LMFDB - Embedded newform 1850.2.b.b.149.2

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} +2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -2.00000i q^{12} -2.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} +6.00000i q^{17} -1.00000i q^{18} -2.00000 q^{19} -4.00000 q^{21} +2.00000 q^{24} +2.00000 q^{26} +4.00000i q^{27} -2.00000i q^{28} -6.00000 q^{29} -10.0000 q^{31} +1.00000i q^{32} -6.00000 q^{34} +1.00000 q^{36} +1.00000i q^{37} -2.00000i q^{38} +4.00000 q^{39} -6.00000 q^{41} -4.00000i q^{42} +4.00000i q^{43} -6.00000i q^{47} +2.00000i q^{48} +3.00000 q^{49} -12.0000 q^{51} +2.00000i q^{52} -6.00000i q^{53} -4.00000 q^{54} +2.00000 q^{56} -4.00000i q^{57} -6.00000i q^{58} +6.00000 q^{59} -10.0000 q^{61} -10.0000i q^{62} -2.00000i q^{63} -1.00000 q^{64} +2.00000i q^{67} -6.00000i q^{68} +1.00000i q^{72} -2.00000i q^{73} -1.00000 q^{74} +2.00000 q^{76} +4.00000i q^{78} +10.0000 q^{79} -11.0000 q^{81} -6.00000i q^{82} +6.00000i q^{83} +4.00000 q^{84} -4.00000 q^{86} -12.0000i q^{87} +6.00000 q^{89} +4.00000 q^{91} -20.0000i q^{93} +6.00000 q^{94} -2.00000 q^{96} +2.00000i q^{97} +3.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 4 q^{6} - 2 q^{9} - 4 q^{14} + 2 q^{16} - 4 q^{19} - 8 q^{21} + 4 q^{24} + 4 q^{26} - 12 q^{29} - 20 q^{31} - 12 q^{34} + 2 q^{36} + 8 q^{39} - 12 q^{41} + 6 q^{49} - 24 q^{51} - 8 q^{54}+ \cdots - 4 q^{96}+O(q^{100}) $ 2 * q - 2 * q^4 - 4 * q^6 - 2 * q^9 - 4 * q^14 + 2 * q^16 - 4 * q^19 - 8 * q^21 + 4 * q^24 + 4 * q^26 - 12 * q^29 - 20 * q^31 - 12 * q^34 + 2 * q^36 + 8 * q^39 - 12 * q^41 + 6 * q^49 - 24 * q^51 - 8 * q^54 + 4 * q^56 + 12 * q^59 - 20 * q^61 - 2 * q^64 - 2 * q^74 + 4 * q^76 + 20 * q^79 - 22 * q^81 + 8 * q^84 - 8 * q^86 + 12 * q^89 + 8 * q^91 + 12 * q^94 - 4 * q^96

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	$0.195913\pi$
$3$	2.00000i	1.15470i	−0.816497	+	0.577350i	$0.804087\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−2.00000	−0.816497
$7$	2.00000i	0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$	2.00000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	− 2.00000i	− 0.577350i
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	6.00000i	1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$	6.00000i	1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$	− 1.00000i	− 0.235702i
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	−4.00000	−0.872872
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	2.00000	0.408248
$25$	0	0
$26$	2.00000	0.392232
$27$	4.00000i	0.769800i
$28$	− 2.00000i	− 0.377964i
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	−6.00000	−1.02899
$35$	0	0
$36$	1.00000	0.166667
$37$	1.00000i	0.164399i
$37$	1.00000i	0.164399i
$38$	− 2.00000i	− 0.324443i
$39$	4.00000	0.640513
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	− 4.00000i	− 0.617213i
$43$	4.00000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 6.00000i	− 0.875190i	−0.899172	−	0.437595i	$-0.855830\pi$
$47$	− 6.00000i	− 0.875190i	0.899172	−	0.437595i	$-0.144170\pi$
$48$	2.00000i	0.288675i
$49$	3.00000	0.428571
$50$	0	0
$51$	−12.0000	−1.68034
$52$	2.00000i	0.277350i
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	−4.00000	−0.544331
$55$	0	0
$56$	2.00000	0.267261
$57$	− 4.00000i	− 0.529813i
$58$	− 6.00000i	− 0.787839i
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	− 10.0000i	− 1.27000i
$63$	− 2.00000i	− 0.251976i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	2.00000i	0.244339i	0.992509	+	0.122169i	$0.0389851\pi$
$67$	2.00000i	0.244339i	−0.992509	+	0.122169i	$0.961015\pi$
$68$	− 6.00000i	− 0.727607i
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	1.00000i	0.117851i
$73$	− 2.00000i	− 0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$	− 2.00000i	− 0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$	−1.00000	−0.116248
$75$	0	0
$76$	2.00000	0.229416
$77$	0	0
$78$	4.00000i	0.452911i
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	− 6.00000i	− 0.662589i
$83$	6.00000i	0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$	6.00000i	0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$	4.00000	0.436436
$85$	0	0
$86$	−4.00000	−0.431331
$87$	− 12.0000i	− 1.28654i
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	0	0
$93$	− 20.0000i	− 2.07390i
$94$	6.00000	0.618853
$95$	0	0
$96$	−2.00000	−0.204124
$97$	2.00000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	2.00000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	3.00000i	0.303046i
$99$	0	0
$100$	0	0
$101$	18.0000	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$102$	− 12.0000i	− 1.18818i
$103$	4.00000i	0.394132i	0.980390	+	0.197066i	$0.0631413\pi$
$103$	4.00000i	0.394132i	−0.980390	+	0.197066i	$0.936859\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	6.00000	0.582772
$107$	6.00000i	0.580042i	0.957020	+	0.290021i	$0.0936623\pi$
$107$	6.00000i	0.580042i	−0.957020	+	0.290021i	$0.906338\pi$
$108$	− 4.00000i	− 0.384900i
$109$	−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
$111$	−2.00000	−0.189832
$112$	2.00000i	0.188982i
$113$	− 6.00000i	− 0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$	− 6.00000i	− 0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	4.00000	0.374634
$115$	0	0
$116$	6.00000	0.557086
$117$	2.00000i	0.184900i
$118$	6.00000i	0.552345i
$119$	−12.0000	−1.10004
$120$	0	0
$121$	−11.0000	−1.00000
$122$	− 10.0000i	− 0.905357i
$123$	− 12.0000i	− 1.08200i
$124$	10.0000	0.898027
$125$	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	$0.971717\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−8.00000	−0.704361
$130$	0	0
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	0	0
$133$	− 4.00000i	− 0.346844i
$134$	−2.00000	−0.172774
$135$	0	0
$136$	6.00000	0.514496
$137$	− 6.00000i	− 0.512615i	−0.966595	−	0.256307i	$-0.917494\pi$
$137$	− 6.00000i	− 0.512615i	0.966595	−	0.256307i	$-0.0825059\pi$
$138$	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
$141$	12.0000	1.01058
$142$	0	0
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	2.00000	0.165521
$147$	6.00000i	0.494872i
$148$	− 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
$151$	20.0000	1.62758	0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000	1.62758	0.813788	+	0.581161i	$0.197401\pi$
$152$	2.00000i	0.162221i
$153$	− 6.00000i	− 0.485071i
$154$	0	0
$155$	0	0
$156$	−4.00000	−0.320256
$157$	− 10.0000i	− 0.798087i	−0.916932	−	0.399043i	$-0.869342\pi$
$157$	− 10.0000i	− 0.798087i	0.916932	−	0.399043i	$-0.130658\pi$
$158$	10.0000i	0.795557i
$159$	12.0000	0.951662
$160$	0	0
$161$	0	0
$162$	− 11.0000i	− 0.864242i
$163$	16.0000i	1.25322i	0.779334	+	0.626608i	$0.215557\pi$
$163$	16.0000i	1.25322i	−0.779334	+	0.626608i	$0.784443\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	−6.00000	−0.465690
$167$	12.0000i	0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$	12.0000i	0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$	4.00000i	0.308607i
$169$	9.00000	0.692308
$170$	0	0
$171$	2.00000	0.152944
$172$	− 4.00000i	− 0.304997i
$173$	18.0000i	1.36851i	0.729241	+	0.684257i	$0.239873\pi$
$173$	18.0000i	1.36851i	−0.729241	+	0.684257i	$0.760127\pi$
$174$	12.0000	0.909718
$175$	0	0
$176$	0	0
$177$	12.0000i	0.901975i
$178$	6.00000i	0.449719i
$179$	−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
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	4.00000i	0.296500i
$183$	− 20.0000i	− 1.47844i
$184$	0	0
$185$	0	0
$186$	20.0000	1.46647
$187$	0	0
$188$	6.00000i	0.437595i
$189$	−8.00000	−0.581914
$190$	0	0
$191$	−6.00000	−0.434145	−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000	−0.434145	−0.217072	+	0.976156i	$0.569651\pi$
$192$	− 2.00000i	− 0.144338i
$193$	− 2.00000i	− 0.143963i	−0.997406	−	0.0719816i	$-0.977068\pi$
$193$	− 2.00000i	− 0.143963i	0.997406	−	0.0719816i	$-0.0229323\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	−3.00000	−0.214286
$197$	− 18.0000i	− 1.28245i	−0.767354	−	0.641223i	$-0.778427\pi$
$197$	− 18.0000i	− 1.28245i	0.767354	−	0.641223i	$-0.221573\pi$
$198$	0	0
$199$	22.0000	1.55954	0.779769	−	0.626067i	$-0.215336\pi$
$199$	22.0000	1.55954	0.779769	+	0.626067i	$0.215336\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	18.0000i	1.26648i
$203$	− 12.0000i	− 0.842235i
$204$	12.0000	0.840168
$205$	0	0
$206$	−4.00000	−0.278693
$207$	0	0
$208$	− 2.00000i	− 0.138675i
$209$	0	0
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	6.00000i	0.412082i
$213$	0	0
$214$	−6.00000	−0.410152
$215$	0	0
$216$	4.00000	0.272166
$217$	− 20.0000i	− 1.35769i
$218$	− 14.0000i	− 0.948200i
$219$	4.00000	0.270295
$220$	0	0
$221$	12.0000	0.807207
$222$	− 2.00000i	− 0.134231i
$223$	− 14.0000i	− 0.937509i	−0.883328	−	0.468755i	$-0.844703\pi$
$223$	− 14.0000i	− 0.937509i	0.883328	−	0.468755i	$-0.155297\pi$
$224$	−2.00000	−0.133631
$225$	0	0
$226$	6.00000	0.399114
$227$	24.0000i	1.59294i	0.604681	+	0.796468i	$0.293301\pi$
$227$	24.0000i	1.59294i	−0.604681	+	0.796468i	$0.706699\pi$
$228$	4.00000i	0.264906i
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	0	0
$232$	6.00000i	0.393919i
$233$	− 18.0000i	− 1.17922i	−0.807688	−	0.589610i	$-0.799282\pi$
$233$	− 18.0000i	− 1.17922i	0.807688	−	0.589610i	$-0.200718\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	−6.00000	−0.390567
$237$	20.0000i	1.29914i
$238$	− 12.0000i	− 0.777844i
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	− 11.0000i	− 0.707107i
$243$	− 10.0000i	− 0.641500i
$244$	10.0000	0.640184
$245$	0	0
$246$	12.0000	0.765092
$247$	4.00000i	0.254514i
$248$	10.0000i	0.635001i
$249$	−12.0000	−0.760469
$250$	0	0
$251$	18.0000	1.13615	0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000	1.13615	0.568075	+	0.822977i	$0.307688\pi$
$252$	2.00000i	0.125988i
$253$	0	0
$254$	−2.00000	−0.125491
$255$	0	0
$256$	1.00000	0.0625000
$257$	6.00000i	0.374270i	0.982334	+	0.187135i	$0.0599201\pi$
$257$	6.00000i	0.374270i	−0.982334	+	0.187135i	$0.940080\pi$
$258$	− 8.00000i	− 0.498058i
$259$	−2.00000	−0.124274
$260$	0	0
$261$	6.00000	0.371391
$262$	− 6.00000i	− 0.370681i
$263$	6.00000i	0.369976i	0.982741	+	0.184988i	$0.0592246\pi$
$263$	6.00000i	0.369976i	−0.982741	+	0.184988i	$0.940775\pi$
$264$	0	0
$265$	0	0
$266$	4.00000	0.245256
$267$	12.0000i	0.734388i
$268$	− 2.00000i	− 0.122169i
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	6.00000i	0.363803i
$273$	8.00000i	0.484182i
$274$	6.00000	0.362473
$275$	0	0
$276$	0	0
$277$	26.0000i	1.56219i	0.624413	+	0.781094i	$0.285338\pi$
$277$	26.0000i	1.56219i	−0.624413	+	0.781094i	$0.714662\pi$
$278$	4.00000i	0.239904i
$279$	10.0000	0.598684
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	12.0000i	0.714590i
$283$	16.0000i	0.951101i	0.879688	+	0.475551i	$0.157751\pi$
$283$	16.0000i	0.951101i	−0.879688	+	0.475551i	$0.842249\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 12.0000i	− 0.708338i
$288$	− 1.00000i	− 0.0589256i
$289$	−19.0000	−1.11765
$290$	0	0
$291$	−4.00000	−0.234484
$292$	2.00000i	0.117041i
$293$	− 6.00000i	− 0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$	− 6.00000i	− 0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$	−6.00000	−0.349927
$295$	0	0
$296$	1.00000	0.0581238
$297$	0	0
$298$	− 18.0000i	− 1.04271i
$299$	0	0
$300$	0	0
$301$	−8.00000	−0.461112
$302$	20.0000i	1.15087i
$303$	36.0000i	2.06815i
$304$	−2.00000	−0.114708
$305$	0	0
$306$	6.00000	0.342997
$307$	26.0000i	1.48390i	0.670456	+	0.741949i	$0.266098\pi$
$307$	26.0000i	1.48390i	−0.670456	+	0.741949i	$0.733902\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	0	0
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	− 4.00000i	− 0.226455i
$313$	22.0000i	1.24351i	0.783210	+	0.621757i	$0.213581\pi$
$313$	22.0000i	1.24351i	−0.783210	+	0.621757i	$0.786419\pi$
$314$	10.0000	0.564333
$315$	0	0
$316$	−10.0000	−0.562544
$317$	− 18.0000i	− 1.01098i	−0.862832	−	0.505490i	$-0.831312\pi$
$317$	− 18.0000i	− 1.01098i	0.862832	−	0.505490i	$-0.168688\pi$
$318$	12.0000i	0.672927i
$319$	0	0
$320$	0	0
$321$	−12.0000	−0.669775
$322$	0	0
$323$	− 12.0000i	− 0.667698i
$324$	11.0000	0.611111
$325$	0	0
$326$	−16.0000	−0.886158
$327$	− 28.0000i	− 1.54840i
$328$	6.00000i	0.331295i
$329$	12.0000	0.661581
$330$	0	0
$331$	−10.0000	−0.549650	−0.274825	−	0.961494i	$-0.588620\pi$
$331$	−10.0000	−0.549650	−0.274825	+	0.961494i	$0.588620\pi$
$332$	− 6.00000i	− 0.329293i
$333$	− 1.00000i	− 0.0547997i
$334$	−12.0000	−0.656611
$335$	0	0
$336$	−4.00000	−0.218218
$337$	2.00000i	0.108947i	0.998515	+	0.0544735i	$0.0173480\pi$
$337$	2.00000i	0.108947i	−0.998515	+	0.0544735i	$0.982652\pi$
$338$	9.00000i	0.489535i
$339$	12.0000	0.651751
$340$	0	0
$341$	0	0
$342$	2.00000i	0.108148i
$343$	20.0000i	1.07990i
$344$	4.00000	0.215666
$345$	0	0
$346$	−18.0000	−0.967686
$347$	− 12.0000i	− 0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$	− 12.0000i	− 0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$	12.0000i	0.643268i
$349$	−26.0000	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$350$	0	0
$351$	8.00000	0.427008
$352$	0	0
$353$	30.0000i	1.59674i	0.602168	+	0.798369i	$0.294304\pi$
$353$	30.0000i	1.59674i	−0.602168	+	0.798369i	$0.705696\pi$
$354$	−12.0000	−0.637793
$355$	0	0
$356$	−6.00000	−0.317999
$357$	− 24.0000i	− 1.27021i
$358$	− 6.00000i	− 0.317110i
$359$	−36.0000	−1.90001	−0.950004	−	0.312239i	$-0.898921\pi$
$359$	−36.0000	−1.90001	−0.950004	+	0.312239i	$0.898921\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	− 22.0000i	− 1.15629i
$363$	− 22.0000i	− 1.15470i
$364$	−4.00000	−0.209657
$365$	0	0
$366$	20.0000	1.04542
$367$	2.00000i	0.104399i	0.998637	+	0.0521996i	$0.0166232\pi$
$367$	2.00000i	0.104399i	−0.998637	+	0.0521996i	$0.983377\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	0	0
$371$	12.0000	0.623009
$372$	20.0000i	1.03695i
$373$	34.0000i	1.76045i	0.474554	+	0.880227i	$0.342610\pi$
$373$	34.0000i	1.76045i	−0.474554	+	0.880227i	$0.657390\pi$
$374$	0	0
$375$	0	0
$376$	−6.00000	−0.309426
$377$	12.0000i	0.618031i
$378$	− 8.00000i	− 0.411476i
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	0	0
$381$	−4.00000	−0.204926
$382$	− 6.00000i	− 0.306987i
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	2.00000	0.102062
$385$	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
$392$	− 3.00000i	− 0.151523i
$393$	− 12.0000i	− 0.605320i
$394$	18.0000	0.906827
$395$	0	0
$396$	0	0
$397$	14.0000i	0.702640i	0.936255	+	0.351320i	$0.114267\pi$
$397$	14.0000i	0.702640i	−0.936255	+	0.351320i	$0.885733\pi$
$398$	22.0000i	1.10276i
$399$	8.00000	0.400501
$400$	0	0
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	− 4.00000i	− 0.199502i
$403$	20.0000i	0.996271i
$404$	−18.0000	−0.895533
$405$	0	0
$406$	12.0000	0.595550
$407$	0	0
$408$	12.0000i	0.594089i
$409$	−2.00000	−0.0988936	−0.0494468	−	0.998777i	$-0.515746\pi$
$409$	−2.00000	−0.0988936	−0.0494468	+	0.998777i	$0.515746\pi$
$410$	0	0
$411$	12.0000	0.591916
$412$	− 4.00000i	− 0.197066i
$413$	12.0000i	0.590481i
$414$	0	0
$415$	0	0
$416$	2.00000	0.0980581
$417$	8.00000i	0.391762i
$418$	0	0
$419$	−24.0000	−1.17248	−0.586238	−	0.810139i	$-0.699392\pi$
$419$	−24.0000	−1.17248	−0.586238	+	0.810139i	$0.699392\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	20.0000i	0.973585i
$423$	6.00000i	0.291730i
$424$	−6.00000	−0.291386
$425$	0	0
$426$	0	0
$427$	− 20.0000i	− 0.967868i
$428$	− 6.00000i	− 0.290021i
$429$	0	0
$430$	0	0
$431$	30.0000	1.44505	0.722525	−	0.691345i	$-0.242982\pi$
$431$	30.0000	1.44505	0.722525	+	0.691345i	$0.242982\pi$
$432$	4.00000i	0.192450i
$433$	− 2.00000i	− 0.0961139i	−0.998845	−	0.0480569i	$-0.984697\pi$
$433$	− 2.00000i	− 0.0961139i	0.998845	−	0.0480569i	$-0.0153029\pi$
$434$	20.0000	0.960031
$435$	0	0
$436$	14.0000	0.670478
$437$	0	0
$438$	4.00000i	0.191127i
$439$	22.0000	1.05000	0.525001	−	0.851101i	$-0.324065\pi$
$439$	22.0000	1.05000	0.525001	+	0.851101i	$0.324065\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	12.0000i	0.570782i
$443$	− 6.00000i	− 0.285069i	−0.989790	−	0.142534i	$-0.954475\pi$
$443$	− 6.00000i	− 0.285069i	0.989790	−	0.142534i	$-0.0455251\pi$
$444$	2.00000	0.0949158
$445$	0	0
$446$	14.0000	0.662919
$447$	− 36.0000i	− 1.70274i
$448$	− 2.00000i	− 0.0944911i
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	0	0
$451$	0	0
$452$	6.00000i	0.282216i
$453$	40.0000i	1.87936i
$454$	−24.0000	−1.12638
$455$	0	0
$456$	−4.00000	−0.187317
$457$	26.0000i	1.21623i	0.793849	+	0.608114i	$0.208074\pi$
$457$	26.0000i	1.21623i	−0.793849	+	0.608114i	$0.791926\pi$
$458$	10.0000i	0.467269i
$459$	−24.0000	−1.12022
$460$	0	0
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	0	0
$463$	40.0000i	1.85896i	0.368875	+	0.929479i	$0.379743\pi$
$463$	40.0000i	1.85896i	−0.368875	+	0.929479i	$0.620257\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	18.0000	0.833834
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	− 2.00000i	− 0.0924500i
$469$	−4.00000	−0.184703
$470$	0	0
$471$	20.0000	0.921551
$472$	− 6.00000i	− 0.276172i
$473$	0	0
$474$	−20.0000	−0.918630
$475$	0	0
$476$	12.0000	0.550019
$477$	6.00000i	0.274721i
$478$	− 6.00000i	− 0.274434i
$479$	18.0000	0.822441	0.411220	−	0.911536i	$-0.365103\pi$
$479$	18.0000	0.822441	0.411220	+	0.911536i	$0.365103\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	− 22.0000i	− 1.00207i
$483$	0	0
$484$	11.0000	0.500000
$485$	0	0
$486$	10.0000	0.453609
$487$	20.0000i	0.906287i	0.891438	+	0.453143i	$0.149697\pi$
$487$	20.0000i	0.906287i	−0.891438	+	0.453143i	$0.850303\pi$
$488$	10.0000i	0.452679i
$489$	−32.0000	−1.44709
$490$	0	0
$491$	36.0000	1.62466	0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000	1.62466	0.812329	+	0.583200i	$0.198200\pi$
$492$	12.0000i	0.541002i
$493$	− 36.0000i	− 1.62136i
$494$	−4.00000	−0.179969
$495$	0	0
$496$	−10.0000	−0.449013
$497$	0	0
$498$	− 12.0000i	− 0.537733i
$499$	−14.0000	−0.626726	−0.313363	−	0.949633i	$-0.601456\pi$
$499$	−14.0000	−0.626726	−0.313363	+	0.949633i	$0.601456\pi$
$500$	0	0
$501$	−24.0000	−1.07224
$502$	18.0000i	0.803379i
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	−2.00000	−0.0890871
$505$	0	0
$506$	0	0
$507$	18.0000i	0.799408i
$508$	− 2.00000i	− 0.0887357i
$509$	−30.0000	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$510$	0	0
$511$	4.00000	0.176950
$512$	1.00000i	0.0441942i
$513$	− 8.00000i	− 0.353209i
$514$	−6.00000	−0.264649
$515$	0	0
$516$	8.00000	0.352180
$517$	0	0
$518$	− 2.00000i	− 0.0878750i
$519$	−36.0000	−1.58022
$520$	0	0
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	6.00000i	0.262613i
$523$	4.00000i	0.174908i	0.996169	+	0.0874539i	$0.0278730\pi$
$523$	4.00000i	0.174908i	−0.996169	+	0.0874539i	$0.972127\pi$
$524$	6.00000	0.262111
$525$	0	0
$526$	−6.00000	−0.261612
$527$	− 60.0000i	− 2.61364i
$528$	0	0
$529$	23.0000	1.00000
$530$	0	0
$531$	−6.00000	−0.260378
$532$	4.00000i	0.173422i
$533$	12.0000i	0.519778i
$534$	−12.0000	−0.519291
$535$	0	0
$536$	2.00000	0.0863868
$537$	− 12.0000i	− 0.517838i
$538$	6.00000i	0.258678i
$539$	0	0
$540$	0	0
$541$	14.0000	0.601907	0.300954	−	0.953639i	$-0.402695\pi$
$541$	14.0000	0.601907	0.300954	+	0.953639i	$0.402695\pi$
$542$	− 16.0000i	− 0.687259i
$543$	− 44.0000i	− 1.88822i
$544$	−6.00000	−0.257248
$545$	0	0
$546$	−8.00000	−0.342368
$547$	44.0000i	1.88130i	0.339372	+	0.940652i	$0.389785\pi$
$547$	44.0000i	1.88130i	−0.339372	+	0.940652i	$0.610215\pi$
$548$	6.00000i	0.256307i
$549$	10.0000	0.426790
$550$	0	0
$551$	12.0000	0.511217
$552$	0	0
$553$	20.0000i	0.850487i
$554$	−26.0000	−1.10463
$555$	0	0
$556$	−4.00000	−0.169638
$557$	− 30.0000i	− 1.27114i	−0.772043	−	0.635570i	$-0.780765\pi$
$557$	− 30.0000i	− 1.27114i	0.772043	−	0.635570i	$-0.219235\pi$
$558$	10.0000i	0.423334i
$559$	8.00000	0.338364
$560$	0	0
$561$	0	0
$562$	18.0000i	0.759284i
$563$	− 36.0000i	− 1.51722i	−0.651546	−	0.758610i	$-0.725879\pi$
$563$	− 36.0000i	− 1.51722i	0.651546	−	0.758610i	$-0.274121\pi$
$564$	−12.0000	−0.505291
$565$	0	0
$566$	−16.0000	−0.672530
$567$	− 22.0000i	− 0.923913i
$568$	0	0
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	0	0
$573$	− 12.0000i	− 0.501307i
$574$	12.0000	0.500870
$575$	0	0
$576$	1.00000	0.0416667
$577$	38.0000i	1.58196i	0.611842	+	0.790980i	$0.290429\pi$
$577$	38.0000i	1.58196i	−0.611842	+	0.790980i	$0.709571\pi$
$578$	− 19.0000i	− 0.790296i
$579$	4.00000	0.166234
$580$	0	0
$581$	−12.0000	−0.497844
$582$	− 4.00000i	− 0.165805i
$583$	0	0
$584$	−2.00000	−0.0827606
$585$	0	0
$586$	6.00000	0.247858
$587$	− 36.0000i	− 1.48588i	−0.669359	−	0.742940i	$-0.733431\pi$
$587$	− 36.0000i	− 1.48588i	0.669359	−	0.742940i	$-0.266569\pi$
$588$	− 6.00000i	− 0.247436i
$589$	20.0000	0.824086
$590$	0	0
$591$	36.0000	1.48084
$592$	1.00000i	0.0410997i
$593$	6.00000i	0.246390i	0.992382	+	0.123195i	$0.0393141\pi$
$593$	6.00000i	0.246390i	−0.992382	+	0.123195i	$0.960686\pi$
$594$	0	0
$595$	0	0
$596$	18.0000	0.737309
$597$	44.0000i	1.80080i
$598$	0	0
$599$	36.0000	1.47092	0.735460	−	0.677568i	$-0.236966\pi$
$599$	36.0000	1.47092	0.735460	+	0.677568i	$0.236966\pi$
$600$	0	0
$601$	38.0000	1.55005	0.775026	−	0.631929i	$-0.217737\pi$
$601$	38.0000	1.55005	0.775026	+	0.631929i	$0.217737\pi$
$602$	− 8.00000i	− 0.326056i
$603$	− 2.00000i	− 0.0814463i
$604$	−20.0000	−0.813788
$605$	0	0
$606$	−36.0000	−1.46240
$607$	32.0000i	1.29884i	0.760430	+	0.649420i	$0.224988\pi$
$607$	32.0000i	1.29884i	−0.760430	+	0.649420i	$0.775012\pi$
$608$	− 2.00000i	− 0.0811107i
$609$	24.0000	0.972529
$610$	0	0
$611$	−12.0000	−0.485468
$612$	6.00000i	0.242536i
$613$	34.0000i	1.37325i	0.727013	+	0.686624i	$0.240908\pi$
$613$	34.0000i	1.37325i	−0.727013	+	0.686624i	$0.759092\pi$
$614$	−26.0000	−1.04927
$615$	0	0
$616$	0	0
$617$	− 30.0000i	− 1.20775i	−0.797077	−	0.603877i	$-0.793622\pi$
$617$	− 30.0000i	− 1.20775i	0.797077	−	0.603877i	$-0.206378\pi$
$618$	− 8.00000i	− 0.321807i
$619$	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
$622$	− 18.0000i	− 0.721734i
$623$	12.0000i	0.480770i
$624$	4.00000	0.160128
$625$	0	0
$626$	−22.0000	−0.879297
$627$	0	0
$628$	10.0000i	0.399043i
$629$	−6.00000	−0.239236
$630$	0	0
$631$	14.0000	0.557331	0.278666	−	0.960388i	$-0.410108\pi$
$631$	14.0000	0.557331	0.278666	+	0.960388i	$0.410108\pi$
$632$	− 10.0000i	− 0.397779i
$633$	40.0000i	1.58986i
$634$	18.0000	0.714871
$635$	0	0
$636$	−12.0000	−0.475831
$637$	− 6.00000i	− 0.237729i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	− 12.0000i	− 0.473602i
$643$	− 32.0000i	− 1.26196i	−0.775800	−	0.630978i	$-0.782654\pi$
$643$	− 32.0000i	− 1.26196i	0.775800	−	0.630978i	$-0.217346\pi$
$644$	0	0
$645$	0	0
$646$	12.0000	0.472134
$647$	48.0000i	1.88707i	0.331266	+	0.943537i	$0.392524\pi$
$647$	48.0000i	1.88707i	−0.331266	+	0.943537i	$0.607476\pi$
$648$	11.0000i	0.432121i
$649$	0	0
$650$	0	0
$651$	40.0000	1.56772
$652$	− 16.0000i	− 0.626608i
$653$	18.0000i	0.704394i	0.935926	+	0.352197i	$0.114565\pi$
$653$	18.0000i	0.704394i	−0.935926	+	0.352197i	$0.885435\pi$
$654$	28.0000	1.09489
$655$	0	0
$656$	−6.00000	−0.234261
$657$	2.00000i	0.0780274i
$658$	12.0000i	0.467809i
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	0	0
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	− 10.0000i	− 0.388661i
$663$	24.0000i	0.932083i
$664$	6.00000	0.232845
$665$	0	0
$666$	1.00000	0.0387492
$667$	0	0
$668$	− 12.0000i	− 0.464294i
$669$	28.0000	1.08254
$670$	0	0
$671$	0	0
$672$	− 4.00000i	− 0.154303i
$673$	− 26.0000i	− 1.00223i	−0.865382	−	0.501113i	$-0.832924\pi$
$673$	− 26.0000i	− 1.00223i	0.865382	−	0.501113i	$-0.167076\pi$
$674$	−2.00000	−0.0770371
$675$	0	0
$676$	−9.00000	−0.346154
$677$	− 42.0000i	− 1.61419i	−0.590421	−	0.807096i	$-0.701038\pi$
$677$	− 42.0000i	− 1.61419i	0.590421	−	0.807096i	$-0.298962\pi$
$678$	12.0000i	0.460857i
$679$	−4.00000	−0.153506
$680$	0	0
$681$	−48.0000	−1.83936
$682$	0	0
$683$	− 36.0000i	− 1.37750i	−0.724998	−	0.688751i	$-0.758159\pi$
$683$	− 36.0000i	− 1.37750i	0.724998	−	0.688751i	$-0.241841\pi$
$684$	−2.00000	−0.0764719
$685$	0	0
$686$	−20.0000	−0.763604
$687$	20.0000i	0.763048i
$688$	4.00000i	0.152499i
$689$	−12.0000	−0.457164
$690$	0	0
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	− 18.0000i	− 0.684257i
$693$	0	0
$694$	12.0000	0.455514
$695$	0	0
$696$	−12.0000	−0.454859
$697$	− 36.0000i	− 1.36360i
$698$	− 26.0000i	− 0.984115i
$699$	36.0000	1.36165
$700$	0	0
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	8.00000i	0.301941i
$703$	− 2.00000i	− 0.0754314i
$704$	0	0
$705$	0	0
$706$	−30.0000	−1.12906
$707$	36.0000i	1.35392i
$708$	− 12.0000i	− 0.450988i
$709$	−38.0000	−1.42712	−0.713560	−	0.700594i	$-0.752918\pi$
$709$	−38.0000	−1.42712	−0.713560	+	0.700594i	$0.752918\pi$
$710$	0	0
$711$	−10.0000	−0.375029
$712$	− 6.00000i	− 0.224860i
$713$	0	0
$714$	24.0000	0.898177
$715$	0	0
$716$	6.00000	0.224231
$717$	− 12.0000i	− 0.448148i
$718$	− 36.0000i	− 1.34351i
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	− 15.0000i	− 0.558242i
$723$	− 44.0000i	− 1.63638i
$724$	22.0000	0.817624
$725$	0	0
$726$	22.0000	0.816497
$727$	32.0000i	1.18681i	0.804902	+	0.593407i	$0.202218\pi$
$727$	32.0000i	1.18681i	−0.804902	+	0.593407i	$0.797782\pi$
$728$	− 4.00000i	− 0.148250i
$729$	−13.0000	−0.481481
$730$	0	0
$731$	−24.0000	−0.887672
$732$	20.0000i	0.739221i
$733$	− 14.0000i	− 0.517102i	−0.965998	−	0.258551i	$-0.916755\pi$
$733$	− 14.0000i	− 0.517102i	0.965998	−	0.258551i	$-0.0832450\pi$
$734$	−2.00000	−0.0738213
$735$	0	0
$736$	0	0
$737$	0	0
$738$	6.00000i	0.220863i
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	−8.00000	−0.293887
$742$	12.0000i	0.440534i
$743$	− 18.0000i	− 0.660356i	−0.943919	−	0.330178i	$-0.892891\pi$
$743$	− 18.0000i	− 0.660356i	0.943919	−	0.330178i	$-0.107109\pi$
$744$	−20.0000	−0.733236
$745$	0	0
$746$	−34.0000	−1.24483
$747$	− 6.00000i	− 0.219529i
$748$	0	0
$749$	−12.0000	−0.438470
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	− 6.00000i	− 0.218797i
$753$	36.0000i	1.31191i
$754$	−12.0000	−0.437014
$755$	0	0
$756$	8.00000	0.290957
$757$	− 10.0000i	− 0.363456i	−0.983349	−	0.181728i	$-0.941831\pi$
$757$	− 10.0000i	− 0.363456i	0.983349	−	0.181728i	$-0.0581691\pi$
$758$	16.0000i	0.581146i
$759$	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	$0.534685\pi$
$762$	− 4.00000i	− 0.144905i
$763$	− 28.0000i	− 1.01367i
$764$	6.00000	0.217072
$765$	0	0
$766$	0	0
$767$	− 12.0000i	− 0.433295i
$768$	2.00000i	0.0721688i
$769$	−26.0000	−0.937584	−0.468792	−	0.883309i	$-0.655311\pi$
$769$	−26.0000	−0.937584	−0.468792	+	0.883309i	$0.655311\pi$
$770$	0	0
$771$	−12.0000	−0.432169
$772$	2.00000i	0.0719816i
$773$	18.0000i	0.647415i	0.946157	+	0.323708i	$0.104929\pi$
$773$	18.0000i	0.647415i	−0.946157	+	0.323708i	$0.895071\pi$
$774$	4.00000	0.143777
$775$	0	0
$776$	2.00000	0.0717958
$777$	− 4.00000i	− 0.143499i
$778$	− 6.00000i	− 0.215110i
$779$	12.0000	0.429945
$780$	0	0
$781$	0	0
$782$	0	0
$783$	− 24.0000i	− 0.857690i
$784$	3.00000	0.107143
$785$	0	0
$786$	12.0000	0.428026
$787$	− 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
$789$	−12.0000	−0.427211
$790$	0	0
$791$	12.0000	0.426671
$792$	0	0
$793$	20.0000i	0.710221i
$794$	−14.0000	−0.496841
$795$	0	0
$796$	−22.0000	−0.779769
$797$	− 30.0000i	− 1.06265i	−0.847167	−	0.531327i	$-0.821693\pi$
$797$	− 30.0000i	− 1.06265i	0.847167	−	0.531327i	$-0.178307\pi$
$798$	8.00000i	0.283197i
$799$	36.0000	1.27359
$800$	0	0
$801$	−6.00000	−0.212000
$802$	− 6.00000i	− 0.211867i
$803$	0	0
$804$	4.00000	0.141069
$805$	0	0
$806$	−20.0000	−0.704470
$807$	12.0000i	0.422420i
$808$	− 18.0000i	− 0.633238i
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	12.0000i	0.421117i
$813$	− 32.0000i	− 1.12229i
$814$	0	0
$815$	0	0
$816$	−12.0000	−0.420084
$817$	− 8.00000i	− 0.279885i
$818$	− 2.00000i	− 0.0699284i
$819$	−4.00000	−0.139771
$820$	0	0
$821$	−42.0000	−1.46581	−0.732905	−	0.680331i	$-0.761836\pi$
$821$	−42.0000	−1.46581	−0.732905	+	0.680331i	$0.761836\pi$
$822$	12.0000i	0.418548i
$823$	34.0000i	1.18517i	0.805510	+	0.592583i	$0.201892\pi$
$823$	34.0000i	1.18517i	−0.805510	+	0.592583i	$0.798108\pi$
$824$	4.00000	0.139347
$825$	0	0
$826$	−12.0000	−0.417533
$827$	24.0000i	0.834562i	0.908778	+	0.417281i	$0.137017\pi$
$827$	24.0000i	0.834562i	−0.908778	+	0.417281i	$0.862983\pi$
$828$	0	0
$829$	34.0000	1.18087	0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000	1.18087	0.590434	+	0.807086i	$0.298956\pi$
$830$	0	0
$831$	−52.0000	−1.80386
$832$	2.00000i	0.0693375i
$833$	18.0000i	0.623663i
$834$	−8.00000	−0.277017
$835$	0	0
$836$	0	0
$837$	− 40.0000i	− 1.38260i
$838$	− 24.0000i	− 0.829066i
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	− 10.0000i	− 0.344623i
$843$	36.0000i	1.23991i
$844$	−20.0000	−0.688428
$845$	0	0
$846$	−6.00000	−0.206284
$847$	− 22.0000i	− 0.755929i
$848$	− 6.00000i	− 0.206041i
$849$	−32.0000	−1.09824
$850$	0	0
$851$	0	0
$852$	0	0
$853$	10.0000i	0.342393i	0.985237	+	0.171197i	$0.0547634\pi$
$853$	10.0000i	0.342393i	−0.985237	+	0.171197i	$0.945237\pi$
$854$	20.0000	0.684386
$855$	0	0
$856$	6.00000	0.205076
$857$	42.0000i	1.43469i	0.696717	+	0.717346i	$0.254643\pi$
$857$	42.0000i	1.43469i	−0.696717	+	0.717346i	$0.745357\pi$
$858$	0	0
$859$	−50.0000	−1.70598	−0.852989	−	0.521929i	$-0.825213\pi$
$859$	−50.0000	−1.70598	−0.852989	+	0.521929i	$0.825213\pi$
$860$	0	0
$861$	24.0000	0.817918
$862$	30.0000i	1.02180i
$863$	− 54.0000i	− 1.83818i	−0.394046	−	0.919091i	$-0.628925\pi$
$863$	− 54.0000i	− 1.83818i	0.394046	−	0.919091i	$-0.371075\pi$
$864$	−4.00000	−0.136083
$865$	0	0
$866$	2.00000	0.0679628
$867$	− 38.0000i	− 1.29055i
$868$	20.0000i	0.678844i
$869$	0	0
$870$	0	0
$871$	4.00000	0.135535
$872$	14.0000i	0.474100i
$873$	− 2.00000i	− 0.0676897i
$874$	0	0
$875$	0	0
$876$	−4.00000	−0.135147
$877$	− 10.0000i	− 0.337676i	−0.985644	−	0.168838i	$-0.945999\pi$
$877$	− 10.0000i	− 0.337676i	0.985644	−	0.168838i	$-0.0540015\pi$
$878$	22.0000i	0.742464i
$879$	12.0000	0.404750
$880$	0	0
$881$	−42.0000	−1.41502	−0.707508	−	0.706705i	$-0.750181\pi$
$881$	−42.0000	−1.41502	−0.707508	+	0.706705i	$0.750181\pi$
$882$	− 3.00000i	− 0.101015i
$883$	− 56.0000i	− 1.88455i	−0.334840	−	0.942275i	$-0.608682\pi$
$883$	− 56.0000i	− 1.88455i	0.334840	−	0.942275i	$-0.391318\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	6.00000	0.201574
$887$	− 42.0000i	− 1.41022i	−0.709097	−	0.705111i	$-0.750897\pi$
$887$	− 42.0000i	− 1.41022i	0.709097	−	0.705111i	$-0.249103\pi$
$888$	2.00000i	0.0671156i
$889$	−4.00000	−0.134156
$890$	0	0
$891$	0	0
$892$	14.0000i	0.468755i
$893$	12.0000i	0.401565i
$894$	36.0000	1.20402
$895$	0	0
$896$	2.00000	0.0668153
$897$	0	0
$898$	30.0000i	1.00111i
$899$	60.0000	2.00111
$900$	0	0
$901$	36.0000	1.19933
$902$	0	0
$903$	− 16.0000i	− 0.532447i
$904$	−6.00000	−0.199557
$905$	0	0
$906$	−40.0000	−1.32891
$907$	8.00000i	0.265636i	0.991140	+	0.132818i	$0.0424025\pi$
$907$	8.00000i	0.265636i	−0.991140	+	0.132818i	$0.957597\pi$
$908$	− 24.0000i	− 0.796468i
$909$	−18.0000	−0.597022
$910$	0	0
$911$	−30.0000	−0.993944	−0.496972	−	0.867766i	$-0.665555\pi$
$911$	−30.0000	−0.993944	−0.496972	+	0.867766i	$0.665555\pi$
$912$	− 4.00000i	− 0.132453i
$913$	0	0
$914$	−26.0000	−0.860004
$915$	0	0
$916$	−10.0000	−0.330409
$917$	− 12.0000i	− 0.396275i
$918$	− 24.0000i	− 0.792118i
$919$	46.0000	1.51740	0.758700	−	0.651440i	$-0.225835\pi$
$919$	46.0000	1.51740	0.758700	+	0.651440i	$0.225835\pi$
$920$	0	0
$921$	−52.0000	−1.71346
$922$	6.00000i	0.197599i
$923$	0	0
$924$	0	0
$925$	0	0
$926$	−40.0000	−1.31448
$927$	− 4.00000i	− 0.131377i
$928$	− 6.00000i	− 0.196960i
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	18.0000i	0.589610i
$933$	− 36.0000i	− 1.17859i
$934$	0	0
$935$	0	0
$936$	2.00000	0.0653720
$937$	26.0000i	0.849383i	0.905338	+	0.424691i	$0.139617\pi$
$937$	26.0000i	0.849383i	−0.905338	+	0.424691i	$0.860383\pi$
$938$	− 4.00000i	− 0.130605i
$939$	−44.0000	−1.43589
$940$	0	0
$941$	−30.0000	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$942$	20.0000i	0.651635i
$943$	0	0
$944$	6.00000	0.195283
$945$	0	0
$946$	0	0
$947$	48.0000i	1.55979i	0.625910	+	0.779895i	$0.284728\pi$
$947$	48.0000i	1.55979i	−0.625910	+	0.779895i	$0.715272\pi$
$948$	− 20.0000i	− 0.649570i
$949$	−4.00000	−0.129845
$950$	0	0
$951$	36.0000	1.16738
$952$	12.0000i	0.388922i
$953$	− 42.0000i	− 1.36051i	−0.732974	−	0.680257i	$-0.761868\pi$
$953$	− 42.0000i	− 1.36051i	0.732974	−	0.680257i	$-0.238132\pi$
$954$	−6.00000	−0.194257
$955$	0	0
$956$	6.00000	0.194054
$957$	0	0
$958$	18.0000i	0.581554i
$959$	12.0000	0.387500
$960$	0	0
$961$	69.0000	2.22581
$962$	2.00000i	0.0644826i
$963$	− 6.00000i	− 0.193347i
$964$	22.0000	0.708572
$965$	0	0
$966$	0	0
$967$	− 4.00000i	− 0.128631i	−0.997930	−	0.0643157i	$-0.979514\pi$
$967$	− 4.00000i	− 0.128631i	0.997930	−	0.0643157i	$-0.0204865\pi$
$968$	11.0000i	0.353553i
$969$	24.0000	0.770991
$970$	0	0
$971$	−36.0000	−1.15529	−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000	−1.15529	−0.577647	+	0.816286i	$0.696029\pi$
$972$	10.0000i	0.320750i
$973$	8.00000i	0.256468i
$974$	−20.0000	−0.640841
$975$	0	0
$976$	−10.0000	−0.320092
$977$	18.0000i	0.575871i	0.957650	+	0.287936i	$0.0929689\pi$
$977$	18.0000i	0.575871i	−0.957650	+	0.287936i	$0.907031\pi$
$978$	− 32.0000i	− 1.02325i
$979$	0	0
$980$	0	0
$981$	14.0000	0.446986
$982$	36.0000i	1.14881i
$983$	− 18.0000i	− 0.574111i	−0.957914	−	0.287055i	$-0.907324\pi$
$983$	− 18.0000i	− 0.574111i	0.957914	−	0.287055i	$-0.0926764\pi$
$984$	−12.0000	−0.382546
$985$	0	0
$986$	36.0000	1.14647
$987$	24.0000i	0.763928i
$988$	− 4.00000i	− 0.127257i
$989$	0	0
$990$	0	0
$991$	−34.0000	−1.08005	−0.540023	−	0.841650i	$-0.681584\pi$
$991$	−34.0000	−1.08005	−0.540023	+	0.841650i	$0.681584\pi$
$992$	− 10.0000i	− 0.317500i
$993$	− 20.0000i	− 0.634681i
$994$	0	0
$995$	0	0
$996$	12.0000	0.380235
$997$	− 10.0000i	− 0.316703i	−0.987383	−	0.158352i	$-0.949382\pi$
$997$	− 10.0000i	− 0.316703i	0.987383	−	0.158352i	$-0.0506179\pi$
$998$	− 14.0000i	− 0.443162i
$999$	−4.00000	−0.126554

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.a.d.1.1	✓	1	5.3	odd	4
1850.2.a.f.1.1		1	5.2	odd	4
1850.2.b.b.149.1		2	5.4	even	2	inner
1850.2.b.b.149.2		2	1.1	even	1	trivial
2960.2.a.m.1.1		1	20.3	even	4
3330.2.a.d.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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -2.00000i q^{12} -2.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} +6.00000i q^{17} -1.00000i q^{18} -2.00000 q^{19} -4.00000 q^{21} +2.00000 q^{24} +2.00000 q^{26} +4.00000i q^{27} -2.00000i q^{28} -6.00000 q^{29} -10.0000 q^{31} +1.00000i q^{32} -6.00000 q^{34} +1.00000 q^{36} +1.00000i q^{37} -2.00000i q^{38} +4.00000 q^{39} -6.00000 q^{41} -4.00000i q^{42} +4.00000i q^{43} -6.00000i q^{47} +2.00000i q^{48} +3.00000 q^{49} -12.0000 q^{51} +2.00000i q^{52} -6.00000i q^{53} -4.00000 q^{54} +2.00000 q^{56} -4.00000i q^{57} -6.00000i q^{58} +6.00000 q^{59} -10.0000 q^{61} -10.0000i q^{62} -2.00000i q^{63} -1.00000 q^{64} +2.00000i q^{67} -6.00000i q^{68} +1.00000i q^{72} -2.00000i q^{73} -1.00000 q^{74} +2.00000 q^{76} +4.00000i q^{78} +10.0000 q^{79} -11.0000 q^{81} -6.00000i q^{82} +6.00000i q^{83} +4.00000 q^{84} -4.00000 q^{86} -12.0000i q^{87} +6.00000 q^{89} +4.00000 q^{91} -20.0000i q^{93} +6.00000 q^{94} -2.00000 q^{96} +2.00000i q^{97} +3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 4 q^{6} - 2 q^{9} - 4 q^{14} + 2 q^{16} - 4 q^{19} - 8 q^{21} + 4 q^{24} + 4 q^{26} - 12 q^{29} - 20 q^{31} - 12 q^{34} + 2 q^{36} + 8 q^{39} - 12 q^{41} + 6 q^{49} - 24 q^{51} - 8 q^{54}+ \cdots - 4 q^{96}+O(q^{100}) \) 2 * q - 2 * q^4 - 4 * q^6 - 2 * q^9 - 4 * q^14 + 2 * q^16 - 4 * q^19 - 8 * q^21 + 4 * q^24 + 4 * q^26 - 12 * q^29 - 20 * q^31 - 12 * q^34 + 2 * q^36 + 8 * q^39 - 12 * q^41 + 6 * q^49 - 24 * q^51 - 8 * q^54 + 4 * q^56 + 12 * q^59 - 20 * q^61 - 2 * q^64 - 2 * q^74 + 4 * q^76 + 20 * q^79 - 22 * q^81 + 8 * q^84 - 8 * q^86 + 12 * q^89 + 8 * q^91 + 12 * q^94 - 4 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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