LMFDB - Embedded newform 1900.2.c.f.1749.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,2,Mod(1749,1900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1900, 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("1900.1749");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1900 = 2^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1900.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:	$15.1715763840$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.6594624.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 9x^{4} + 18x^{2} + 1 $ x^6 + 9x^4 + 18x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1749.6
Root			$-1.69963i$ of defining polynomial
Character	$\chi$	$=$	1900.1749
Dual form			1900.2.c.f.1749.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.58836i q^{3} -2.81089i q^{7} -3.69963 q^{9} +O(q^{10})$	$q+2.58836i q^{3} -2.81089i q^{7} -3.69963 q^{9} +1.58836 q^{11} +0.888736i q^{13} +3.98762i q^{17} -1.00000 q^{19} +7.27561 q^{21} +3.30037i q^{23} -1.81089i q^{27} -6.98762 q^{29} -4.51052 q^{31} +4.11126i q^{33} +2.41164i q^{37} -2.30037 q^{39} +5.09888 q^{41} +8.17673i q^{43} +9.08650i q^{47} -0.901116 q^{49} -10.3214 q^{51} +7.79851i q^{53} -2.58836i q^{57} +2.22253 q^{59} -9.90978 q^{61} +10.3993i q^{63} +7.56360i q^{67} -8.54256 q^{69} -0.777472 q^{71} -0.876356i q^{73} -4.46472i q^{77} +8.94182 q^{79} -6.41164 q^{81} +11.6996i q^{83} -18.0865i q^{87} +5.27561 q^{89} +2.49814 q^{91} -11.6749i q^{93} -7.24219i q^{97} -5.87636 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 10 q^{9}+O(q^{10}) $ 6 * q - 10 * q^9	$ 6 q - 10 q^{9} - 2 q^{11} - 6 q^{19} - 16 q^{21} - 6 q^{29} - 2 q^{31} - 26 q^{39} - 6 q^{41} - 42 q^{49} - 24 q^{51} + 12 q^{59} - 10 q^{61} - 18 q^{69} - 6 q^{71} - 4 q^{79} - 50 q^{81} - 28 q^{89} - 46 q^{91}+O(q^{100}) $ 6 * q - 10 * q^9 - 2 * q^11 - 6 * q^19 - 16 * q^21 - 6 * q^29 - 2 * q^31 - 26 * q^39 - 6 * q^41 - 42 * q^49 - 24 * q^51 + 12 * q^59 - 10 * q^61 - 18 * q^69 - 6 * q^71 - 4 * q^79 - 50 * q^81 - 28 * q^89 - 46 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times$.

$n$	$77$	$401$	$951$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	0	0
$2$	0	0
$3$	2.58836i	1.49439i	0.664603	+	0.747196i	$0.268601\pi$
$3$	2.58836i	1.49439i	−0.664603	+	0.747196i	$0.731399\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.81089i	− 1.06242i	−0.847241	−	0.531209i	$-0.821738\pi$
$7$	− 2.81089i	− 1.06242i	0.847241	−	0.531209i	$-0.178262\pi$
$8$	0	0
$9$	−3.69963	−1.23321
$10$	0	0
$11$	1.58836	0.478910	0.239455	−	0.970907i	$-0.423031\pi$
$11$	1.58836	0.478910	0.239455	+	0.970907i	$0.423031\pi$
$12$	0	0
$13$	0.888736i	0.246491i	0.992376	+	0.123245i	$0.0393303\pi$
$13$	0.888736i	0.246491i	−0.992376	+	0.123245i	$0.960670\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.98762i	0.967140i	0.875306	+	0.483570i	$0.160660\pi$
$17$	3.98762i	0.967140i	−0.875306	+	0.483570i	$0.839340\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	7.27561	1.58767
$22$	0	0
$23$	3.30037i	0.688175i	0.938938	+	0.344088i	$0.111812\pi$
$23$	3.30037i	0.688175i	−0.938938	+	0.344088i	$0.888188\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.81089i	− 0.348506i
$28$	0	0
$29$	−6.98762	−1.29757	−0.648784	−	0.760972i	$-0.724722\pi$
$29$	−6.98762	−1.29757	−0.648784	+	0.760972i	$0.724722\pi$
$30$	0	0
$31$	−4.51052	−0.810113	−0.405057	−	0.914292i	$-0.632748\pi$
$31$	−4.51052	−0.810113	−0.405057	+	0.914292i	$0.632748\pi$
$32$	0	0
$33$	4.11126i	0.715679i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.41164i	0.396471i	0.980154	+	0.198235i	$0.0635210\pi$
$37$	2.41164i	0.396471i	−0.980154	+	0.198235i	$0.936479\pi$
$38$	0	0
$39$	−2.30037	−0.368354
$40$	0	0
$41$	5.09888	0.796312	0.398156	−	0.917318i	$-0.369650\pi$
$41$	5.09888	0.796312	0.398156	+	0.917318i	$0.369650\pi$
$42$	0	0
$43$	8.17673i	1.24694i	0.781848	+	0.623470i	$0.214278\pi$
$43$	8.17673i	1.24694i	−0.781848	+	0.623470i	$0.785722\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.08650i	1.32540i	0.748884	+	0.662701i	$0.230590\pi$
$47$	9.08650i	1.32540i	−0.748884	+	0.662701i	$0.769410\pi$
$48$	0	0
$49$	−0.901116	−0.128731
$50$	0	0
$51$	−10.3214	−1.44529
$52$	0	0
$53$	7.79851i	1.07121i	0.844469	+	0.535604i	$0.179916\pi$
$53$	7.79851i	1.07121i	−0.844469	+	0.535604i	$0.820084\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 2.58836i	− 0.342837i
$58$	0	0
$59$	2.22253	0.289349	0.144674	−	0.989479i	$-0.453787\pi$
$59$	2.22253	0.289349	0.144674	+	0.989479i	$0.453787\pi$
$60$	0	0
$61$	−9.90978	−1.26882	−0.634408	−	0.772998i	$-0.718756\pi$
$61$	−9.90978	−1.26882	−0.634408	+	0.772998i	$0.718756\pi$
$62$	0	0
$63$	10.3993i	1.31018i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	7.56360i	0.924041i	0.886869	+	0.462021i	$0.152875\pi$
$67$	7.56360i	0.924041i	−0.886869	+	0.462021i	$0.847125\pi$
$68$	0	0
$69$	−8.54256	−1.02840
$70$	0	0
$71$	−0.777472	−0.0922689	−0.0461345	−	0.998935i	$-0.514690\pi$
$71$	−0.777472	−0.0922689	−0.0461345	+	0.998935i	$0.514690\pi$
$72$	0	0
$73$	− 0.876356i	− 0.102570i	−0.998684	−	0.0512849i	$-0.983668\pi$
$73$	− 0.876356i	− 0.102570i	0.998684	−	0.0512849i	$-0.0163316\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 4.46472i	− 0.508802i
$78$	0	0
$79$	8.94182	1.00603	0.503017	−	0.864277i	$-0.332223\pi$
$79$	8.94182	1.00603	0.503017	+	0.864277i	$0.332223\pi$
$80$	0	0
$81$	−6.41164	−0.712404
$82$	0	0
$83$	11.6996i	1.28420i	0.766620	+	0.642101i	$0.221937\pi$
$83$	11.6996i	1.28420i	−0.766620	+	0.642101i	$0.778063\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 18.0865i	− 1.93908i
$88$	0	0
$89$	5.27561	0.559214	0.279607	−	0.960115i	$-0.409796\pi$
$89$	5.27561	0.559214	0.279607	+	0.960115i	$0.409796\pi$
$90$	0	0
$91$	2.49814	0.261876
$92$	0	0
$93$	− 11.6749i	− 1.21063i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 7.24219i	− 0.735333i	−0.929958	−	0.367667i	$-0.880157\pi$
$97$	− 7.24219i	− 0.735333i	0.929958	−	0.367667i	$-0.119843\pi$
$98$	0	0
$99$	−5.87636	−0.590596
$100$	0	0
$101$	−0.143307	−0.0142596	−0.00712981	−	0.999975i	$-0.502270\pi$
$101$	−0.143307	−0.0142596	−0.00712981	+	0.999975i	$0.502270\pi$
$102$	0	0
$103$	− 11.8319i	− 1.16584i	−0.812531	−	0.582918i	$-0.801911\pi$
$103$	− 11.8319i	− 1.16584i	0.812531	−	0.582918i	$-0.198089\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 10.9222i	− 1.05588i	−0.849280	−	0.527942i	$-0.822964\pi$
$107$	− 10.9222i	− 1.05588i	0.849280	−	0.527942i	$-0.177036\pi$
$108$	0	0
$109$	−15.6749	−1.50138	−0.750690	−	0.660655i	$-0.770279\pi$
$109$	−15.6749	−1.50138	−0.750690	+	0.660655i	$0.770279\pi$
$110$	0	0
$111$	−6.24219	−0.592483
$112$	0	0
$113$	6.33379i	0.595833i	0.954592	+	0.297917i	$0.0962917\pi$
$113$	6.33379i	0.595833i	−0.954592	+	0.297917i	$0.903708\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 3.28799i	− 0.303975i
$118$	0	0
$119$	11.2088	1.02751
$120$	0	0
$121$	−8.47710	−0.770645
$122$	0	0
$123$	13.1978i	1.19000i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 11.7207i	− 1.04004i	−0.854154	−	0.520021i	$-0.825924\pi$
$127$	− 11.7207i	− 1.04004i	0.854154	−	0.520021i	$-0.174076\pi$
$128$	0	0
$129$	−21.1643	−1.86342
$130$	0	0
$131$	−5.45234	−0.476373	−0.238187	−	0.971219i	$-0.576553\pi$
$131$	−5.45234	−0.476373	−0.238187	+	0.971219i	$0.576553\pi$
$132$	0	0
$133$	2.81089i	0.243735i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 14.8764i	− 1.27097i	−0.772112	−	0.635486i	$-0.780800\pi$
$137$	− 14.8764i	− 1.27097i	0.772112	−	0.635486i	$-0.219200\pi$
$138$	0	0
$139$	21.9294	1.86003	0.930015	−	0.367520i	$-0.119793\pi$
$139$	21.9294	1.86003	0.930015	+	0.367520i	$0.119793\pi$
$140$	0	0
$141$	−23.5192	−1.98067
$142$	0	0
$143$	1.41164i	0.118047i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 2.33242i	− 0.192374i
$148$	0	0
$149$	−10.1447	−0.831085	−0.415542	−	0.909574i	$-0.636408\pi$
$149$	−10.1447	−0.831085	−0.415542	+	0.909574i	$0.636408\pi$
$150$	0	0
$151$	−9.66621	−0.786625	−0.393312	−	0.919405i	$-0.628671\pi$
$151$	−9.66621	−0.786625	−0.393312	+	0.919405i	$0.628671\pi$
$152$	0	0
$153$	− 14.7527i	− 1.19269i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	8.25457i	0.658787i	0.944193	+	0.329393i	$0.106844\pi$
$157$	8.25457i	0.658787i	−0.944193	+	0.329393i	$0.893156\pi$
$158$	0	0
$159$	−20.1854	−1.60081
$160$	0	0
$161$	9.27699	0.731129
$162$	0	0
$163$	9.46472i	0.741334i	0.928766	+	0.370667i	$0.120871\pi$
$163$	9.46472i	0.741334i	−0.928766	+	0.370667i	$0.879129\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.92216i	0.148741i	0.997231	+	0.0743705i	$0.0236947\pi$
$167$	1.92216i	0.148741i	−0.997231	+	0.0743705i	$0.976305\pi$
$168$	0	0
$169$	12.2101	0.939242
$170$	0	0
$171$	3.69963	0.282918
$172$	0	0
$173$	− 22.3411i	− 1.69856i	−0.527942	−	0.849280i	$-0.677036\pi$
$173$	− 22.3411i	− 1.69856i	0.527942	−	0.849280i	$-0.322964\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	5.75271i	0.432400i
$178$	0	0
$179$	21.7738	1.62745	0.813723	−	0.581252i	$-0.197437\pi$
$179$	21.7738	1.62745	0.813723	+	0.581252i	$0.197437\pi$
$180$	0	0
$181$	−7.00000	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000	−0.520306	−0.260153	+	0.965567i	$0.583773\pi$
$182$	0	0
$183$	− 25.6501i	− 1.89611i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	6.33379i	0.463173i
$188$	0	0
$189$	−5.09022	−0.370259
$190$	0	0
$191$	3.21015	0.232278	0.116139	−	0.993233i	$-0.462948\pi$
$191$	3.21015	0.232278	0.116139	+	0.993233i	$0.462948\pi$
$192$	0	0
$193$	− 0.522900i	− 0.0376392i	−0.999823	−	0.0188196i	$-0.994009\pi$
$193$	− 0.522900i	− 0.0376392i	0.999823	−	0.0188196i	$-0.00599081\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.60940i	0.185912i	0.995670	+	0.0929562i	$0.0296317\pi$
$197$	2.60940i	0.185912i	−0.995670	+	0.0929562i	$0.970368\pi$
$198$	0	0
$199$	12.8960	0.914175	0.457087	−	0.889422i	$-0.348893\pi$
$199$	12.8960	0.914175	0.457087	+	0.889422i	$0.348893\pi$
$200$	0	0
$201$	−19.5774	−1.38088
$202$	0	0
$203$	19.6414i	1.37856i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 12.2101i	− 0.848664i
$208$	0	0
$209$	−1.58836	−0.109869
$210$	0	0
$211$	23.3535	1.60772	0.803859	−	0.594820i	$-0.202777\pi$
$211$	23.3535	1.60772	0.803859	+	0.594820i	$0.202777\pi$
$212$	0	0
$213$	− 2.01238i	− 0.137886i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	12.6786i	0.860679i
$218$	0	0
$219$	2.26833	0.153279
$220$	0	0
$221$	−3.54394	−0.238391
$222$	0	0
$223$	− 3.50914i	− 0.234990i	−0.993073	−	0.117495i	$-0.962514\pi$
$223$	− 3.50914i	− 0.234990i	0.993073	−	0.117495i	$-0.0374863\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	12.0531i	0.799991i	0.916517	+	0.399996i	$0.130988\pi$
$227$	12.0531i	0.799991i	−0.916517	+	0.399996i	$0.869012\pi$
$228$	0	0
$229$	−21.0407	−1.39041	−0.695204	−	0.718812i	$-0.744686\pi$
$229$	−21.0407	−1.39041	−0.695204	+	0.718812i	$0.744686\pi$
$230$	0	0
$231$	11.5563	0.760350
$232$	0	0
$233$	23.4720i	1.53770i	0.639428	+	0.768851i	$0.279171\pi$
$233$	23.4720i	1.53770i	−0.639428	+	0.768851i	$0.720829\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	23.1447i	1.50341i
$238$	0	0
$239$	−0.263233	−0.0170271	−0.00851355	−	0.999964i	$-0.502710\pi$
$239$	−0.263233	−0.0170271	−0.00851355	+	0.999964i	$0.502710\pi$
$240$	0	0
$241$	−19.8974	−1.28170	−0.640852	−	0.767664i	$-0.721419\pi$
$241$	−19.8974	−1.28170	−0.640852	+	0.767664i	$0.721419\pi$
$242$	0	0
$243$	− 22.0283i	− 1.41312i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 0.888736i	− 0.0565489i
$248$	0	0
$249$	−30.2829	−1.91910
$250$	0	0
$251$	15.1964	0.959188	0.479594	−	0.877491i	$-0.340784\pi$
$251$	15.1964	0.959188	0.479594	+	0.877491i	$0.340784\pi$
$252$	0	0
$253$	5.24219i	0.329574i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 21.8072i	− 1.36029i	−0.733076	−	0.680147i	$-0.761916\pi$
$257$	− 21.8072i	− 1.36029i	0.733076	−	0.680147i	$-0.238084\pi$
$258$	0	0
$259$	6.77885	0.421217
$260$	0	0
$261$	25.8516	1.60017
$262$	0	0
$263$	19.8850i	1.22616i	0.790020	+	0.613081i	$0.210070\pi$
$263$	19.8850i	1.22616i	−0.790020	+	0.613081i	$0.789930\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	13.6552i	0.835685i
$268$	0	0
$269$	31.2880	1.90766	0.953831	−	0.300343i	$-0.0971011\pi$
$269$	31.2880	1.90766	0.953831	+	0.300343i	$0.0971011\pi$
$270$	0	0
$271$	−11.3411	−0.688921	−0.344461	−	0.938801i	$-0.611938\pi$
$271$	−11.3411	−0.688921	−0.344461	+	0.938801i	$0.611938\pi$
$272$	0	0
$273$	6.46610i	0.391346i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	18.1520i	1.09065i	0.838226	+	0.545323i	$0.183593\pi$
$277$	18.1520i	1.09065i	−0.838226	+	0.545323i	$0.816407\pi$
$278$	0	0
$279$	16.6872	0.999039
$280$	0	0
$281$	−6.42030	−0.383003	−0.191501	−	0.981492i	$-0.561336\pi$
$281$	−6.42030	−0.383003	−0.191501	+	0.981492i	$0.561336\pi$
$282$	0	0
$283$	19.2051i	1.14162i	0.821082	+	0.570811i	$0.193371\pi$
$283$	19.2051i	1.14162i	−0.821082	+	0.570811i	$0.806629\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 14.3324i	− 0.846016i
$288$	0	0
$289$	1.09888	0.0646403
$290$	0	0
$291$	18.7454	1.09888
$292$	0	0
$293$	− 32.5192i	− 1.89979i	−0.312568	−	0.949895i	$-0.601189\pi$
$293$	− 32.5192i	− 1.89979i	0.312568	−	0.949895i	$-0.398811\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 2.87636i	− 0.166903i
$298$	0	0
$299$	−2.93316	−0.169629
$300$	0	0
$301$	22.9839	1.32477
$302$	0	0
$303$	− 0.370932i	− 0.0213095i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	26.7810i	1.52847i	0.644935	+	0.764237i	$0.276884\pi$
$307$	26.7810i	1.52847i	−0.644935	+	0.764237i	$0.723116\pi$
$308$	0	0
$309$	30.6253	1.74222
$310$	0	0
$311$	22.2880	1.26384	0.631918	−	0.775035i	$-0.282268\pi$
$311$	22.2880	1.26384	0.631918	+	0.775035i	$0.282268\pi$
$312$	0	0
$313$	22.9629i	1.29794i	0.760815	+	0.648969i	$0.224799\pi$
$313$	22.9629i	1.29794i	−0.760815	+	0.648969i	$0.775201\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	28.1840i	1.58297i	0.611187	+	0.791486i	$0.290692\pi$
$317$	28.1840i	1.58297i	−0.611187	+	0.791486i	$0.709308\pi$
$318$	0	0
$319$	−11.0989	−0.621418
$320$	0	0
$321$	28.2705	1.57791
$322$	0	0
$323$	− 3.98762i	− 0.221877i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 40.5723i	− 2.24365i
$328$	0	0
$329$	25.5412	1.40813
$330$	0	0
$331$	19.1716	1.05377	0.526884	−	0.849937i	$-0.323360\pi$
$331$	19.1716	1.05377	0.526884	+	0.849937i	$0.323360\pi$
$332$	0	0
$333$	− 8.92216i	− 0.488931i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 14.2632i	− 0.776968i	−0.921456	−	0.388484i	$-0.872999\pi$
$337$	− 14.2632i	− 0.776968i	0.921456	−	0.388484i	$-0.127001\pi$
$338$	0	0
$339$	−16.3942	−0.890409
$340$	0	0
$341$	−7.16435	−0.387971
$342$	0	0
$343$	− 17.1433i	− 0.925652i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 1.90249i	− 0.102131i	−0.998695	−	0.0510656i	$-0.983738\pi$
$347$	− 1.90249i	− 0.102131i	0.998695	−	0.0510656i	$-0.0162618\pi$
$348$	0	0
$349$	0.556321	0.0297792	0.0148896	−	0.999889i	$-0.495260\pi$
$349$	0.556321	0.0297792	0.0148896	+	0.999889i	$0.495260\pi$
$350$	0	0
$351$	1.60940	0.0859037
$352$	0	0
$353$	− 13.1075i	− 0.697644i	−0.937189	−	0.348822i	$-0.886582\pi$
$353$	− 13.1075i	− 0.697644i	0.937189	−	0.348822i	$-0.113418\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	29.0124i	1.53550i
$358$	0	0
$359$	21.0197	1.10938	0.554688	−	0.832059i	$-0.312838\pi$
$359$	21.0197	1.10938	0.554688	+	0.832059i	$0.312838\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	− 21.9418i	− 1.15165i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	30.8021i	1.60786i	0.594727	+	0.803928i	$0.297260\pi$
$367$	30.8021i	1.60786i	−0.594727	+	0.803928i	$0.702740\pi$
$368$	0	0
$369$	−18.8640	−0.982019
$370$	0	0
$371$	21.9208	1.13807
$372$	0	0
$373$	− 10.0902i	− 0.522452i	−0.965278	−	0.261226i	$-0.915873\pi$
$373$	− 10.0902i	− 0.522452i	0.965278	−	0.261226i	$-0.0841268\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 6.21015i	− 0.319839i
$378$	0	0
$379$	−31.1286	−1.59897	−0.799484	−	0.600687i	$-0.794894\pi$
$379$	−31.1286	−1.59897	−0.799484	+	0.600687i	$0.794894\pi$
$380$	0	0
$381$	30.3374	1.55423
$382$	0	0
$383$	11.4895i	0.587085i	0.955946	+	0.293542i	$0.0948342\pi$
$383$	11.4895i	0.587085i	−0.955946	+	0.293542i	$0.905166\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 30.2509i	− 1.53774i
$388$	0	0
$389$	35.0146	1.77531	0.887655	−	0.460510i	$-0.152333\pi$
$389$	35.0146	1.77531	0.887655	+	0.460510i	$0.152333\pi$
$390$	0	0
$391$	−13.1606	−0.665562
$392$	0	0
$393$	− 14.1126i	− 0.711889i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	5.88874i	0.295547i	0.989021	+	0.147774i	$0.0472107\pi$
$397$	5.88874i	0.295547i	−0.989021	+	0.147774i	$0.952789\pi$
$398$	0	0
$399$	−7.27561	−0.364236
$400$	0	0
$401$	4.23491	0.211481	0.105741	−	0.994394i	$-0.466279\pi$
$401$	4.23491	0.211481	0.105741	+	0.994394i	$0.466279\pi$
$402$	0	0
$403$	− 4.00866i	− 0.199686i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.83056i	0.189874i
$408$	0	0
$409$	7.05308	0.348753	0.174376	−	0.984679i	$-0.444209\pi$
$409$	7.05308	0.348753	0.174376	+	0.984679i	$0.444209\pi$
$410$	0	0
$411$	38.5054	1.89933
$412$	0	0
$413$	− 6.24729i	− 0.307409i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	56.7614i	2.77962i
$418$	0	0
$419$	−17.7724	−0.868237	−0.434119	−	0.900856i	$-0.642940\pi$
$419$	−17.7724	−0.868237	−0.434119	+	0.900856i	$0.642940\pi$
$420$	0	0
$421$	−5.81818	−0.283561	−0.141780	−	0.989898i	$-0.545283\pi$
$421$	−5.81818	−0.283561	−0.141780	+	0.989898i	$0.545283\pi$
$422$	0	0
$423$	− 33.6167i	− 1.63450i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	27.8553i	1.34801i
$428$	0	0
$429$	−3.65383	−0.176408
$430$	0	0
$431$	−23.1854	−1.11680	−0.558400	−	0.829572i	$-0.688585\pi$
$431$	−23.1854	−1.11680	−0.558400	+	0.829572i	$0.688585\pi$
$432$	0	0
$433$	− 12.3462i	− 0.593319i	−0.954983	−	0.296660i	$-0.904127\pi$
$433$	− 12.3462i	− 0.593319i	0.954983	−	0.296660i	$-0.0958727\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 3.30037i	− 0.157878i
$438$	0	0
$439$	−23.8502	−1.13831	−0.569154	−	0.822231i	$-0.692729\pi$
$439$	−23.8502	−1.13831	−0.569154	+	0.822231i	$0.692729\pi$
$440$	0	0
$441$	3.33379	0.158752
$442$	0	0
$443$	32.6291i	1.55025i	0.631806	+	0.775127i	$0.282314\pi$
$443$	32.6291i	1.55025i	−0.631806	+	0.775127i	$0.717686\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 26.2581i	− 1.24197i
$448$	0	0
$449$	−29.4152	−1.38819	−0.694095	−	0.719884i	$-0.744195\pi$
$449$	−29.4152	−1.38819	−0.694095	+	0.719884i	$0.744195\pi$
$450$	0	0
$451$	8.09888	0.381362
$452$	0	0
$453$	− 25.0197i	− 1.17553i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	26.2880i	1.22970i	0.788644	+	0.614850i	$0.210784\pi$
$457$	26.2880i	1.22970i	−0.788644	+	0.614850i	$0.789216\pi$
$458$	0	0
$459$	7.22115	0.337054
$460$	0	0
$461$	−12.5068	−0.582500	−0.291250	−	0.956647i	$-0.594071\pi$
$461$	−12.5068	−0.582500	−0.291250	+	0.956647i	$0.594071\pi$
$462$	0	0
$463$	− 23.7083i	− 1.10182i	−0.834565	−	0.550909i	$-0.814281\pi$
$463$	− 23.7083i	− 1.10182i	0.834565	−	0.550909i	$-0.185719\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 30.4734i	− 1.41014i	−0.709138	−	0.705070i	$-0.750916\pi$
$467$	− 30.4734i	− 1.41014i	0.709138	−	0.705070i	$-0.249084\pi$
$468$	0	0
$469$	21.2605	0.981718
$470$	0	0
$471$	−21.3658	−0.984486
$472$	0	0
$473$	12.9876i	0.597171i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 28.8516i	− 1.32102i
$478$	0	0
$479$	12.6676	0.578797	0.289398	−	0.957209i	$-0.406545\pi$
$479$	12.6676	0.578797	0.289398	+	0.957209i	$0.406545\pi$
$480$	0	0
$481$	−2.14331	−0.0977264
$482$	0	0
$483$	24.0122i	1.09259i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 43.2051i	− 1.95781i	−0.204321	−	0.978904i	$-0.565499\pi$
$487$	− 43.2051i	− 1.95781i	0.204321	−	0.978904i	$-0.434501\pi$
$488$	0	0
$489$	−24.4981	−1.10784
$490$	0	0
$491$	18.4065	0.830676	0.415338	−	0.909667i	$-0.363663\pi$
$491$	18.4065	0.830676	0.415338	+	0.909667i	$0.363663\pi$
$492$	0	0
$493$	− 27.8640i	− 1.25493i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.18539i	0.0980281i
$498$	0	0
$499$	13.9208	0.623180	0.311590	−	0.950217i	$-0.399139\pi$
$499$	13.9208	0.623180	0.311590	+	0.950217i	$0.399139\pi$
$500$	0	0
$501$	−4.97524	−0.222277
$502$	0	0
$503$	− 18.8777i	− 0.841717i	−0.907126	−	0.420858i	$-0.861729\pi$
$503$	− 18.8777i	− 0.841717i	0.907126	−	0.420858i	$-0.138271\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	31.6043i	1.40360i
$508$	0	0
$509$	7.63554	0.338439	0.169220	−	0.985578i	$-0.445875\pi$
$509$	7.63554	0.338439	0.169220	+	0.985578i	$0.445875\pi$
$510$	0	0
$511$	−2.46334	−0.108972
$512$	0	0
$513$	1.81089i	0.0799528i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	14.4327i	0.634748i
$518$	0	0
$519$	57.8268	2.53832
$520$	0	0
$521$	−23.3324	−1.02221	−0.511106	−	0.859518i	$-0.670764\pi$
$521$	−23.3324	−1.02221	−0.511106	+	0.859518i	$0.670764\pi$
$522$	0	0
$523$	− 13.6735i	− 0.597900i	−0.954269	−	0.298950i	$-0.903364\pi$
$523$	− 13.6735i	− 0.597900i	0.954269	−	0.298950i	$-0.0966364\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 17.9862i	− 0.783493i
$528$	0	0
$529$	12.1075	0.526415
$530$	0	0
$531$	−8.22253	−0.356827
$532$	0	0
$533$	4.53156i	0.196284i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	56.3584i	2.43204i
$538$	0	0
$539$	−1.43130	−0.0616504
$540$	0	0
$541$	−35.4472	−1.52400	−0.761998	−	0.647579i	$-0.775781\pi$
$541$	−35.4472	−1.52400	−0.761998	+	0.647579i	$0.775781\pi$
$542$	0	0
$543$	− 18.1185i	− 0.777541i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 13.8392i	− 0.591722i	−0.955231	−	0.295861i	$-0.904393\pi$
$547$	− 13.8392i	− 0.591722i	0.955231	−	0.295861i	$-0.0956066\pi$
$548$	0	0
$549$	36.6625	1.56472
$550$	0	0
$551$	6.98762	0.297683
$552$	0	0
$553$	− 25.1345i	− 1.06883i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 30.8777i	− 1.30833i	−0.756351	−	0.654166i	$-0.773020\pi$
$557$	− 30.8777i	− 1.30833i	0.756351	−	0.654166i	$-0.226980\pi$
$558$	0	0
$559$	−7.26695	−0.307359
$560$	0	0
$561$	−16.3942	−0.692162
$562$	0	0
$563$	− 6.38688i	− 0.269175i	−0.990902	−	0.134587i	$-0.957029\pi$
$563$	− 6.38688i	− 0.269175i	0.990902	−	0.134587i	$-0.0429709\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	18.0224i	0.756870i
$568$	0	0
$569$	43.2334	1.81244	0.906219	−	0.422809i	$-0.138956\pi$
$569$	43.2334	1.81244	0.906219	+	0.422809i	$0.138956\pi$
$570$	0	0
$571$	−33.8726	−1.41753	−0.708763	−	0.705447i	$-0.750746\pi$
$571$	−33.8726	−1.41753	−0.708763	+	0.705447i	$0.750746\pi$
$572$	0	0
$573$	8.30903i	0.347115i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 15.0975i	− 0.628517i	−0.949337	−	0.314259i	$-0.898244\pi$
$577$	− 15.0975i	− 0.628517i	0.949337	−	0.314259i	$-0.101756\pi$
$578$	0	0
$579$	1.35346	0.0562477
$580$	0	0
$581$	32.8864	1.36436
$582$	0	0
$583$	12.3869i	0.513012i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 40.7651i	− 1.68256i	−0.540603	−	0.841278i	$-0.681804\pi$
$587$	− 40.7651i	− 1.68256i	0.540603	−	0.841278i	$-0.318196\pi$
$588$	0	0
$589$	4.51052	0.185853
$590$	0	0
$591$	−6.75409	−0.277826
$592$	0	0
$593$	− 5.77609i	− 0.237196i	−0.992942	−	0.118598i	$-0.962160\pi$
$593$	− 5.77609i	− 0.237196i	0.992942	−	0.118598i	$-0.0378399\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	33.3796i	1.36614i
$598$	0	0
$599$	14.0087	0.572378	0.286189	−	0.958173i	$-0.407611\pi$
$599$	14.0087	0.572378	0.286189	+	0.958173i	$0.407611\pi$
$600$	0	0
$601$	−23.9222	−0.975805	−0.487903	−	0.872898i	$-0.662238\pi$
$601$	−23.9222	−0.975805	−0.487903	+	0.872898i	$0.662238\pi$
$602$	0	0
$603$	− 27.9825i	− 1.13954i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	18.1026i	0.734762i	0.930071	+	0.367381i	$0.119745\pi$
$607$	18.1026i	0.734762i	−0.930071	+	0.367381i	$0.880255\pi$
$608$	0	0
$609$	−50.8392	−2.06011
$610$	0	0
$611$	−8.07550	−0.326700
$612$	0	0
$613$	41.9701i	1.69516i	0.530669	+	0.847579i	$0.321941\pi$
$613$	41.9701i	1.69516i	−0.530669	+	0.847579i	$0.678059\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 15.9112i	− 0.640559i	−0.947323	−	0.320279i	$-0.896223\pi$
$617$	− 15.9112i	− 0.640559i	0.947323	−	0.320279i	$-0.103777\pi$
$618$	0	0
$619$	27.8726	1.12030	0.560148	−	0.828393i	$-0.310744\pi$
$619$	27.8726	1.12030	0.560148	+	0.828393i	$0.310744\pi$
$620$	0	0
$621$	5.97662	0.239833
$622$	0	0
$623$	− 14.8292i	− 0.594118i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 4.11126i	− 0.164188i
$628$	0	0
$629$	−9.61669	−0.383442
$630$	0	0
$631$	33.3214	1.32650	0.663252	−	0.748396i	$-0.269176\pi$
$631$	33.3214	1.32650	0.663252	+	0.748396i	$0.269176\pi$
$632$	0	0
$633$	60.4472i	2.40256i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 0.800854i	− 0.0317310i
$638$	0	0
$639$	2.87636	0.113787
$640$	0	0
$641$	19.7985	0.781994	0.390997	−	0.920392i	$-0.372130\pi$
$641$	19.7985	0.781994	0.390997	+	0.920392i	$0.372130\pi$
$642$	0	0
$643$	− 22.0000i	− 0.867595i	−0.901010	−	0.433798i	$-0.857173\pi$
$643$	− 22.0000i	− 0.867595i	0.901010	−	0.433798i	$-0.142827\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 29.3818i	− 1.15512i	−0.816349	−	0.577558i	$-0.804006\pi$
$647$	− 29.3818i	− 1.15512i	0.816349	−	0.577558i	$-0.195994\pi$
$648$	0	0
$649$	3.53018	0.138572
$650$	0	0
$651$	−32.8168	−1.28619
$652$	0	0
$653$	− 18.8306i	− 0.736897i	−0.929648	−	0.368448i	$-0.879889\pi$
$653$	− 18.8306i	− 0.736897i	0.929648	−	0.368448i	$-0.120111\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	3.24219i	0.126490i
$658$	0	0
$659$	44.7293	1.74241	0.871204	−	0.490922i	$-0.163340\pi$
$659$	44.7293	1.74241	0.871204	+	0.490922i	$0.163340\pi$
$660$	0	0
$661$	3.28799	0.127888	0.0639440	−	0.997953i	$-0.479632\pi$
$661$	3.28799	0.127888	0.0639440	+	0.997953i	$0.479632\pi$
$662$	0	0
$663$	− 9.17301i	− 0.356250i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 23.0617i	− 0.892954i
$668$	0	0
$669$	9.08294	0.351167
$670$	0	0
$671$	−15.7403	−0.607649
$672$	0	0
$673$	1.83703i	0.0708123i	0.999373	+	0.0354061i	$0.0112725\pi$
$673$	1.83703i	0.0708123i	−0.999373	+	0.0354061i	$0.988728\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 26.2953i	− 1.01061i	−0.862941	−	0.505305i	$-0.831380\pi$
$677$	− 26.2953i	− 1.01061i	0.862941	−	0.505305i	$-0.168620\pi$
$678$	0	0
$679$	−20.3570	−0.781231
$680$	0	0
$681$	−31.1978	−1.19550
$682$	0	0
$683$	8.29899i	0.317552i	0.987315	+	0.158776i	$0.0507548\pi$
$683$	8.29899i	0.317552i	−0.987315	+	0.158776i	$0.949245\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 54.4610i	− 2.07782i
$688$	0	0
$689$	−6.93082	−0.264043
$690$	0	0
$691$	38.2123	1.45367	0.726833	−	0.686814i	$-0.240991\pi$
$691$	38.2123	1.45367	0.726833	+	0.686814i	$0.240991\pi$
$692$	0	0
$693$	16.5178i	0.627459i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	20.3324i	0.770145i
$698$	0	0
$699$	−60.7541	−2.29793
$700$	0	0
$701$	13.2212	0.499356	0.249678	−	0.968329i	$-0.419675\pi$
$701$	13.2212	0.499356	0.249678	+	0.968329i	$0.419675\pi$
$702$	0	0
$703$	− 2.41164i	− 0.0909566i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0.402822i	0.0151497i
$708$	0	0
$709$	43.7266	1.64219	0.821093	−	0.570794i	$-0.193365\pi$
$709$	43.7266	1.64219	0.821093	+	0.570794i	$0.193365\pi$
$710$	0	0
$711$	−33.0814	−1.24065
$712$	0	0
$713$	− 14.8864i	− 0.557500i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 0.681342i	− 0.0254452i
$718$	0	0
$719$	−32.1716	−1.19980	−0.599900	−	0.800075i	$-0.704793\pi$
$719$	−32.1716	−1.19980	−0.599900	+	0.800075i	$0.704793\pi$
$720$	0	0
$721$	−33.2583	−1.23860
$722$	0	0
$723$	− 51.5017i	− 1.91537i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	6.25967i	0.232158i	0.993240	+	0.116079i	$0.0370326\pi$
$727$	6.25967i	0.232158i	−0.993240	+	0.116079i	$0.962967\pi$
$728$	0	0
$729$	37.7824	1.39935
$730$	0	0
$731$	−32.6057	−1.20596
$732$	0	0
$733$	4.81955i	0.178014i	0.996031	+	0.0890071i	$0.0283694\pi$
$733$	4.81955i	0.178014i	−0.996031	+	0.0890071i	$0.971631\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	12.0138i	0.442532i
$738$	0	0
$739$	−29.3969	−1.08138	−0.540692	−	0.841221i	$-0.681837\pi$
$739$	−29.3969	−1.08138	−0.540692	+	0.841221i	$0.681837\pi$
$740$	0	0
$741$	2.30037	0.0845063
$742$	0	0
$743$	16.9890i	0.623266i	0.950203	+	0.311633i	$0.100876\pi$
$743$	16.9890i	0.623266i	−0.950203	+	0.311633i	$0.899124\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 43.2843i	− 1.58369i
$748$	0	0
$749$	−30.7010	−1.12179
$750$	0	0
$751$	7.98252	0.291286	0.145643	−	0.989337i	$-0.453475\pi$
$751$	7.98252	0.291286	0.145643	+	0.989337i	$0.453475\pi$
$752$	0	0
$753$	39.3338i	1.43340i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	14.6749i	0.533367i	0.963784	+	0.266684i	$0.0859279\pi$
$757$	14.6749i	0.533367i	−0.963784	+	0.266684i	$0.914072\pi$
$758$	0	0
$759$	−13.5687	−0.492513
$760$	0	0
$761$	−9.85807	−0.357355	−0.178677	−	0.983908i	$-0.557182\pi$
$761$	−9.85807	−0.357355	−0.178677	+	0.983908i	$0.557182\pi$
$762$	0	0
$763$	44.0604i	1.59509i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.97524i	0.0713218i
$768$	0	0
$769$	17.7948	0.641697	0.320848	−	0.947131i	$-0.396032\pi$
$769$	17.7948	0.641697	0.320848	+	0.947131i	$0.396032\pi$
$770$	0	0
$771$	56.4449	2.03281
$772$	0	0
$773$	− 2.34617i	− 0.0843859i	−0.999109	−	0.0421930i	$-0.986566\pi$
$773$	− 2.34617i	− 0.0843859i	0.999109	−	0.0421930i	$-0.0134344\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	17.5461i	0.629464i
$778$	0	0
$779$	−5.09888	−0.182686
$780$	0	0
$781$	−1.23491	−0.0441885
$782$	0	0
$783$	12.6538i	0.452211i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	27.2422i	0.971079i	0.874215	+	0.485540i	$0.161377\pi$
$787$	27.2422i	0.971079i	−0.874215	+	0.485540i	$0.838623\pi$
$788$	0	0
$789$	−51.4697	−1.83237
$790$	0	0
$791$	17.8036	0.633023
$792$	0	0
$793$	− 8.80717i	− 0.312752i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 12.2261i	− 0.433070i	−0.976275	−	0.216535i	$-0.930524\pi$
$797$	− 12.2261i	− 0.433070i	0.976275	−	0.216535i	$-0.0694756\pi$
$798$	0	0
$799$	−36.2335	−1.28185
$800$	0	0
$801$	−19.5178	−0.689628
$802$	0	0
$803$	− 1.39197i	− 0.0491216i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	80.9847i	2.85080i
$808$	0	0
$809$	−9.40063	−0.330509	−0.165254	−	0.986251i	$-0.552844\pi$
$809$	−9.40063	−0.330509	−0.165254	+	0.986251i	$0.552844\pi$
$810$	0	0
$811$	37.9729	1.33341	0.666704	−	0.745322i	$-0.267704\pi$
$811$	37.9729	1.33341	0.666704	+	0.745322i	$0.267704\pi$
$812$	0	0
$813$	− 29.3548i	− 1.02952i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 8.17673i	− 0.286068i
$818$	0	0
$819$	−9.24219	−0.322948
$820$	0	0
$821$	−14.0283	−0.489592	−0.244796	−	0.969575i	$-0.578721\pi$
$821$	−14.0283	−0.489592	−0.244796	+	0.969575i	$0.578721\pi$
$822$	0	0
$823$	7.66621i	0.267227i	0.991034	+	0.133614i	$0.0426581\pi$
$823$	7.66621i	0.267227i	−0.991034	+	0.133614i	$0.957342\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	24.0122i	0.834987i	0.908680	+	0.417493i	$0.137091\pi$
$827$	24.0122i	0.834987i	−0.908680	+	0.417493i	$0.862909\pi$
$828$	0	0
$829$	−19.7527	−0.686040	−0.343020	−	0.939328i	$-0.611450\pi$
$829$	−19.7527	−0.686040	−0.343020	+	0.939328i	$0.611450\pi$
$830$	0	0
$831$	−46.9839	−1.62985
$832$	0	0
$833$	− 3.59331i	− 0.124501i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	8.16807i	0.282330i
$838$	0	0
$839$	27.5736	0.951948	0.475974	−	0.879459i	$-0.342096\pi$
$839$	27.5736	0.951948	0.475974	+	0.879459i	$0.342096\pi$
$840$	0	0
$841$	19.8268	0.683684
$842$	0	0
$843$	− 16.6181i	− 0.572357i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	23.8282i	0.818747i
$848$	0	0
$849$	−49.7097	−1.70603
$850$	0	0
$851$	−7.95930	−0.272841
$852$	0	0
$853$	44.5599i	1.52570i	0.646575	+	0.762851i	$0.276201\pi$
$853$	44.5599i	1.52570i	−0.646575	+	0.762851i	$0.723799\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 42.4400i	− 1.44972i	−0.688895	−	0.724861i	$-0.741904\pi$
$857$	− 42.4400i	− 1.44972i	0.688895	−	0.724861i	$-0.258096\pi$
$858$	0	0
$859$	−56.9998	−1.94481	−0.972405	−	0.233300i	$-0.925048\pi$
$859$	−56.9998	−1.94481	−0.972405	+	0.233300i	$0.925048\pi$
$860$	0	0
$861$	37.0975	1.26428
$862$	0	0
$863$	46.0210i	1.56657i	0.621660	+	0.783287i	$0.286459\pi$
$863$	46.0210i	1.56657i	−0.621660	+	0.783287i	$0.713541\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	2.84431i	0.0965979i
$868$	0	0
$869$	14.2029	0.481799
$870$	0	0
$871$	−6.72205	−0.227768
$872$	0	0
$873$	26.7934i	0.906820i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 45.5475i	− 1.53803i	−0.639231	−	0.769015i	$-0.720747\pi$
$877$	− 45.5475i	− 1.53803i	0.639231	−	0.769015i	$-0.279253\pi$
$878$	0	0
$879$	84.1715	2.83903
$880$	0	0
$881$	41.0283	1.38228	0.691140	−	0.722721i	$-0.257109\pi$
$881$	41.0283	1.38228	0.691140	+	0.722721i	$0.257109\pi$
$882$	0	0
$883$	− 3.67625i	− 0.123716i	−0.998085	−	0.0618578i	$-0.980297\pi$
$883$	− 3.67625i	− 0.123716i	0.998085	−	0.0618578i	$-0.0197025\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 9.00591i	− 0.302389i	−0.988504	−	0.151194i	$-0.951688\pi$
$887$	− 9.00591i	− 0.302389i	0.988504	−	0.151194i	$-0.0483119\pi$
$888$	0	0
$889$	−32.9455	−1.10496
$890$	0	0
$891$	−10.1840	−0.341177
$892$	0	0
$893$	− 9.08650i	− 0.304068i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 7.59208i	− 0.253492i
$898$	0	0
$899$	31.5178	1.05118
$900$	0	0
$901$	−31.0975	−1.03601
$902$	0	0
$903$	59.4907i	1.97973i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	14.5687i	0.483746i	0.970308	+	0.241873i	$0.0777617\pi$
$907$	14.5687i	0.483746i	−0.970308	+	0.241873i	$0.922238\pi$
$908$	0	0
$909$	0.530184	0.0175851
$910$	0	0
$911$	18.3694	0.608605	0.304303	−	0.952575i	$-0.401577\pi$
$911$	18.3694	0.608605	0.304303	+	0.952575i	$0.401577\pi$
$912$	0	0
$913$	18.5833i	0.615016i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	15.3259i	0.506107i
$918$	0	0
$919$	−26.6130	−0.877881	−0.438940	−	0.898516i	$-0.644646\pi$
$919$	−26.6130	−0.877881	−0.438940	+	0.898516i	$0.644646\pi$
$920$	0	0
$921$	−69.3191	−2.28414
$922$	0	0
$923$	− 0.690967i	− 0.0227435i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	43.7738i	1.43772i
$928$	0	0
$929$	−13.2939	−0.436159	−0.218079	−	0.975931i	$-0.569979\pi$
$929$	−13.2939	−0.436159	−0.218079	+	0.975931i	$0.569979\pi$
$930$	0	0
$931$	0.901116	0.0295329
$932$	0	0
$933$	57.6894i	1.88867i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 45.2980i	− 1.47982i	−0.672705	−	0.739911i	$-0.734868\pi$
$937$	− 45.2980i	− 1.47982i	0.672705	−	0.739911i	$-0.265132\pi$
$938$	0	0
$939$	−59.4362	−1.93963
$940$	0	0
$941$	41.5192	1.35349	0.676743	−	0.736219i	$-0.263391\pi$
$941$	41.5192	1.35349	0.676743	+	0.736219i	$0.263391\pi$
$942$	0	0
$943$	16.8282i	0.548002i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 5.65658i	− 0.183814i	−0.995768	−	0.0919071i	$-0.970704\pi$
$947$	− 5.65658i	− 0.183814i	0.995768	−	0.0919071i	$-0.0292963\pi$
$948$	0	0
$949$	0.778849	0.0252825
$950$	0	0
$951$	−72.9505	−2.36558
$952$	0	0
$953$	33.2967i	1.07858i	0.842119	+	0.539292i	$0.181308\pi$
$953$	33.2967i	1.07858i	−0.842119	+	0.539292i	$0.818692\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 28.7280i	− 0.928643i
$958$	0	0
$959$	−41.8158	−1.35030
$960$	0	0
$961$	−10.6552	−0.343716
$962$	0	0
$963$	40.4079i	1.30213i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 36.4647i	− 1.17263i	−0.810084	−	0.586313i	$-0.800579\pi$
$967$	− 36.4647i	− 1.17263i	0.810084	−	0.586313i	$-0.199421\pi$
$968$	0	0
$969$	10.3214	0.331572
$970$	0	0
$971$	22.9047	0.735046	0.367523	−	0.930014i	$-0.380206\pi$
$971$	22.9047	0.735046	0.367523	+	0.930014i	$0.380206\pi$
$972$	0	0
$973$	− 61.6413i	− 1.97613i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	6.57736i	0.210428i	0.994450	+	0.105214i	$0.0335528\pi$
$977$	6.57736i	0.210428i	−0.994450	+	0.105214i	$0.966447\pi$
$978$	0	0
$979$	8.37959	0.267813
$980$	0	0
$981$	57.9912	1.85152
$982$	0	0
$983$	25.4451i	0.811571i	0.913968	+	0.405786i	$0.133002\pi$
$983$	25.4451i	0.811571i	−0.913968	+	0.405786i	$0.866998\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	66.1099i	2.10430i
$988$	0	0
$989$	−26.9862	−0.858113
$990$	0	0
$991$	41.5090	1.31858	0.659288	−	0.751890i	$-0.270858\pi$
$991$	41.5090	1.31858	0.659288	+	0.751890i	$0.270858\pi$
$992$	0	0
$993$	49.6232i	1.57474i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	15.4486i	0.489263i	0.969616	+	0.244631i	$0.0786669\pi$
$997$	15.4486i	0.489263i	−0.969616	+	0.244631i	$0.921333\pi$
$998$	0	0
$999$	4.36721	0.138173

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1900.2.c.f.1749.6		6
5.2	odd	4		1900.2.a.i.1.3	yes	3
5.3	odd	4		1900.2.a.g.1.1	✓	3
5.4	even	2	inner	1900.2.c.f.1749.1		6
20.3	even	4		7600.2.a.ca.1.3		3
20.7	even	4		7600.2.a.bl.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1900.2.a.g.1.1	✓	3	5.3	odd	4
1900.2.a.i.1.3	yes	3	5.2	odd	4
1900.2.c.f.1749.1		6	5.4	even	2	inner
1900.2.c.f.1749.6		6	1.1	even	1	trivial
7600.2.a.bl.1.1		3	20.7	even	4
7600.2.a.ca.1.3		3	20.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1900.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.58836i q^{3} -2.81089i q^{7} -3.69963 q^{9} +O(q^{10})\)	\(q+2.58836i q^{3} -2.81089i q^{7} -3.69963 q^{9} +1.58836 q^{11} +0.888736i q^{13} +3.98762i q^{17} -1.00000 q^{19} +7.27561 q^{21} +3.30037i q^{23} -1.81089i q^{27} -6.98762 q^{29} -4.51052 q^{31} +4.11126i q^{33} +2.41164i q^{37} -2.30037 q^{39} +5.09888 q^{41} +8.17673i q^{43} +9.08650i q^{47} -0.901116 q^{49} -10.3214 q^{51} +7.79851i q^{53} -2.58836i q^{57} +2.22253 q^{59} -9.90978 q^{61} +10.3993i q^{63} +7.56360i q^{67} -8.54256 q^{69} -0.777472 q^{71} -0.876356i q^{73} -4.46472i q^{77} +8.94182 q^{79} -6.41164 q^{81} +11.6996i q^{83} -18.0865i q^{87} +5.27561 q^{89} +2.49814 q^{91} -11.6749i q^{93} -7.24219i q^{97} -5.87636 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 10 q^{9}+O(q^{10}) \) 6 * q - 10 * q^9	\( 6 q - 10 q^{9} - 2 q^{11} - 6 q^{19} - 16 q^{21} - 6 q^{29} - 2 q^{31} - 26 q^{39} - 6 q^{41} - 42 q^{49} - 24 q^{51} + 12 q^{59} - 10 q^{61} - 18 q^{69} - 6 q^{71} - 4 q^{79} - 50 q^{81} - 28 q^{89} - 46 q^{91}+O(q^{100}) \) 6 * q - 10 * q^9 - 2 * q^11 - 6 * q^19 - 16 * q^21 - 6 * q^29 - 2 * q^31 - 26 * q^39 - 6 * q^41 - 42 * q^49 - 24 * q^51 + 12 * q^59 - 10 * q^61 - 18 * q^69 - 6 * q^71 - 4 * q^79 - 50 * q^81 - 28 * q^89 - 46 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more