	0
$221$	0	0
$222$	2.00000i	0.134231i
$223$	− 23.0000i	− 1.54019i	−0.637927	−	0.770097i	$-0.720208\pi$
$223$	− 23.0000i	− 1.54019i	0.637927	−	0.770097i	$-0.279792\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	7.00000	0.465633
$227$	13.0000i	0.862840i	0.902151	+	0.431420i	$0.141987\pi$
$227$	13.0000i	0.862840i	−0.902151	+	0.431420i	$0.858013\pi$
$228$	12.0000i	0.794719i
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	0	0
$231$	6.00000	0.394771
$232$	3.00000i	0.196960i
$233$	− 4.00000i	− 0.262049i	−0.991379	−	0.131024i	$-0.958173\pi$
$233$	− 4.00000i	− 0.262049i	0.991379	−	0.131024i	$-0.0418266\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 16.0000i	− 1.03931i
$238$	3.00000i	0.194461i
$239$	−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
$241$	24.0000	1.54598	0.772988	−	0.634421i	$-0.218761\pi$
$241$	24.0000	1.54598	0.772988	+	0.634421i	$0.218761\pi$
$242$	2.00000i	0.128565i
$243$	10.0000i	0.641500i
$244$	15.0000	0.960277
$245$	0	0
$246$	−6.00000	−0.382546
$247$	0	0
$248$	3.00000i	0.190500i
$249$	8.00000	0.506979
$250$	0	0
$251$	6.00000	0.378717	0.189358	−	0.981908i	$-0.439359\pi$
$251$	6.00000	0.378717	0.189358	+	0.981908i	$0.439359\pi$
$252$	1.00000i	0.0629941i
$253$	− 6.00000i	− 0.377217i
$254$	−4.00000	−0.250982
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 22.0000i	− 1.37232i	−0.727450	−	0.686161i	$-0.759294\pi$
$257$	− 22.0000i	− 1.37232i	0.727450	−	0.686161i	$-0.240706\pi$
$258$	− 2.00000i	− 0.124515i
$259$	1.00000	0.0621370
$260$	0	0
$261$	−3.00000	−0.185695
$262$	12.0000i	0.741362i
$263$	1.00000i	0.0616626i	0.999525	+	0.0308313i	$0.00981547\pi$
$263$	1.00000i	0.0616626i	−0.999525	+	0.0308313i	$0.990185\pi$
$264$	6.00000	0.369274
$265$	0	0
$266$	6.00000	0.367884
$267$	− 36.0000i	− 2.20316i
$268$	0	0
$269$	10.0000	0.609711	0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000	0.609711	0.304855	+	0.952399i	$0.401392\pi$
$270$	0	0
$271$	−10.0000	−0.607457	−0.303728	−	0.952759i	$-0.598232\pi$
$271$	−10.0000	−0.607457	−0.303728	+	0.952759i	$0.598232\pi$
$272$	3.00000i	0.181902i
$273$	0	0
$274$	8.00000	0.483298
$275$	0	0
$276$	4.00000	0.240772
$277$	− 20.0000i	− 1.20168i	−0.799368	−	0.600842i	$-0.794832\pi$
$277$	− 20.0000i	− 1.20168i	0.799368	−	0.600842i	$-0.205168\pi$
$278$	3.00000i	0.179928i
$279$	−3.00000	−0.179605
$280$	0	0
$281$	−24.0000	−1.43172	−0.715860	−	0.698244i	$-0.753965\pi$
$281$	−24.0000	−1.43172	−0.715860	+	0.698244i	$0.753965\pi$
$282$	− 8.00000i	− 0.476393i
$283$	20.0000i	1.18888i	0.804141	+	0.594438i	$0.202626\pi$
$283$	20.0000i	1.18888i	−0.804141	+	0.594438i	$0.797374\pi$
$284$	2.00000	0.118678
$285$	0	0
$286$	0	0
$287$	3.00000i	0.177084i
$288$	1.00000i	0.0589256i
$289$	8.00000	0.470588
$290$	0	0
$291$	−14.0000	−0.820695
$292$	0	0
$293$	− 3.00000i	− 0.175262i	−0.996153	−	0.0876309i	$-0.972070\pi$
$293$	− 3.00000i	− 0.175262i	0.996153	−	0.0876309i	$-0.0279296\pi$
$294$	−12.0000	−0.699854
$295$	0	0
$296$	1.00000	0.0581238
$297$	− 12.0000i	− 0.696311i
$298$	18.0000i	1.04271i
$299$	0	0
$300$	0	0
$301$	−1.00000	−0.0576390
$302$	− 16.0000i	− 0.920697i
$303$	20.0000i	1.14897i
$304$	6.00000	0.344124
$305$	0	0
$306$	−3.00000	−0.171499
$307$	34.0000i	1.94048i	0.242140	+	0.970241i	$0.422151\pi$
$307$	34.0000i	1.94048i	−0.242140	+	0.970241i	$0.577849\pi$
$308$	− 3.00000i	− 0.170941i
$309$	−16.0000	−0.910208
$310$	0	0
$311$	25.0000	1.41762	0.708810	−	0.705399i	$-0.249232\pi$
$311$	25.0000	1.41762	0.708810	+	0.705399i	$0.249232\pi$
$312$	0	0
$313$	− 26.0000i	− 1.46961i	−0.678280	−	0.734803i	$-0.737274\pi$
$313$	− 26.0000i	− 1.46961i	0.678280	−	0.734803i	$-0.262726\pi$
$314$	−3.00000	−0.169300
$315$	0	0
$316$	−8.00000	−0.450035
$317$	21.0000i	1.17948i	0.807594	+	0.589739i	$0.200769\pi$
$317$	21.0000i	1.17948i	−0.807594	+	0.589739i	$0.799231\pi$
$318$	26.0000i	1.45801i
$319$	9.00000	0.503903
$320$	0	0
$321$	4.00000	0.223258
$322$	− 2.00000i	− 0.111456i
$323$	18.0000i	1.00155i
$324$	11.0000	0.611111
$325$	0	0
$326$	5.00000	0.276924
$327$	− 6.00000i	− 0.331801i
$328$	3.00000i	0.165647i
$329$	−4.00000	−0.220527
$330$	0	0
$331$	30.0000	1.64895	0.824475	−	0.565899i	$-0.191471\pi$
$331$	30.0000	1.64895	0.824475	+	0.565899i	$0.191471\pi$
$332$	− 4.00000i	− 0.219529i
$333$	1.00000i	0.0547997i
$334$	12.0000	0.656611
$335$	0	0
$336$	2.00000	0.109109
$337$	12.0000i	0.653682i	0.945079	+	0.326841i	$0.105984\pi$
$337$	12.0000i	0.653682i	−0.945079	+	0.326841i	$0.894016\pi$
$338$	− 13.0000i	− 0.707107i
$339$	14.0000	0.760376
$340$	0	0
$341$	9.00000	0.487377
$342$	6.00000i	0.324443i
$343$	13.0000i	0.701934i
$344$	−1.00000	−0.0539164
$345$	0	0
$346$	−9.00000	−0.483843
$347$	− 28.0000i	− 1.50312i	−0.659665	−	0.751559i	$-0.729302\pi$
$347$	− 28.0000i	− 1.50312i	0.659665	−	0.751559i	$-0.270698\pi$
$348$	6.00000i	0.321634i
$349$	22.0000	1.17763	0.588817	−	0.808267i	$-0.299594\pi$
$349$	22.0000	1.17763	0.588817	+	0.808267i	$0.299594\pi$
$350$	0	0
$351$	0	0
$352$	− 3.00000i	− 0.159901i
$353$	− 19.0000i	− 1.01127i	−0.862748	−	0.505634i	$-0.831259\pi$
$353$	− 19.0000i	− 1.01127i	0.862748	−	0.505634i	$-0.168741\pi$
$354$	0	0
$355$	0	0
$356$	−18.0000	−0.953998
$357$	6.00000i	0.317554i
$358$	24.0000i	1.26844i
$359$	−16.0000	−0.844448	−0.422224	−	0.906492i	$-0.638750\pi$
$359$	−16.0000	−0.844448	−0.422224	+	0.906492i	$0.638750\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	− 2.00000i	− 0.105118i
$363$	4.00000i	0.209946i
$364$	0	0
$365$	0	0
$366$	30.0000	1.56813
$367$	13.0000i	0.678594i	0.940679	+	0.339297i	$0.110189\pi$
$367$	13.0000i	0.678594i	−0.940679	+	0.339297i	$0.889811\pi$
$368$	− 2.00000i	− 0.104257i
$369$	−3.00000	−0.156174
$370$	0	0
$371$	13.0000	0.674926
$372$	6.00000i	0.311086i
$373$	− 22.0000i	− 1.13912i	−0.821951	−	0.569558i	$-0.807114\pi$
$373$	− 22.0000i	− 1.13912i	0.821951	−	0.569558i	$-0.192886\pi$
$374$	9.00000	0.465379
$375$	0	0
$376$	−4.00000	−0.206284
$377$	0	0
$378$	− 4.00000i	− 0.205738i
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	0	0
$381$	−8.00000	−0.409852
$382$	− 21.0000i	− 1.07445i
$383$	12.0000i	0.613171i	0.951843	+	0.306586i	$0.0991866\pi$
$383$	12.0000i	0.613171i	−0.951843	+	0.306586i	$0.900813\pi$
$384$	2.00000	0.102062
$385$	0	0
$386$	10.0000	0.508987
$387$	− 1.00000i	− 0.0508329i
$388$	7.00000i	0.355371i
$389$	−15.0000	−0.760530	−0.380265	−	0.924878i	$-0.624167\pi$
$389$	−15.0000	−0.760530	−0.380265	+	0.924878i	$0.624167\pi$
$390$	0	0
$391$	6.00000	0.303433
$392$	6.00000i	0.303046i
$393$	24.0000i	1.21064i
$394$	−18.0000	−0.906827
$395$	0	0
$396$	3.00000	0.150756
$397$	18.0000i	0.903394i	0.892171	+	0.451697i	$0.149181\pi$
$397$	18.0000i	0.903394i	−0.892171	+	0.451697i	$0.850819\pi$
$398$	8.00000i	0.401004i
$399$	12.0000	0.600751
$400$	0	0
$401$	34.0000	1.69788	0.848939	−	0.528490i	$-0.177242\pi$
$401$	34.0000	1.69788	0.848939	+	0.528490i	$0.177242\pi$
$402$	0	0
$403$	0	0
$404$	10.0000	0.497519
$405$	0	0
$406$	3.00000	0.148888
$407$	− 3.00000i	− 0.148704i
$408$	6.00000i	0.297044i
$409$	0	0		−	1.00000i	$-0.5\pi$
$409$	0	0			1.00000i	$0.5\pi$
$410$	0	0
$411$	16.0000	0.789222
$412$	8.00000i	0.394132i
$413$	0	0
$414$	2.00000	0.0982946
$415$	0	0
$416$	0	0
$417$	6.00000i	0.293821i
$418$	− 18.0000i	− 0.880409i
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	34.0000	1.65706	0.828529	−	0.559946i	$-0.189178\pi$
$421$	34.0000	1.65706	0.828529	+	0.559946i	$0.189178\pi$
$422$	3.00000i	0.146038i
$423$	− 4.00000i	− 0.194487i
$424$	13.0000	0.631336
$425$	0	0
$426$	4.00000	0.193801
$427$	− 15.0000i	− 0.725901i
$428$	− 2.00000i	− 0.0966736i
$429$	0	0
$430$	0	0
$431$	−31.0000	−1.49322	−0.746609	−	0.665263i	$-0.768319\pi$
$431$	−31.0000	−1.49322	−0.746609	+	0.665263i	$0.768319\pi$
$432$	− 4.00000i	− 0.192450i
$433$	22.0000i	1.05725i	0.848855	+	0.528626i	$0.177293\pi$
$433$	22.0000i	1.05725i	−0.848855	+	0.528626i	$0.822707\pi$
$434$	3.00000	0.144005
$435$	0	0
$436$	−3.00000	−0.143674
$437$	− 12.0000i	− 0.574038i
$438$	0	0
$439$	−37.0000	−1.76591	−0.882957	−	0.469454i	$-0.844451\pi$
$439$	−37.0000	−1.76591	−0.882957	+	0.469454i	$0.844451\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	0	0
$443$	34.0000i	1.61539i	0.589601	+	0.807694i	$0.299285\pi$
$443$	34.0000i	1.61539i	−0.589601	+	0.807694i	$0.700715\pi$
$444$	2.00000	0.0949158
$445$	0	0
$446$	−23.0000	−1.08908
$447$	36.0000i	1.70274i
$448$	− 1.00000i	− 0.0472456i
$449$	10.0000	0.471929	0.235965	−	0.971762i	$-0.424175\pi$
$449$	10.0000	0.471929	0.235965	+	0.971762i	$0.424175\pi$
$450$	0	0
$451$	9.00000	0.423793
$452$	− 7.00000i	− 0.329252i
$453$	− 32.0000i	− 1.50349i
$454$	13.0000	0.610120
$455$	0	0
$456$	12.0000	0.561951
$457$	9.00000i	0.421002i	0.977594	+	0.210501i	$0.0675096\pi$
$457$	9.00000i	0.421002i	−0.977594	+	0.210501i	$0.932490\pi$
$458$	− 6.00000i	− 0.280362i
$459$	12.0000	0.560112
$460$	0	0
$461$	−1.00000	−0.0465746	−0.0232873	−	0.999729i	$-0.507413\pi$
$461$	−1.00000	−0.0465746	−0.0232873	+	0.999729i	$0.507413\pi$
$462$	− 6.00000i	− 0.279145i
$463$	− 12.0000i	− 0.557687i	−0.960337	−	0.278844i	$-0.910049\pi$
$463$	− 12.0000i	− 0.557687i	0.960337	−	0.278844i	$-0.0899511\pi$
$464$	3.00000	0.139272
$465$	0	0
$466$	−4.00000	−0.185296
$467$	− 37.0000i	− 1.71216i	−0.516847	−	0.856078i	$-0.672894\pi$
$467$	− 37.0000i	− 1.71216i	0.516847	−	0.856078i	$-0.327106\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−6.00000	−0.276465
$472$	0	0
$473$	3.00000i	0.137940i
$474$	−16.0000	−0.734904
$475$	0	0
$476$	3.00000	0.137505
$477$	13.0000i	0.595229i
$478$	9.00000i	0.411650i
$479$	8.00000	0.365529	0.182765	−	0.983157i	$-0.441495\pi$
$479$	8.00000	0.365529	0.182765	+	0.983157i	$0.441495\pi$
$480$	0	0
$481$	0	0
$482$	− 24.0000i	− 1.09317i
$483$	− 4.00000i	− 0.182006i
$484$	2.00000	0.0909091
$485$	0	0
$486$	10.0000	0.453609
$487$	34.0000i	1.54069i	0.637629	+	0.770344i	$0.279915\pi$
$487$	34.0000i	1.54069i	−0.637629	+	0.770344i	$0.720085\pi$
$488$	− 15.0000i	− 0.679018i
$489$	10.0000	0.452216
$490$	0	0
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	6.00000i	0.270501i
$493$	9.00000i	0.405340i
$494$	0	0
$495$	0	0
$496$	3.00000	0.134704
$497$	− 2.00000i	− 0.0897123i
$498$	− 8.00000i	− 0.358489i
$499$	22.0000	0.984855	0.492428	−	0.870353i	$-0.336110\pi$
$499$	22.0000	0.984855	0.492428	+	0.870353i	$0.336110\pi$
$500$	0	0
$501$	24.0000	1.07224
$502$	− 6.00000i	− 0.267793i
$503$	24.0000i	1.07011i	0.844818	+	0.535054i	$0.179709\pi$
$503$	24.0000i	1.07011i	−0.844818	+	0.535054i	$0.820291\pi$
$504$	1.00000	0.0445435
$505$	0	0
$506$	−6.00000	−0.266733
$507$	− 26.0000i	− 1.15470i
$508$	4.00000i	0.177471i
$509$	−36.0000	−1.59567	−0.797836	−	0.602875i	$-0.794022\pi$
$509$	−36.0000	−1.59567	−0.797836	+	0.602875i	$0.794022\pi$
$510$	0	0
$511$	0	0
$512$	− 1.00000i	− 0.0441942i
$513$	− 24.0000i	− 1.05963i
$514$	−22.0000	−0.970378
$515$	0	0
$516$	−2.00000	−0.0880451
$517$	12.0000i	0.527759i
$518$	− 1.00000i	− 0.0439375i
$519$	−18.0000	−0.790112
$520$	0	0
$521$	−23.0000	−1.00765	−0.503824	−	0.863806i	$-0.668074\pi$
$521$	−23.0000	−1.00765	−0.503824	+	0.863806i	$0.668074\pi$
$522$	3.00000i	0.131306i
$523$	44.0000i	1.92399i	0.273075	+	0.961993i	$0.411959\pi$
$523$	44.0000i	1.92399i	−0.273075	+	0.961993i	$0.588041\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	1.00000	0.0436021
$527$	9.00000i	0.392046i
$528$	− 6.00000i	− 0.261116i
$529$	19.0000	0.826087
$530$	0	0
$531$	0	0
$532$	− 6.00000i	− 0.260133i
$533$	0	0
$534$	−36.0000	−1.55787
$535$	0	0
$536$	0	0
$537$	48.0000i	2.07135i
$538$	− 10.0000i	− 0.431131i
$539$	18.0000	0.775315
$540$	0	0
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	10.0000i	0.429537i
$543$	− 4.00000i	− 0.171656i
$544$	3.00000	0.128624
$545$	0	0
$546$	0	0
$547$	23.0000i	0.983409i	0.870762	+	0.491704i	$0.163626\pi$
$547$	23.0000i	0.983409i	−0.870762	+	0.491704i	$0.836374\pi$
$548$	− 8.00000i	− 0.341743i
$549$	15.0000	0.640184
$550$	0	0
$551$	18.0000	0.766826
$552$	− 4.00000i	− 0.170251i
$553$	8.00000i	0.340195i
$554$	−20.0000	−0.849719
$555$	0	0
$556$	3.00000	0.127228
$557$	− 40.0000i	− 1.69485i	−0.530912	−	0.847427i	$-0.678150\pi$
$557$	− 40.0000i	− 1.69485i	0.530912	−	0.847427i	$-0.321850\pi$
$558$	3.00000i	0.127000i
$559$	0	0
$560$	0	0
$561$	18.0000	0.759961
$562$	24.0000i	1.01238i
$563$	11.0000i	0.463595i	0.972764	+	0.231797i	$0.0744606\pi$
$563$	11.0000i	0.463595i	−0.972764	+	0.231797i	$0.925539\pi$
$564$	−8.00000	−0.336861
$565$	0	0
$566$	20.0000	0.840663
$567$	− 11.0000i	− 0.461957i
$568$	− 2.00000i	− 0.0839181i
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	0	0
$571$	−31.0000	−1.29731	−0.648655	−	0.761083i	$-0.724668\pi$
$571$	−31.0000	−1.29731	−0.648655	+	0.761083i	$0.724668\pi$
$572$	0	0
$573$	− 42.0000i	− 1.75458i
$574$	3.00000	0.125218
$575$	0	0
$576$	1.00000	0.0416667
$577$	30.0000i	1.24892i	0.781058	+	0.624458i	$0.214680\pi$
$577$	30.0000i	1.24892i	−0.781058	+	0.624458i	$0.785320\pi$
$578$	− 8.00000i	− 0.332756i
$579$	20.0000	0.831172
$580$	0	0
$581$	−4.00000	−0.165948
$582$	14.0000i	0.580319i
$583$	− 39.0000i	− 1.61521i
$584$	0	0
$585$	0	0
$586$	−3.00000	−0.123929
$587$	35.0000i	1.44460i	0.691577	+	0.722302i	$0.256916\pi$
$587$	35.0000i	1.44460i	−0.691577	+	0.722302i	$0.743084\pi$
$588$	12.0000i	0.494872i
$589$	18.0000	0.741677
$590$	0	0
$591$	−36.0000	−1.48084
$592$	− 1.00000i	− 0.0410997i
$593$	− 12.0000i	− 0.492781i	−0.969171	−	0.246390i	$-0.920755\pi$
$593$	− 12.0000i	− 0.492781i	0.969171	−	0.246390i	$-0.0792446\pi$
$594$	−12.0000	−0.492366
$595$	0	0
$596$	18.0000	0.737309
$597$	16.0000i	0.654836i
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	21.0000	0.856608	0.428304	−	0.903635i	$-0.359111\pi$
$601$	21.0000	0.856608	0.428304	+	0.903635i	$0.359111\pi$
$602$	1.00000i	0.0407570i
$603$	0	0
$604$	−16.0000	−0.651031
$605$	0	0
$606$	20.0000	0.812444
$607$	22.0000i	0.892952i	0.894795	+	0.446476i	$0.147321\pi$
$607$	22.0000i	0.892952i	−0.894795	+	0.446476i	$0.852679\pi$
$608$	− 6.00000i	− 0.243332i
$609$	6.00000	0.243132
$610$	0	0
$611$	0	0
$612$	3.00000i	0.121268i
$613$	− 11.0000i	− 0.444286i	−0.975014	−	0.222143i	$-0.928695\pi$
$613$	− 11.0000i	− 0.444286i	0.975014	−	0.222143i	$-0.0713052\pi$
$614$	34.0000	1.37213
$615$	0	0
$616$	−3.00000	−0.120873
$617$	36.0000i	1.44931i	0.689114	+	0.724653i	$0.258000\pi$
$617$	36.0000i	1.44931i	−0.689114	+	0.724653i	$0.742000\pi$
$618$	16.0000i	0.643614i
$619$	37.0000	1.48716	0.743578	−	0.668649i	$-0.233127\pi$
$619$	37.0000	1.48716	0.743578	+	0.668649i	$0.233127\pi$
$620$	0	0
$621$	−8.00000	−0.321029
$622$	− 25.0000i	− 1.00241i
$623$	18.0000i	0.721155i
$624$	0	0
$625$	0	0
$626$	−26.0000	−1.03917
$627$	− 36.0000i	− 1.43770i
$628$	3.00000i	0.119713i
$629$	3.00000	0.119618
$630$	0	0
$631$	−41.0000	−1.63218	−0.816092	−	0.577922i	$-0.803864\pi$
$631$	−41.0000	−1.63218	−0.816092	+	0.577922i	$0.803864\pi$
$632$	8.00000i	0.318223i
$633$	6.00000i	0.238479i
$634$	21.0000	0.834017
$635$	0	0
$636$	26.0000	1.03097
$637$	0	0
$638$	− 9.00000i	− 0.356313i
$639$	2.00000	0.0791188
$640$	0	0
$641$	−33.0000	−1.30342	−0.651711	−	0.758468i	$-0.725948\pi$
$641$	−33.0000	−1.30342	−0.651711	+	0.758468i	$0.725948\pi$
$642$	− 4.00000i	− 0.157867i
$643$	− 11.0000i	− 0.433798i	−0.976194	−	0.216899i	$-0.930406\pi$
$643$	− 11.0000i	− 0.433798i	0.976194	−	0.216899i	$-0.0695942\pi$
$644$	−2.00000	−0.0788110
$645$	0	0
$646$	18.0000	0.708201
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	− 11.0000i	− 0.432121i
$649$	0	0
$650$	0	0
$651$	6.00000	0.235159
$652$	− 5.00000i	− 0.195815i
$653$	− 30.0000i	− 1.17399i	−0.809590	−	0.586995i	$-0.800311\pi$
$653$	− 30.0000i	− 1.17399i	0.809590	−	0.586995i	$-0.199689\pi$
$654$	−6.00000	−0.234619
$655$	0	0
$656$	3.00000	0.117130
$657$	0	0
$658$	4.00000i	0.155936i
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	−13.0000	−0.505641	−0.252821	−	0.967513i	$-0.581358\pi$
$661$	−13.0000	−0.505641	−0.252821	+	0.967513i	$0.581358\pi$
$662$	− 30.0000i	− 1.16598i
$663$	0	0
$664$	−4.00000	−0.155230
$665$	0	0
$666$	1.00000	0.0387492
$667$	− 6.00000i	− 0.232321i
$668$	− 12.0000i	− 0.464294i
$669$	−46.0000	−1.77846
$670$	0	0
$671$	−45.0000	−1.73721
$672$	− 2.00000i	− 0.0771517i
$673$	34.0000i	1.31060i	0.755367	+	0.655302i	$0.227459\pi$
$673$	34.0000i	1.31060i	−0.755367	+	0.655302i	$0.772541\pi$
$674$	12.0000	0.462223
$675$	0	0
$676$	−13.0000	−0.500000
$677$	34.0000i	1.30673i	0.757045	+	0.653363i	$0.226642\pi$
$677$	34.0000i	1.30673i	−0.757045	+	0.653363i	$0.773358\pi$
$678$	− 14.0000i	− 0.537667i
$679$	7.00000	0.268635
$680$	0	0
$681$	26.0000	0.996322
$682$	− 9.00000i	− 0.344628i
$683$	31.0000i	1.18618i	0.805135	+	0.593091i	$0.202093\pi$
$683$	31.0000i	1.18618i	−0.805135	+	0.593091i	$0.797907\pi$
$684$	6.00000	0.229416
$685$	0	0
$686$	13.0000	0.496342
$687$	− 12.0000i	− 0.457829i
$688$	1.00000i	0.0381246i
$689$	0	0
$690$	0	0
$691$	5.00000	0.190209	0.0951045	−	0.995467i	$-0.469681\pi$
$691$	5.00000	0.190209	0.0951045	+	0.995467i	$0.469681\pi$
$692$	9.00000i	0.342129i
$693$	− 3.00000i	− 0.113961i
$694$	−28.0000	−1.06287
$695$	0	0
$696$	6.00000	0.227429
$697$	9.00000i	0.340899i
$698$	− 22.0000i	− 0.832712i
$699$	−8.00000	−0.302588
$700$	0	0
$701$	−22.0000	−0.830929	−0.415464	−	0.909610i	$-0.636381\pi$
$701$	−22.0000	−0.830929	−0.415464	+	0.909610i	$0.636381\pi$
$702$	0	0
$703$	− 6.00000i	− 0.226294i
$704$	−3.00000	−0.113067
$705$	0	0
$706$	−19.0000	−0.715074
$707$	− 10.0000i	− 0.376089i
$708$	0	0
$709$	19.0000	0.713560	0.356780	−	0.934188i	$-0.383875\pi$
$709$	19.0000	0.713560	0.356780	+	0.934188i	$0.383875\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	18.0000i	0.674579i
$713$	− 6.00000i	− 0.224702i
$714$	6.00000	0.224544
$715$	0	0
$716$	24.0000	0.896922
$717$	18.0000i	0.672222i
$718$	16.0000i	0.597115i
$719$	18.0000	0.671287	0.335643	−	0.941989i	$-0.391046\pi$
$719$	18.0000	0.671287	0.335643	+	0.941989i	$0.391046\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	− 17.0000i	− 0.632674i
$723$	− 48.0000i	− 1.78514i
$724$	−2.00000	−0.0743294
$725$	0	0
$726$	4.00000	0.148454
$727$	52.0000i	1.92857i	0.264861	+	0.964287i	$0.414674\pi$
$727$	52.0000i	1.92857i	−0.264861	+	0.964287i	$0.585326\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	−3.00000	−0.110959
$732$	− 30.0000i	− 1.10883i
$733$	41.0000i	1.51437i	0.653201	+	0.757185i	$0.273426\pi$
$733$	41.0000i	1.51437i	−0.653201	+	0.757185i	$0.726574\pi$
$734$	13.0000	0.479839
$735$	0	0
$736$	−2.00000	−0.0737210
$737$	0	0
$738$	3.00000i	0.110432i
$739$	−27.0000	−0.993211	−0.496606	−	0.867976i	$-0.665420\pi$
$739$	−27.0000	−0.993211	−0.496606	+	0.867976i	$0.665420\pi$
$740$	0	0
$741$	0	0
$742$	− 13.0000i	− 0.477245i
$743$	− 3.00000i	− 0.110059i	−0.998485	−	0.0550297i	$-0.982475\pi$
$743$	− 3.00000i	− 0.110059i	0.998485	−	0.0550297i	$-0.0175253\pi$
$744$	6.00000	0.219971
$745$	0	0
$746$	−22.0000	−0.805477
$747$	− 4.00000i	− 0.146352i
$748$	− 9.00000i	− 0.329073i
$749$	−2.00000	−0.0730784
$750$	0	0
$751$	−22.0000	−0.802791	−0.401396	−	0.915905i	$-0.631475\pi$
$751$	−22.0000	−0.802791	−0.401396	+	0.915905i	$0.631475\pi$
$752$	4.00000i	0.145865i
$753$	− 12.0000i	− 0.437304i
$754$	0	0
$755$	0	0
$756$	−4.00000	−0.145479
$757$	− 50.0000i	− 1.81728i	−0.417579	−	0.908640i	$-0.637121\pi$
$757$	− 50.0000i	− 1.81728i	0.417579	−	0.908640i	$-0.362879\pi$
$758$	16.0000i	0.581146i
$759$	−12.0000	−0.435572
$760$	0	0
$761$	−21.0000	−0.761249	−0.380625	−	0.924730i	$-0.624291\pi$
$761$	−21.0000	−0.761249	−0.380625	+	0.924730i	$0.624291\pi$
$762$	8.00000i	0.289809i
$763$	3.00000i	0.108607i
$764$	−21.0000	−0.759753
$765$	0	0
$766$	12.0000	0.433578
$767$	0	0
$768$	− 2.00000i	− 0.0721688i
$769$	−8.00000	−0.288487	−0.144244	−	0.989542i	$-0.546075\pi$
$769$	−8.00000	−0.288487	−0.144244	+	0.989542i	$0.546075\pi$
$770$	0	0
$771$	−44.0000	−1.58462
$772$	− 10.0000i	− 0.359908i
$773$	41.0000i	1.47467i	0.675529	+	0.737334i	$0.263915\pi$
$773$	41.0000i	1.47467i	−0.675529	+	0.737334i	$0.736085\pi$
$774$	−1.00000	−0.0359443
$775$	0	0
$776$	7.00000	0.251285
$777$	− 2.00000i	− 0.0717496i
$778$	15.0000i	0.537776i
$779$	18.0000	0.644917
$780$	0	0
$781$	−6.00000	−0.214697
$782$	− 6.00000i	− 0.214560i
$783$	− 12.0000i	− 0.428845i
$784$	6.00000	0.214286
$785$	0	0
$786$	24.0000	0.856052
$787$	− 10.0000i	− 0.356462i	−0.983989	−	0.178231i	$-0.942963\pi$
$787$	− 10.0000i	− 0.356462i	0.983989	−	0.178231i	$-0.0570374\pi$
$788$	18.0000i	0.641223i
$789$	2.00000	0.0712019
$790$	0	0
$791$	−7.00000	−0.248891
$792$	− 3.00000i	− 0.106600i
$793$	0	0
$794$	18.0000	0.638796
$795$	0	0
$796$	8.00000	0.283552
$797$	− 26.0000i	− 0.920967i	−0.887668	−	0.460484i	$-0.847676\pi$
$797$	− 26.0000i	− 0.920967i	0.887668	−	0.460484i	$-0.152324\pi$
$798$	− 12.0000i	− 0.424795i
$799$	−12.0000	−0.424529
$800$	0	0
$801$	−18.0000	−0.635999
$802$	− 34.0000i	− 1.20058i
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 20.0000i	− 0.704033i
$808$	− 10.0000i	− 0.351799i
$809$	−34.0000	−1.19538	−0.597688	−	0.801729i	$-0.703914\pi$
$809$	−34.0000	−1.19538	−0.597688	+	0.801729i	$0.703914\pi$
$810$	0	0
$811$	−16.0000	−0.561836	−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000	−0.561836	−0.280918	+	0.959732i	$0.590639\pi$
$812$	− 3.00000i	− 0.105279i
$813$	20.0000i	0.701431i
$814$	−3.00000	−0.105150
$815$	0	0
$816$	6.00000	0.210042
$817$	6.00000i	0.209913i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	8.00000	0.279202	0.139601	−	0.990208i	$-0.455418\pi$
$821$	8.00000	0.279202	0.139601	+	0.990208i	$0.455418\pi$
$822$	− 16.0000i	− 0.558064i
$823$	− 24.0000i	− 0.836587i	−0.908312	−	0.418294i	$-0.862628\pi$
$823$	− 24.0000i	− 0.836587i	0.908312	−	0.418294i	$-0.137372\pi$
$824$	8.00000	0.278693
$825$	0	0
$826$	0	0
$827$	− 11.0000i	− 0.382507i	−0.981541	−	0.191254i	$-0.938745\pi$
$827$	− 11.0000i	− 0.382507i	0.981541	−	0.191254i	$-0.0612553\pi$
$828$	− 2.00000i	− 0.0695048i
$829$	39.0000	1.35453	0.677263	−	0.735741i	$-0.263166\pi$
$829$	39.0000	1.35453	0.677263	+	0.735741i	$0.263166\pi$
$830$	0	0
$831$	−40.0000	−1.38758
$832$	0	0
$833$	18.0000i	0.623663i
$834$	6.00000	0.207763
$835$	0	0
$836$	−18.0000	−0.622543
$837$	− 12.0000i	− 0.414781i
$838$	12.0000i	0.414533i
$839$	−36.0000	−1.24286	−0.621429	−	0.783470i	$-0.713448\pi$
$839$	−36.0000	−1.24286	−0.621429	+	0.783470i	$0.713448\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	− 34.0000i	− 1.17172i
$843$	48.0000i	1.65321i
$844$	3.00000	0.103264
$845$	0	0
$846$	−4.00000	−0.137523
$847$	− 2.00000i	− 0.0687208i
$848$	− 13.0000i	− 0.446422i
$849$	40.0000	1.37280
$850$	0	0
$851$	−2.00000	−0.0685591
$852$	− 4.00000i	− 0.137038i
$853$	− 22.0000i	− 0.753266i	−0.926363	−	0.376633i	$-0.877082\pi$
$853$	− 22.0000i	− 0.753266i	0.926363	−	0.376633i	$-0.122918\pi$
$854$	−15.0000	−0.513289
$855$	0	0
$856$	−2.00000	−0.0683586
$857$	33.0000i	1.12726i	0.826028	+	0.563629i	$0.190595\pi$
$857$	33.0000i	1.12726i	−0.826028	+	0.563629i	$0.809405\pi$
$858$	0	0
$859$	36.0000	1.22830	0.614152	−	0.789188i	$-0.289498\pi$
$859$	36.0000	1.22830	0.614152	+	0.789188i	$0.289498\pi$
$860$	0	0
$861$	6.00000	0.204479
$862$	31.0000i	1.05586i
$863$	− 33.0000i	− 1.12333i	−0.827364	−	0.561667i	$-0.810160\pi$
$863$	− 33.0000i	− 1.12333i	0.827364	−	0.561667i	$-0.189840\pi$
$864$	−4.00000	−0.136083
$865$	0	0
$866$	22.0000	0.747590
$867$	− 16.0000i	− 0.543388i
$868$	− 3.00000i	− 0.101827i
$869$	24.0000	0.814144
$870$	0	0
$871$	0	0
$872$	3.00000i	0.101593i
$873$	7.00000i	0.236914i
$874$	−12.0000	−0.405906
$875$	0	0
$876$	0	0
$877$	− 43.0000i	− 1.45201i	−0.687691	−	0.726003i	$-0.741376\pi$
$877$	− 43.0000i	− 1.45201i	0.687691	−	0.726003i	$-0.258624\pi$
$878$	37.0000i	1.24869i
$879$	−6.00000	−0.202375
$880$	0	0
$881$	−9.00000	−0.303218	−0.151609	−	0.988441i	$-0.548445\pi$
$881$	−9.00000	−0.303218	−0.151609	+	0.988441i	$0.548445\pi$
$882$	6.00000i	0.202031i
$883$	− 29.0000i	− 0.975928i	−0.872864	−	0.487964i	$-0.837740\pi$
$883$	− 29.0000i	− 0.975928i	0.872864	−	0.487964i	$-0.162260\pi$
$884$	0	0
$885$	0	0
$886$	34.0000	1.14225
$887$	− 33.0000i	− 1.10803i	−0.832506	−	0.554016i	$-0.813095\pi$
$887$	− 33.0000i	− 1.10803i	0.832506	−	0.554016i	$-0.186905\pi$
$888$	− 2.00000i	− 0.0671156i
$889$	4.00000	0.134156
$890$	0	0
$891$	−33.0000	−1.10554
$892$	23.0000i	0.770097i
$893$	24.0000i	0.803129i
$894$	36.0000	1.20402
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	− 10.0000i	− 0.333704i
$899$	9.00000	0.300167
$900$	0	0
$901$	39.0000	1.29928
$902$	− 9.00000i	− 0.299667i
$903$	2.00000i	0.0665558i
$904$	−7.00000	−0.232817
$905$	0	0
$906$	−32.0000	−1.06313
$907$	− 12.0000i	− 0.398453i	−0.979953	−	0.199227i	$-0.936157\pi$
$907$	− 12.0000i	− 0.398453i	0.979953	−	0.199227i	$-0.0638430\pi$
$908$	− 13.0000i	− 0.431420i
$909$	10.0000	0.331679
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	− 12.0000i	− 0.397360i
$913$	12.0000i	0.397142i
$914$	9.00000	0.297694
$915$	0	0
$916$	−6.00000	−0.198246
$917$	− 12.0000i	− 0.396275i
$918$	− 12.0000i	− 0.396059i
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	0	0
$921$	68.0000	2.24068
$922$	1.00000i	0.0329332i
$923$	0	0
$924$	−6.00000	−0.197386
$925$	0	0
$926$	−12.0000	−0.394344
$927$	8.00000i	0.262754i
$928$	− 3.00000i	− 0.0984798i
$929$	3.00000	0.0984268	0.0492134	−	0.998788i	$-0.484329\pi$
$929$	3.00000	0.0984268	0.0492134	+	0.998788i	$0.484329\pi$
$930$	0	0
$931$	36.0000	1.17985
$932$	4.00000i	0.131024i
$933$	− 50.0000i	− 1.63693i
$934$	−37.0000	−1.21068
$935$	0	0
$936$	0	0
$937$	− 14.0000i	− 0.457360i	−0.973502	−	0.228680i	$-0.926559\pi$
$937$	− 14.0000i	− 0.457360i	0.973502	−	0.228680i	$-0.0734410\pi$
$938$	0	0
$939$	−52.0000	−1.69696
$940$	0	0
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	6.00000i	0.195491i
$943$	− 6.00000i	− 0.195387i
$944$	0	0
$945$	0	0
$946$	3.00000	0.0975384
$947$	9.00000i	0.292461i	0.989251	+	0.146230i	$0.0467141\pi$
$947$	9.00000i	0.292461i	−0.989251	+	0.146230i	$0.953286\pi$
$948$	16.0000i	0.519656i
$949$	0	0
$950$	0	0
$951$	42.0000	1.36194
$952$	− 3.00000i	− 0.0972306i
$953$	− 36.0000i	− 1.16615i	−0.812417	−	0.583077i	$-0.801849\pi$
$953$	− 36.0000i	− 1.16615i	0.812417	−	0.583077i	$-0.198151\pi$
$954$	13.0000	0.420891
$955$	0	0
$956$	9.00000	0.291081
$957$	− 18.0000i	− 0.581857i
$958$	− 8.00000i	− 0.258468i
$959$	−8.00000	−0.258333
$960$	0	0
$961$	−22.0000	−0.709677
$962$	0	0
$963$	− 2.00000i	− 0.0644491i
$964$	−24.0000	−0.772988
$965$	0	0
$966$	−4.00000	−0.128698
$967$	48.0000i	1.54358i	0.635880	+	0.771788i	$0.280637\pi$
$967$	48.0000i	1.54358i	−0.635880	+	0.771788i	$0.719363\pi$
$968$	− 2.00000i	− 0.0642824i
$969$	36.0000	1.15649
$970$	0	0
$971$	9.00000	0.288824	0.144412	−	0.989518i	$-0.453871\pi$
$971$	9.00000	0.288824	0.144412	+	0.989518i	$0.453871\pi$
$972$	− 10.0000i	− 0.320750i
$973$	− 3.00000i	− 0.0961756i
$974$	34.0000	1.08943
$975$	0	0
$976$	−15.0000	−0.480138
$977$	35.0000i	1.11975i	0.828577	+	0.559875i	$0.189151\pi$
$977$	35.0000i	1.11975i	−0.828577	+	0.559875i	$0.810849\pi$
$978$	− 10.0000i	− 0.319765i
$979$	54.0000	1.72585
$980$	0	0
$981$	−3.00000	−0.0957826
$982$	20.0000i	0.638226i
$983$	27.0000i	0.861166i	0.902551	+	0.430583i	$0.141692\pi$
$983$	27.0000i	0.861166i	−0.902551	+	0.430583i	$0.858308\pi$
$984$	6.00000	0.191273
$985$	0	0
$986$	9.00000	0.286618
$987$	8.00000i	0.254643i
$988$	0	0
$989$	2.00000	0.0635963
$990$	0	0
$991$	33.0000	1.04828	0.524140	−	0.851632i	$-0.324387\pi$
$991$	33.0000	1.04828	0.524140	+	0.851632i	$0.324387\pi$
$992$	− 3.00000i	− 0.0952501i
$993$	− 60.0000i	− 1.90404i
$994$	−2.00000	−0.0634361
$995$	0	0
$996$	−8.00000	−0.253490
$997$	− 12.0000i	− 0.380044i	−0.981780	−	0.190022i	$-0.939144\pi$
$997$	− 12.0000i	− 0.380044i	0.981780	−	0.190022i	$-0.0608559\pi$
$998$	− 22.0000i	− 0.696398i
$999$	−4.00000	−0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1850.2.b.c.149.1		2
5.2	odd	4		1850.2.a.i.1.1		1
5.3	odd	4		370.2.a.c.1.1	✓	1
5.4	even	2	inner	1850.2.b.c.149.2		2
15.8	even	4		3330.2.a.p.1.1		1
20.3	even	4		2960.2.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.a.c.1.1	✓	1	5.3	odd	4
1850.2.a.i.1.1		1	5.2	odd	4
1850.2.b.c.149.1		2	1.1	even	1	trivial
1850.2.b.c.149.2		2	5.4	even	2	inner
2960.2.a.c.1.1		1	20.3	even	4
3330.2.a.p.1.1		1	15.8	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:	\( N \)	\(=\)	\( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1850.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} +1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} +1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +3.00000 q^{11} +2.00000i q^{12} +1.00000 q^{14} +1.00000 q^{16} +3.00000i q^{17} +1.00000i q^{18} +6.00000 q^{19} +2.00000 q^{21} -3.00000i q^{22} -2.00000i q^{23} +2.00000 q^{24} -4.00000i q^{27} -1.00000i q^{28} +3.00000 q^{29} +3.00000 q^{31} -1.00000i q^{32} -6.00000i q^{33} +3.00000 q^{34} +1.00000 q^{36} -1.00000i q^{37} -6.00000i q^{38} +3.00000 q^{41} -2.00000i q^{42} +1.00000i q^{43} -3.00000 q^{44} -2.00000 q^{46} +4.00000i q^{47} -2.00000i q^{48} +6.00000 q^{49} +6.00000 q^{51} -13.0000i q^{53} -4.00000 q^{54} -1.00000 q^{56} -12.0000i q^{57} -3.00000i q^{58} -15.0000 q^{61} -3.00000i q^{62} -1.00000i q^{63} -1.00000 q^{64} -6.00000 q^{66} -3.00000i q^{68} -4.00000 q^{69} -2.00000 q^{71} -1.00000i q^{72} -1.00000 q^{74} -6.00000 q^{76} +3.00000i q^{77} +8.00000 q^{79} -11.0000 q^{81} -3.00000i q^{82} +4.00000i q^{83} -2.00000 q^{84} +1.00000 q^{86} -6.00000i q^{87} +3.00000i q^{88} +18.0000 q^{89} +2.00000i q^{92} -6.00000i q^{93} +4.00000 q^{94} -2.00000 q^{96} -7.00000i q^{97} -6.00000i q^{98} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 4 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 4 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} - 4 q^{6} - 2 q^{9} + 6 q^{11} + 2 q^{14} + 2 q^{16} + 12 q^{19} + 4 q^{21} + 4 q^{24} + 6 q^{29} + 6 q^{31} + 6 q^{34} + 2 q^{36} + 6 q^{41} - 6 q^{44} - 4 q^{46} + 12 q^{49} + 12 q^{51} - 8 q^{54} - 2 q^{56} - 30 q^{61} - 2 q^{64} - 12 q^{66} - 8 q^{69} - 4 q^{71} - 2 q^{74} - 12 q^{76} + 16 q^{79} - 22 q^{81} - 4 q^{84} + 2 q^{86} + 36 q^{89} + 8 q^{94} - 4 q^{96} - 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 - 4 * q^6 - 2 * q^9 + 6 * q^11 + 2 * q^14 + 2 * q^16 + 12 * q^19 + 4 * q^21 + 4 * q^24 + 6 * q^29 + 6 * q^31 + 6 * q^34 + 2 * q^36 + 6 * q^41 - 6 * q^44 - 4 * q^46 + 12 * q^49 + 12 * q^51 - 8 * q^54 - 2 * q^56 - 30 * q^61 - 2 * q^64 - 12 * q^66 - 8 * q^69 - 4 * q^71 - 2 * q^74 - 12 * q^76 + 16 * q^79 - 22 * q^81 - 4 * q^84 + 2 * q^86 + 36 * q^89 + 8 * q^94 - 4 * q^96 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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