LMFDB - Embedded newform 5850.2.a.cm.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5850,2,Mod(1,5850)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5850.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5850, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5850.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,2,0,2,0,0,2,2,0,0,0,0,2,2,0,2,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$46.7124851824$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 390)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	5850.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.23607 q^{7} +1.00000 q^{8} +4.47214 q^{11} +1.00000 q^{13} -1.23607 q^{14} +1.00000 q^{16} -2.76393 q^{17} -7.23607 q^{19} +4.47214 q^{22} -7.23607 q^{23} +1.00000 q^{26} -1.23607 q^{28} -9.70820 q^{29} -4.00000 q^{31} +1.00000 q^{32} -2.76393 q^{34} +6.94427 q^{37} -7.23607 q^{38} -12.4721 q^{41} +6.47214 q^{43} +4.47214 q^{44} -7.23607 q^{46} +4.94427 q^{47} -5.47214 q^{49} +1.00000 q^{52} -8.47214 q^{53} -1.23607 q^{56} -9.70820 q^{58} -0.472136 q^{59} +6.94427 q^{61} -4.00000 q^{62} +1.00000 q^{64} -2.76393 q^{68} -6.47214 q^{71} +8.76393 q^{73} +6.94427 q^{74} -7.23607 q^{76} -5.52786 q^{77} -4.00000 q^{79} -12.4721 q^{82} -12.9443 q^{83} +6.47214 q^{86} +4.47214 q^{88} +8.47214 q^{89} -1.23607 q^{91} -7.23607 q^{92} +4.94427 q^{94} +9.70820 q^{97} -5.47214 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{7} + 2 q^{8} + 2 q^{13} + 2 q^{14} + 2 q^{16} - 10 q^{17} - 10 q^{19} - 10 q^{23} + 2 q^{26} + 2 q^{28} - 6 q^{29} - 8 q^{31} + 2 q^{32} - 10 q^{34} - 4 q^{37} - 10 q^{38}+ \cdots - 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^7 + 2 * q^8 + 2 * q^13 + 2 * q^14 + 2 * q^16 - 10 * q^17 - 10 * q^19 - 10 * q^23 + 2 * q^26 + 2 * q^28 - 6 * q^29 - 8 * q^31 + 2 * q^32 - 10 * q^34 - 4 * q^37 - 10 * q^38 - 16 * q^41 + 4 * q^43 - 10 * q^46 - 8 * q^47 - 2 * q^49 + 2 * q^52 - 8 * q^53 + 2 * q^56 - 6 * q^58 + 8 * q^59 - 4 * q^61 - 8 * q^62 + 2 * q^64 - 10 * q^68 - 4 * q^71 + 22 * q^73 - 4 * q^74 - 10 * q^76 - 20 * q^77 - 8 * q^79 - 16 * q^82 - 8 * q^83 + 4 * q^86 + 8 * q^89 + 2 * q^91 - 10 * q^92 - 8 * q^94 + 6 * q^97 - 2 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.23607	−0.467190	−0.233595	−	0.972334i	$-0.575049\pi$
$7$	−1.23607	−0.467190	−0.233595	+	0.972334i	$0.575049\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	0	0
$11$	4.47214	1.34840	0.674200	−	0.738549i	$-0.264489\pi$
$11$	4.47214	1.34840	0.674200	+	0.738549i	$0.264489\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−1.23607	−0.330353
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.76393	−0.670352	−0.335176	−	0.942156i	$-0.608796\pi$
$17$	−2.76393	−0.670352	−0.335176	+	0.942156i	$0.608796\pi$
$18$	0	0
$19$	−7.23607	−1.66007	−0.830034	−	0.557713i	$-0.811679\pi$
$19$	−7.23607	−1.66007	−0.830034	+	0.557713i	$0.811679\pi$
$20$	0	0
$21$	0	0
$22$	4.47214	0.953463
$23$	−7.23607	−1.50882	−0.754412	−	0.656401i	$-0.772078\pi$
$23$	−7.23607	−1.50882	−0.754412	+	0.656401i	$0.772078\pi$
$24$	0	0
$25$	0	0
$26$	1.00000	0.196116
$27$	0	0
$28$	−1.23607	−0.233595
$29$	−9.70820	−1.80277	−0.901384	−	0.433020i	$-0.857448\pi$
$29$	−9.70820	−1.80277	−0.901384	+	0.433020i	$0.857448\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−2.76393	−0.474010
$35$	0	0
$36$	0	0
$37$	6.94427	1.14163	0.570816	−	0.821078i	$-0.306627\pi$
$37$	6.94427	1.14163	0.570816	+	0.821078i	$0.306627\pi$
$38$	−7.23607	−1.17385
$39$	0	0
$40$	0	0
$41$	−12.4721	−1.94782	−0.973910	−	0.226934i	$-0.927130\pi$
$41$	−12.4721	−1.94782	−0.973910	+	0.226934i	$0.927130\pi$
$42$	0	0
$43$	6.47214	0.986991	0.493496	−	0.869748i	$-0.335719\pi$
$43$	6.47214	0.986991	0.493496	+	0.869748i	$0.335719\pi$
$44$	4.47214	0.674200
$45$	0	0
$46$	−7.23607	−1.06690
$47$	4.94427	0.721196	0.360598	−	0.932721i	$-0.382573\pi$
$47$	4.94427	0.721196	0.360598	+	0.932721i	$0.382573\pi$
$48$	0	0
$49$	−5.47214	−0.781734
$50$	0	0
$51$	0	0
$52$	1.00000	0.138675
$53$	−8.47214	−1.16374	−0.581869	−	0.813283i	$-0.697678\pi$
$53$	−8.47214	−1.16374	−0.581869	+	0.813283i	$0.697678\pi$
$54$	0	0
$55$	0	0
$56$	−1.23607	−0.165177
$57$	0	0
$58$	−9.70820	−1.27475
$59$	−0.472136	−0.0614669	−0.0307334	−	0.999528i	$-0.509784\pi$
$59$	−0.472136	−0.0614669	−0.0307334	+	0.999528i	$0.509784\pi$
$60$	0	0
$61$	6.94427	0.889123	0.444561	−	0.895748i	$-0.353360\pi$
$61$	6.94427	0.889123	0.444561	+	0.895748i	$0.353360\pi$
$62$	−4.00000	−0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	−2.76393	−0.335176
$69$	0	0
$70$	0	0
$71$	−6.47214	−0.768101	−0.384051	−	0.923312i	$-0.625471\pi$
$71$	−6.47214	−0.768101	−0.384051	+	0.923312i	$0.625471\pi$
$72$	0	0
$73$	8.76393	1.02574	0.512870	−	0.858466i	$-0.328582\pi$
$73$	8.76393	1.02574	0.512870	+	0.858466i	$0.328582\pi$
$74$	6.94427	0.807255
$75$	0	0
$76$	−7.23607	−0.830034
$77$	−5.52786	−0.629959
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	0	0
$82$	−12.4721	−1.37732
$83$	−12.9443	−1.42082	−0.710409	−	0.703789i	$-0.751490\pi$
$83$	−12.9443	−1.42082	−0.710409	+	0.703789i	$0.751490\pi$
$84$	0	0
$85$	0	0
$86$	6.47214	0.697908
$87$	0	0
$88$	4.47214	0.476731
$89$	8.47214	0.898045	0.449022	−	0.893521i	$-0.351772\pi$
$89$	8.47214	0.898045	0.449022	+	0.893521i	$0.351772\pi$
$90$	0	0
$91$	−1.23607	−0.129575
$92$	−7.23607	−0.754412
$93$	0	0
$94$	4.94427	0.509963
$95$	0	0
$96$	0	0
$97$	9.70820	0.985719	0.492859	−	0.870109i	$-0.335952\pi$
$97$	9.70820	0.985719	0.492859	+	0.870109i	$0.335952\pi$
$98$	−5.47214	−0.552769
$99$	0	0
$100$	0	0
$101$	4.76393	0.474029	0.237014	−	0.971506i	$-0.423831\pi$
$101$	4.76393	0.474029	0.237014	+	0.971506i	$0.423831\pi$
$102$	0	0
$103$	−0.472136	−0.0465209	−0.0232605	−	0.999729i	$-0.507405\pi$
$103$	−0.472136	−0.0465209	−0.0232605	+	0.999729i	$0.507405\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	−8.47214	−0.822887
$107$	10.4721	1.01238	0.506190	−	0.862422i	$-0.331054\pi$
$107$	10.4721	1.01238	0.506190	+	0.862422i	$0.331054\pi$
$108$	0	0
$109$	−15.7082	−1.50457	−0.752287	−	0.658836i	$-0.771049\pi$
$109$	−15.7082	−1.50457	−0.752287	+	0.658836i	$0.771049\pi$
$110$	0	0
$111$	0	0
$112$	−1.23607	−0.116797
$113$	12.6525	1.19024	0.595122	−	0.803635i	$-0.297104\pi$
$113$	12.6525	1.19024	0.595122	+	0.803635i	$0.297104\pi$
$114$	0	0
$115$	0	0
$116$	−9.70820	−0.901384
$117$	0	0
$118$	−0.472136	−0.0434636
$119$	3.41641	0.313182
$120$	0	0
$121$	9.00000	0.818182
$122$	6.94427	0.628705
$123$	0	0
$124$	−4.00000	−0.359211
$125$	0	0
$126$	0	0
$127$	−5.41641	−0.480628	−0.240314	−	0.970695i	$-0.577250\pi$
$127$	−5.41641	−0.480628	−0.240314	+	0.970695i	$0.577250\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	3.70820	0.323987	0.161994	−	0.986792i	$-0.448208\pi$
$131$	3.70820	0.323987	0.161994	+	0.986792i	$0.448208\pi$
$132$	0	0
$133$	8.94427	0.775567
$134$	0	0
$135$	0	0
$136$	−2.76393	−0.237005
$137$	−15.8885	−1.35745	−0.678725	−	0.734393i	$-0.737467\pi$
$137$	−15.8885	−1.35745	−0.678725	+	0.734393i	$0.737467\pi$
$138$	0	0
$139$	2.47214	0.209684	0.104842	−	0.994489i	$-0.466566\pi$
$139$	2.47214	0.209684	0.104842	+	0.994489i	$0.466566\pi$
$140$	0	0
$141$	0	0
$142$	−6.47214	−0.543130
$143$	4.47214	0.373979
$144$	0	0
$145$	0	0
$146$	8.76393	0.725308
$147$	0	0
$148$	6.94427	0.570816
$149$	−22.4721	−1.84099	−0.920495	−	0.390755i	$-0.872214\pi$
$149$	−22.4721	−1.84099	−0.920495	+	0.390755i	$0.872214\pi$
$150$	0	0
$151$	7.41641	0.603539	0.301769	−	0.953381i	$-0.402423\pi$
$151$	7.41641	0.603539	0.301769	+	0.953381i	$0.402423\pi$
$152$	−7.23607	−0.586923
$153$	0	0
$154$	−5.52786	−0.445448
$155$	0	0
$156$	0	0
$157$	−1.05573	−0.0842563	−0.0421281	−	0.999112i	$-0.513414\pi$
$157$	−1.05573	−0.0842563	−0.0421281	+	0.999112i	$0.513414\pi$
$158$	−4.00000	−0.318223
$159$	0	0
$160$	0	0
$161$	8.94427	0.704907
$162$	0	0
$163$	−4.94427	−0.387265	−0.193633	−	0.981074i	$-0.562027\pi$
$163$	−4.94427	−0.387265	−0.193633	+	0.981074i	$0.562027\pi$
$164$	−12.4721	−0.973910
$165$	0	0
$166$	−12.9443	−1.00467
$167$	−7.41641	−0.573899	−0.286949	−	0.957946i	$-0.592641\pi$
$167$	−7.41641	−0.573899	−0.286949	+	0.957946i	$0.592641\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	6.47214	0.493496
$173$	5.05573	0.384380	0.192190	−	0.981358i	$-0.438441\pi$
$173$	5.05573	0.384380	0.192190	+	0.981358i	$0.438441\pi$
$174$	0	0
$175$	0	0
$176$	4.47214	0.337100
$177$	0	0
$178$	8.47214	0.635013
$179$	14.1803	1.05989	0.529944	−	0.848033i	$-0.322213\pi$
$179$	14.1803	1.05989	0.529944	+	0.848033i	$0.322213\pi$
$180$	0	0
$181$	−9.41641	−0.699916	−0.349958	−	0.936765i	$-0.613804\pi$
$181$	−9.41641	−0.699916	−0.349958	+	0.936765i	$0.613804\pi$
$182$	−1.23607	−0.0916235
$183$	0	0
$184$	−7.23607	−0.533450
$185$	0	0
$186$	0	0
$187$	−12.3607	−0.903902
$188$	4.94427	0.360598
$189$	0	0
$190$	0	0
$191$	−13.5279	−0.978842	−0.489421	−	0.872048i	$-0.662792\pi$
$191$	−13.5279	−0.978842	−0.489421	+	0.872048i	$0.662792\pi$
$192$	0	0
$193$	−21.7082	−1.56259	−0.781295	−	0.624161i	$-0.785441\pi$
$193$	−21.7082	−1.56259	−0.781295	+	0.624161i	$0.785441\pi$
$194$	9.70820	0.697008
$195$	0	0
$196$	−5.47214	−0.390867
$197$	−2.94427	−0.209771	−0.104885	−	0.994484i	$-0.533448\pi$
$197$	−2.94427	−0.209771	−0.104885	+	0.994484i	$0.533448\pi$
$198$	0	0
$199$	16.9443	1.20115	0.600574	−	0.799569i	$-0.294939\pi$
$199$	16.9443	1.20115	0.600574	+	0.799569i	$0.294939\pi$
$200$	0	0
$201$	0	0
$202$	4.76393	0.335189
$203$	12.0000	0.842235
$204$	0	0
$205$	0	0
$206$	−0.472136	−0.0328953
$207$	0	0
$208$	1.00000	0.0693375
$209$	−32.3607	−2.23844
$210$	0	0
$211$	−21.8885	−1.50687	−0.753435	−	0.657523i	$-0.771604\pi$
$211$	−21.8885	−1.50687	−0.753435	+	0.657523i	$0.771604\pi$
$212$	−8.47214	−0.581869
$213$	0	0
$214$	10.4721	0.715860
$215$	0	0
$216$	0	0
$217$	4.94427	0.335639
$218$	−15.7082	−1.06389
$219$	0	0
$220$	0	0
$221$	−2.76393	−0.185922
$222$	0	0
$223$	21.2361	1.42207	0.711036	−	0.703155i	$-0.248226\pi$
$223$	21.2361	1.42207	0.711036	+	0.703155i	$0.248226\pi$
$224$	−1.23607	−0.0825883
$225$	0	0
$226$	12.6525	0.841630
$227$	−24.9443	−1.65561	−0.827805	−	0.561016i	$-0.810410\pi$
$227$	−24.9443	−1.65561	−0.827805	+	0.561016i	$0.810410\pi$
$228$	0	0
$229$	−3.70820	−0.245045	−0.122523	−	0.992466i	$-0.539098\pi$
$229$	−3.70820	−0.245045	−0.122523	+	0.992466i	$0.539098\pi$
$230$	0	0
$231$	0	0
$232$	−9.70820	−0.637375
$233$	2.18034	0.142839	0.0714194	−	0.997446i	$-0.477247\pi$
$233$	2.18034	0.142839	0.0714194	+	0.997446i	$0.477247\pi$
$234$	0	0
$235$	0	0
$236$	−0.472136	−0.0307334
$237$	0	0
$238$	3.41641	0.221453
$239$	−13.8885	−0.898375	−0.449188	−	0.893437i	$-0.648287\pi$
$239$	−13.8885	−0.898375	−0.449188	+	0.893437i	$0.648287\pi$
$240$	0	0
$241$	12.4721	0.803401	0.401700	−	0.915771i	$-0.368419\pi$
$241$	12.4721	0.803401	0.401700	+	0.915771i	$0.368419\pi$
$242$	9.00000	0.578542
$243$	0	0
$244$	6.94427	0.444561
$245$	0	0
$246$	0	0
$247$	−7.23607	−0.460420
$248$	−4.00000	−0.254000
$249$	0	0
$250$	0	0
$251$	−15.7082	−0.991493	−0.495747	−	0.868467i	$-0.665105\pi$
$251$	−15.7082	−0.991493	−0.495747	+	0.868467i	$0.665105\pi$
$252$	0	0
$253$	−32.3607	−2.03450
$254$	−5.41641	−0.339856
$255$	0	0
$256$	1.00000	0.0625000
$257$	−5.23607	−0.326617	−0.163308	−	0.986575i	$-0.552217\pi$
$257$	−5.23607	−0.326617	−0.163308	+	0.986575i	$0.552217\pi$
$258$	0	0
$259$	−8.58359	−0.533358
$260$	0	0
$261$	0	0
$262$	3.70820	0.229094
$263$	12.1803	0.751072	0.375536	−	0.926808i	$-0.377459\pi$
$263$	12.1803	0.751072	0.375536	+	0.926808i	$0.377459\pi$
$264$	0	0
$265$	0	0
$266$	8.94427	0.548408
$267$	0	0
$268$	0	0
$269$	−11.2361	−0.685075	−0.342538	−	0.939504i	$-0.611286\pi$
$269$	−11.2361	−0.685075	−0.342538	+	0.939504i	$0.611286\pi$
$270$	0	0
$271$	15.4164	0.936480	0.468240	−	0.883601i	$-0.344888\pi$
$271$	15.4164	0.936480	0.468240	+	0.883601i	$0.344888\pi$
$272$	−2.76393	−0.167588
$273$	0	0
$274$	−15.8885	−0.959862
$275$	0	0
$276$	0	0
$277$	−32.8328	−1.97273	−0.986366	−	0.164564i	$-0.947378\pi$
$277$	−32.8328	−1.97273	−0.986366	+	0.164564i	$0.947378\pi$
$278$	2.47214	0.148269
$279$	0	0
$280$	0	0
$281$	−15.5279	−0.926315	−0.463157	−	0.886276i	$-0.653284\pi$
$281$	−15.5279	−0.926315	−0.463157	+	0.886276i	$0.653284\pi$
$282$	0	0
$283$	15.4164	0.916410	0.458205	−	0.888846i	$-0.348492\pi$
$283$	15.4164	0.916410	0.458205	+	0.888846i	$0.348492\pi$
$284$	−6.47214	−0.384051
$285$	0	0
$286$	4.47214	0.264443
$287$	15.4164	0.910002
$288$	0	0
$289$	−9.36068	−0.550628
$290$	0	0
$291$	0	0
$292$	8.76393	0.512870
$293$	14.9443	0.873054	0.436527	−	0.899691i	$-0.356208\pi$
$293$	14.9443	0.873054	0.436527	+	0.899691i	$0.356208\pi$
$294$	0	0
$295$	0	0
$296$	6.94427	0.403628
$297$	0	0
$298$	−22.4721	−1.30178
$299$	−7.23607	−0.418473
$300$	0	0
$301$	−8.00000	−0.461112
$302$	7.41641	0.426766
$303$	0	0
$304$	−7.23607	−0.415017
$305$	0	0
$306$	0	0
$307$	20.3607	1.16205	0.581023	−	0.813887i	$-0.302653\pi$
$307$	20.3607	1.16205	0.581023	+	0.813887i	$0.302653\pi$
$308$	−5.52786	−0.314979
$309$	0	0
$310$	0	0
$311$	13.5279	0.767095	0.383547	−	0.923521i	$-0.374702\pi$
$311$	13.5279	0.767095	0.383547	+	0.923521i	$0.374702\pi$
$312$	0	0
$313$	7.41641	0.419200	0.209600	−	0.977787i	$-0.432784\pi$
$313$	7.41641	0.419200	0.209600	+	0.977787i	$0.432784\pi$
$314$	−1.05573	−0.0595782
$315$	0	0
$316$	−4.00000	−0.225018
$317$	13.4164	0.753541	0.376770	−	0.926307i	$-0.377035\pi$
$317$	13.4164	0.753541	0.376770	+	0.926307i	$0.377035\pi$
$318$	0	0
$319$	−43.4164	−2.43085
$320$	0	0
$321$	0	0
$322$	8.94427	0.498445
$323$	20.0000	1.11283
$324$	0	0
$325$	0	0
$326$	−4.94427	−0.273838
$327$	0	0
$328$	−12.4721	−0.688659
$329$	−6.11146	−0.336935
$330$	0	0
$331$	−7.23607	−0.397730	−0.198865	−	0.980027i	$-0.563726\pi$
$331$	−7.23607	−0.397730	−0.198865	+	0.980027i	$0.563726\pi$
$332$	−12.9443	−0.710409
$333$	0	0
$334$	−7.41641	−0.405808
$335$	0	0
$336$	0	0
$337$	2.47214	0.134666	0.0673329	−	0.997731i	$-0.478551\pi$
$337$	2.47214	0.134666	0.0673329	+	0.997731i	$0.478551\pi$
$338$	1.00000	0.0543928
$339$	0	0
$340$	0	0
$341$	−17.8885	−0.968719
$342$	0	0
$343$	15.4164	0.832408
$344$	6.47214	0.348954
$345$	0	0
$346$	5.05573	0.271798
$347$	−10.4721	−0.562174	−0.281087	−	0.959682i	$-0.590695\pi$
$347$	−10.4721	−0.562174	−0.281087	+	0.959682i	$0.590695\pi$
$348$	0	0
$349$	−10.7639	−0.576180	−0.288090	−	0.957603i	$-0.593020\pi$
$349$	−10.7639	−0.576180	−0.288090	+	0.957603i	$0.593020\pi$
$350$	0	0
$351$	0	0
$352$	4.47214	0.238366
$353$	9.05573	0.481988	0.240994	−	0.970527i	$-0.422527\pi$
$353$	9.05573	0.481988	0.240994	+	0.970527i	$0.422527\pi$
$354$	0	0
$355$	0	0
$356$	8.47214	0.449022
$357$	0	0
$358$	14.1803	0.749454
$359$	2.47214	0.130474	0.0652372	−	0.997870i	$-0.479220\pi$
$359$	2.47214	0.130474	0.0652372	+	0.997870i	$0.479220\pi$
$360$	0	0
$361$	33.3607	1.75583
$362$	−9.41641	−0.494915
$363$	0	0
$364$	−1.23607	−0.0647876
$365$	0	0
$366$	0	0
$367$	−8.47214	−0.442242	−0.221121	−	0.975246i	$-0.570972\pi$
$367$	−8.47214	−0.442242	−0.221121	+	0.975246i	$0.570972\pi$
$368$	−7.23607	−0.377206
$369$	0	0
$370$	0	0
$371$	10.4721	0.543686
$372$	0	0
$373$	2.00000	0.103556	0.0517780	−	0.998659i	$-0.483511\pi$
$373$	2.00000	0.103556	0.0517780	+	0.998659i	$0.483511\pi$
$374$	−12.3607	−0.639156
$375$	0	0
$376$	4.94427	0.254981
$377$	−9.70820	−0.499998
$378$	0	0
$379$	5.12461	0.263234	0.131617	−	0.991301i	$-0.457983\pi$
$379$	5.12461	0.263234	0.131617	+	0.991301i	$0.457983\pi$
$380$	0	0
$381$	0	0
$382$	−13.5279	−0.692146
$383$	−14.4721	−0.739492	−0.369746	−	0.929133i	$-0.620555\pi$
$383$	−14.4721	−0.739492	−0.369746	+	0.929133i	$0.620555\pi$
$384$	0	0
$385$	0	0
$386$	−21.7082	−1.10492
$387$	0	0
$388$	9.70820	0.492859
$389$	18.6525	0.945718	0.472859	−	0.881138i	$-0.343222\pi$
$389$	18.6525	0.945718	0.472859	+	0.881138i	$0.343222\pi$
$390$	0	0
$391$	20.0000	1.01144
$392$	−5.47214	−0.276385
$393$	0	0
$394$	−2.94427	−0.148330
$395$	0	0
$396$	0	0
$397$	−9.41641	−0.472596	−0.236298	−	0.971681i	$-0.575934\pi$
$397$	−9.41641	−0.472596	−0.236298	+	0.971681i	$0.575934\pi$
$398$	16.9443	0.849340
$399$	0	0
$400$	0	0
$401$	15.5279	0.775425	0.387712	−	0.921780i	$-0.373265\pi$
$401$	15.5279	0.775425	0.387712	+	0.921780i	$0.373265\pi$
$402$	0	0
$403$	−4.00000	−0.199254
$404$	4.76393	0.237014
$405$	0	0
$406$	12.0000	0.595550
$407$	31.0557	1.53938
$408$	0	0
$409$	6.00000	0.296681	0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000	0.296681	0.148340	+	0.988936i	$0.452607\pi$
$410$	0	0
$411$	0	0
$412$	−0.472136	−0.0232605
$413$	0.583592	0.0287167
$414$	0	0
$415$	0	0
$416$	1.00000	0.0490290
$417$	0	0
$418$	−32.3607	−1.58281
$419$	21.5967	1.05507	0.527535	−	0.849533i	$-0.323116\pi$
$419$	21.5967	1.05507	0.527535	+	0.849533i	$0.323116\pi$
$420$	0	0
$421$	−0.291796	−0.0142213	−0.00711064	−	0.999975i	$-0.502263\pi$
$421$	−0.291796	−0.0142213	−0.00711064	+	0.999975i	$0.502263\pi$
$422$	−21.8885	−1.06552
$423$	0	0
$424$	−8.47214	−0.411443
$425$	0	0
$426$	0	0
$427$	−8.58359	−0.415389
$428$	10.4721	0.506190
$429$	0	0
$430$	0	0
$431$	−38.8328	−1.87051	−0.935255	−	0.353973i	$-0.884830\pi$
$431$	−38.8328	−1.87051	−0.935255	+	0.353973i	$0.884830\pi$
$432$	0	0
$433$	−18.4721	−0.887714	−0.443857	−	0.896098i	$-0.646390\pi$
$433$	−18.4721	−0.887714	−0.443857	+	0.896098i	$0.646390\pi$
$434$	4.94427	0.237333
$435$	0	0
$436$	−15.7082	−0.752287
$437$	52.3607	2.50475
$438$	0	0
$439$	12.9443	0.617796	0.308898	−	0.951095i	$-0.400040\pi$
$439$	12.9443	0.617796	0.308898	+	0.951095i	$0.400040\pi$
$440$	0	0
$441$	0	0
$442$	−2.76393	−0.131467
$443$	13.5279	0.642728	0.321364	−	0.946956i	$-0.395859\pi$
$443$	13.5279	0.642728	0.321364	+	0.946956i	$0.395859\pi$
$444$	0	0
$445$	0	0
$446$	21.2361	1.00556
$447$	0	0
$448$	−1.23607	−0.0583987
$449$	8.47214	0.399825	0.199912	−	0.979814i	$-0.435934\pi$
$449$	8.47214	0.399825	0.199912	+	0.979814i	$0.435934\pi$
$450$	0	0
$451$	−55.7771	−2.62644
$452$	12.6525	0.595122
$453$	0	0
$454$	−24.9443	−1.17069
$455$	0	0
$456$	0	0
$457$	9.70820	0.454131	0.227065	−	0.973880i	$-0.427087\pi$
$457$	9.70820	0.454131	0.227065	+	0.973880i	$0.427087\pi$
$458$	−3.70820	−0.173273
$459$	0	0
$460$	0	0
$461$	−6.11146	−0.284639	−0.142319	−	0.989821i	$-0.545456\pi$
$461$	−6.11146	−0.284639	−0.142319	+	0.989821i	$0.545456\pi$
$462$	0	0
$463$	−35.7082	−1.65950	−0.829750	−	0.558135i	$-0.811517\pi$
$463$	−35.7082	−1.65950	−0.829750	+	0.558135i	$0.811517\pi$
$464$	−9.70820	−0.450692
$465$	0	0
$466$	2.18034	0.101002
$467$	2.47214	0.114397	0.0571984	−	0.998363i	$-0.481783\pi$
$467$	2.47214	0.114397	0.0571984	+	0.998363i	$0.481783\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	−0.472136	−0.0217318
$473$	28.9443	1.33086
$474$	0	0
$475$	0	0
$476$	3.41641	0.156591
$477$	0	0
$478$	−13.8885	−0.635247
$479$	36.0000	1.64488	0.822441	−	0.568850i	$-0.192612\pi$
$479$	36.0000	1.64488	0.822441	+	0.568850i	$0.192612\pi$
$480$	0	0
$481$	6.94427	0.316632
$482$	12.4721	0.568090
$483$	0	0
$484$	9.00000	0.409091
$485$	0	0
$486$	0	0
$487$	25.5967	1.15990	0.579950	−	0.814652i	$-0.303072\pi$
$487$	25.5967	1.15990	0.579950	+	0.814652i	$0.303072\pi$
$488$	6.94427	0.314352
$489$	0	0
$490$	0	0
$491$	−6.18034	−0.278915	−0.139457	−	0.990228i	$-0.544536\pi$
$491$	−6.18034	−0.278915	−0.139457	+	0.990228i	$0.544536\pi$
$492$	0	0
$493$	26.8328	1.20849
$494$	−7.23607	−0.325566
$495$	0	0
$496$	−4.00000	−0.179605
$497$	8.00000	0.358849
$498$	0	0
$499$	1.12461	0.0503445	0.0251723	−	0.999683i	$-0.491987\pi$
$499$	1.12461	0.0503445	0.0251723	+	0.999683i	$0.491987\pi$
$500$	0	0
$501$	0	0
$502$	−15.7082	−0.701091
$503$	35.2361	1.57110	0.785549	−	0.618799i	$-0.212380\pi$
$503$	35.2361	1.57110	0.785549	+	0.618799i	$0.212380\pi$
$504$	0	0
$505$	0	0
$506$	−32.3607	−1.43861
$507$	0	0
$508$	−5.41641	−0.240314
$509$	32.9443	1.46023	0.730115	−	0.683325i	$-0.239467\pi$
$509$	32.9443	1.46023	0.730115	+	0.683325i	$0.239467\pi$
$510$	0	0
$511$	−10.8328	−0.479216
$512$	1.00000	0.0441942
$513$	0	0
$514$	−5.23607	−0.230953
$515$	0	0
$516$	0	0
$517$	22.1115	0.972461
$518$	−8.58359	−0.377141
$519$	0	0
$520$	0	0
$521$	32.8328	1.43843	0.719216	−	0.694787i	$-0.244501\pi$
$521$	32.8328	1.43843	0.719216	+	0.694787i	$0.244501\pi$
$522$	0	0
$523$	8.00000	0.349816	0.174908	−	0.984585i	$-0.444037\pi$
$523$	8.00000	0.349816	0.174908	+	0.984585i	$0.444037\pi$
$524$	3.70820	0.161994
$525$	0	0
$526$	12.1803	0.531088
$527$	11.0557	0.481595
$528$	0	0
$529$	29.3607	1.27655
$530$	0	0
$531$	0	0
$532$	8.94427	0.387783
$533$	−12.4721	−0.540228
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	−11.2361	−0.484421
$539$	−24.4721	−1.05409
$540$	0	0
$541$	−9.23607	−0.397090	−0.198545	−	0.980092i	$-0.563622\pi$
$541$	−9.23607	−0.397090	−0.198545	+	0.980092i	$0.563622\pi$
$542$	15.4164	0.662191
$543$	0	0
$544$	−2.76393	−0.118503
$545$	0	0
$546$	0	0
$547$	12.0000	0.513083	0.256541	−	0.966533i	$-0.417417\pi$
$547$	12.0000	0.513083	0.256541	+	0.966533i	$0.417417\pi$
$548$	−15.8885	−0.678725
$549$	0	0
$550$	0	0
$551$	70.2492	2.99272
$552$	0	0
$553$	4.94427	0.210252
$554$	−32.8328	−1.39493
$555$	0	0
$556$	2.47214	0.104842
$557$	43.3050	1.83489	0.917445	−	0.397863i	$-0.130248\pi$
$557$	43.3050	1.83489	0.917445	+	0.397863i	$0.130248\pi$
$558$	0	0
$559$	6.47214	0.273742
$560$	0	0
$561$	0	0
$562$	−15.5279	−0.655003
$563$	−7.41641	−0.312564	−0.156282	−	0.987712i	$-0.549951\pi$
$563$	−7.41641	−0.312564	−0.156282	+	0.987712i	$0.549951\pi$
$564$	0	0
$565$	0	0
$566$	15.4164	0.648000
$567$	0	0
$568$	−6.47214	−0.271565
$569$	−17.0557	−0.715013	−0.357507	−	0.933911i	$-0.616373\pi$
$569$	−17.0557	−0.715013	−0.357507	+	0.933911i	$0.616373\pi$
$570$	0	0
$571$	−35.4164	−1.48213	−0.741065	−	0.671433i	$-0.765679\pi$
$571$	−35.4164	−1.48213	−0.741065	+	0.671433i	$0.765679\pi$
$572$	4.47214	0.186989
$573$	0	0
$574$	15.4164	0.643468
$575$	0	0
$576$	0	0
$577$	−2.87539	−0.119704	−0.0598520	−	0.998207i	$-0.519063\pi$
$577$	−2.87539	−0.119704	−0.0598520	+	0.998207i	$0.519063\pi$
$578$	−9.36068	−0.389353
$579$	0	0
$580$	0	0
$581$	16.0000	0.663792
$582$	0	0
$583$	−37.8885	−1.56918
$584$	8.76393	0.362654
$585$	0	0
$586$	14.9443	0.617342
$587$	37.8885	1.56383	0.781914	−	0.623387i	$-0.214244\pi$
$587$	37.8885	1.56383	0.781914	+	0.623387i	$0.214244\pi$
$588$	0	0
$589$	28.9443	1.19263
$590$	0	0
$591$	0	0
$592$	6.94427	0.285408
$593$	−12.4721	−0.512169	−0.256085	−	0.966654i	$-0.582433\pi$
$593$	−12.4721	−0.512169	−0.256085	+	0.966654i	$0.582433\pi$
$594$	0	0
$595$	0	0
$596$	−22.4721	−0.920495
$597$	0	0
$598$	−7.23607	−0.295905
$599$	41.8885	1.71152	0.855760	−	0.517373i	$-0.173090\pi$
$599$	41.8885	1.71152	0.855760	+	0.517373i	$0.173090\pi$
$600$	0	0
$601$	−13.4164	−0.547267	−0.273633	−	0.961834i	$-0.588225\pi$
$601$	−13.4164	−0.547267	−0.273633	+	0.961834i	$0.588225\pi$
$602$	−8.00000	−0.326056
$603$	0	0
$604$	7.41641	0.301769
$605$	0	0
$606$	0	0
$607$	−25.4164	−1.03162	−0.515810	−	0.856703i	$-0.672509\pi$
$607$	−25.4164	−1.03162	−0.515810	+	0.856703i	$0.672509\pi$
$608$	−7.23607	−0.293461
$609$	0	0
$610$	0	0
$611$	4.94427	0.200024
$612$	0	0
$613$	18.0000	0.727013	0.363507	−	0.931592i	$-0.381579\pi$
$613$	18.0000	0.727013	0.363507	+	0.931592i	$0.381579\pi$
$614$	20.3607	0.821690
$615$	0	0
$616$	−5.52786	−0.222724
$617$	−0.111456	−0.00448706	−0.00224353	−	0.999997i	$-0.500714\pi$
$617$	−0.111456	−0.00448706	−0.00224353	+	0.999997i	$0.500714\pi$
$618$	0	0
$619$	−20.7639	−0.834573	−0.417286	−	0.908775i	$-0.637019\pi$
$619$	−20.7639	−0.834573	−0.417286	+	0.908775i	$0.637019\pi$
$620$	0	0
$621$	0	0
$622$	13.5279	0.542418
$623$	−10.4721	−0.419557
$624$	0	0
$625$	0	0
$626$	7.41641	0.296419
$627$	0	0
$628$	−1.05573	−0.0421281
$629$	−19.1935	−0.765295
$630$	0	0
$631$	−39.4164	−1.56914	−0.784571	−	0.620039i	$-0.787117\pi$
$631$	−39.4164	−1.56914	−0.784571	+	0.620039i	$0.787117\pi$
$632$	−4.00000	−0.159111
$633$	0	0
$634$	13.4164	0.532834
$635$	0	0
$636$	0	0
$637$	−5.47214	−0.216814
$638$	−43.4164	−1.71887
$639$	0	0
$640$	0	0
$641$	−39.3050	−1.55245	−0.776226	−	0.630455i	$-0.782869\pi$
$641$	−39.3050	−1.55245	−0.776226	+	0.630455i	$0.782869\pi$
$642$	0	0
$643$	36.9443	1.45694	0.728470	−	0.685078i	$-0.240232\pi$
$643$	36.9443	1.45694	0.728470	+	0.685078i	$0.240232\pi$
$644$	8.94427	0.352454
$645$	0	0
$646$	20.0000	0.786889
$647$	−9.70820	−0.381669	−0.190834	−	0.981622i	$-0.561119\pi$
$647$	−9.70820	−0.381669	−0.190834	+	0.981622i	$0.561119\pi$
$648$	0	0
$649$	−2.11146	−0.0828819
$650$	0	0
$651$	0	0
$652$	−4.94427	−0.193633
$653$	−14.3607	−0.561977	−0.280988	−	0.959711i	$-0.590662\pi$
$653$	−14.3607	−0.561977	−0.280988	+	0.959711i	$0.590662\pi$
$654$	0	0
$655$	0	0
$656$	−12.4721	−0.486955
$657$	0	0
$658$	−6.11146	−0.238249
$659$	−48.0689	−1.87250	−0.936249	−	0.351337i	$-0.885727\pi$
$659$	−48.0689	−1.87250	−0.936249	+	0.351337i	$0.885727\pi$
$660$	0	0
$661$	−28.2918	−1.10042	−0.550212	−	0.835025i	$-0.685453\pi$
$661$	−28.2918	−1.10042	−0.550212	+	0.835025i	$0.685453\pi$
$662$	−7.23607	−0.281238
$663$	0	0
$664$	−12.9443	−0.502335
$665$	0	0
$666$	0	0
$667$	70.2492	2.72006
$668$	−7.41641	−0.286949
$669$	0	0
$670$	0	0
$671$	31.0557	1.19889
$672$	0	0
$673$	44.0000	1.69608	0.848038	−	0.529936i	$-0.177784\pi$
$673$	44.0000	1.69608	0.848038	+	0.529936i	$0.177784\pi$
$674$	2.47214	0.0952231
$675$	0	0
$676$	1.00000	0.0384615
$677$	−36.4721	−1.40174	−0.700869	−	0.713290i	$-0.747204\pi$
$677$	−36.4721	−1.40174	−0.700869	+	0.713290i	$0.747204\pi$
$678$	0	0
$679$	−12.0000	−0.460518
$680$	0	0
$681$	0	0
$682$	−17.8885	−0.684988
$683$	6.11146	0.233848	0.116924	−	0.993141i	$-0.462697\pi$
$683$	6.11146	0.233848	0.116924	+	0.993141i	$0.462697\pi$
$684$	0	0
$685$	0	0
$686$	15.4164	0.588601
$687$	0	0
$688$	6.47214	0.246748
$689$	−8.47214	−0.322763
$690$	0	0
$691$	13.7082	0.521485	0.260742	−	0.965408i	$-0.416033\pi$
$691$	13.7082	0.521485	0.260742	+	0.965408i	$0.416033\pi$
$692$	5.05573	0.192190
$693$	0	0
$694$	−10.4721	−0.397517
$695$	0	0
$696$	0	0
$697$	34.4721	1.30573
$698$	−10.7639	−0.407421
$699$	0	0
$700$	0	0
$701$	−34.0689	−1.28676	−0.643382	−	0.765545i	$-0.722469\pi$
$701$	−34.0689	−1.28676	−0.643382	+	0.765545i	$0.722469\pi$
$702$	0	0
$703$	−50.2492	−1.89519
$704$	4.47214	0.168550
$705$	0	0
$706$	9.05573	0.340817
$707$	−5.88854	−0.221461
$708$	0	0
$709$	−34.5410	−1.29722	−0.648608	−	0.761123i	$-0.724648\pi$
$709$	−34.5410	−1.29722	−0.648608	+	0.761123i	$0.724648\pi$
$710$	0	0
$711$	0	0
$712$	8.47214	0.317507
$713$	28.9443	1.08397
$714$	0	0
$715$	0	0
$716$	14.1803	0.529944
$717$	0	0
$718$	2.47214	0.0922593
$719$	−3.05573	−0.113959	−0.0569797	−	0.998375i	$-0.518147\pi$
$719$	−3.05573	−0.113959	−0.0569797	+	0.998375i	$0.518147\pi$
$720$	0	0
$721$	0.583592	0.0217341
$722$	33.3607	1.24156
$723$	0	0
$724$	−9.41641	−0.349958
$725$	0	0
$726$	0	0
$727$	18.0000	0.667583	0.333792	−	0.942647i	$-0.391672\pi$
$727$	18.0000	0.667583	0.333792	+	0.942647i	$0.391672\pi$
$728$	−1.23607	−0.0458117
$729$	0	0
$730$	0	0
$731$	−17.8885	−0.661632
$732$	0	0
$733$	6.58359	0.243171	0.121585	−	0.992581i	$-0.461202\pi$
$733$	6.58359	0.243171	0.121585	+	0.992581i	$0.461202\pi$
$734$	−8.47214	−0.312712
$735$	0	0
$736$	−7.23607	−0.266725
$737$	0	0
$738$	0	0
$739$	−13.7082	−0.504264	−0.252132	−	0.967693i	$-0.581132\pi$
$739$	−13.7082	−0.504264	−0.252132	+	0.967693i	$0.581132\pi$
$740$	0	0
$741$	0	0
$742$	10.4721	0.384444
$743$	16.3607	0.600215	0.300108	−	0.953905i	$-0.402977\pi$
$743$	16.3607	0.600215	0.300108	+	0.953905i	$0.402977\pi$
$744$	0	0
$745$	0	0
$746$	2.00000	0.0732252
$747$	0	0
$748$	−12.3607	−0.451951
$749$	−12.9443	−0.472973
$750$	0	0
$751$	6.83282	0.249333	0.124666	−	0.992199i	$-0.460214\pi$
$751$	6.83282	0.249333	0.124666	+	0.992199i	$0.460214\pi$
$752$	4.94427	0.180299
$753$	0	0
$754$	−9.70820	−0.353552
$755$	0	0
$756$	0	0
$757$	−38.0000	−1.38113	−0.690567	−	0.723269i	$-0.742639\pi$
$757$	−38.0000	−1.38113	−0.690567	+	0.723269i	$0.742639\pi$
$758$	5.12461	0.186134
$759$	0	0
$760$	0	0
$761$	−16.4721	−0.597114	−0.298557	−	0.954392i	$-0.596505\pi$
$761$	−16.4721	−0.597114	−0.298557	+	0.954392i	$0.596505\pi$
$762$	0	0
$763$	19.4164	0.702921
$764$	−13.5279	−0.489421
$765$	0	0
$766$	−14.4721	−0.522900
$767$	−0.472136	−0.0170478
$768$	0	0
$769$	48.8328	1.76096	0.880478	−	0.474087i	$-0.157222\pi$
$769$	48.8328	1.76096	0.880478	+	0.474087i	$0.157222\pi$
$770$	0	0
$771$	0	0
$772$	−21.7082	−0.781295
$773$	−14.9443	−0.537508	−0.268754	−	0.963209i	$-0.586612\pi$
$773$	−14.9443	−0.537508	−0.268754	+	0.963209i	$0.586612\pi$
$774$	0	0
$775$	0	0
$776$	9.70820	0.348504
$777$	0	0
$778$	18.6525	0.668724
$779$	90.2492	3.23351
$780$	0	0
$781$	−28.9443	−1.03571
$782$	20.0000	0.715199
$783$	0	0
$784$	−5.47214	−0.195433
$785$	0	0
$786$	0	0
$787$	11.0557	0.394094	0.197047	−	0.980394i	$-0.436865\pi$
$787$	11.0557	0.394094	0.197047	+	0.980394i	$0.436865\pi$
$788$	−2.94427	−0.104885
$789$	0	0
$790$	0	0
$791$	−15.6393	−0.556070
$792$	0	0
$793$	6.94427	0.246598
$794$	−9.41641	−0.334176
$795$	0	0
$796$	16.9443	0.600574
$797$	−26.9443	−0.954415	−0.477208	−	0.878791i	$-0.658351\pi$
$797$	−26.9443	−0.954415	−0.477208	+	0.878791i	$0.658351\pi$
$798$	0	0
$799$	−13.6656	−0.483455
$800$	0	0
$801$	0	0
$802$	15.5279	0.548308
$803$	39.1935	1.38311
$804$	0	0
$805$	0	0
$806$	−4.00000	−0.140894
$807$	0	0
$808$	4.76393	0.167595
$809$	−40.4721	−1.42292	−0.711462	−	0.702724i	$-0.751967\pi$
$809$	−40.4721	−1.42292	−0.711462	+	0.702724i	$0.751967\pi$
$810$	0	0
$811$	17.7082	0.621819	0.310910	−	0.950439i	$-0.399366\pi$
$811$	17.7082	0.621819	0.310910	+	0.950439i	$0.399366\pi$
$812$	12.0000	0.421117
$813$	0	0
$814$	31.0557	1.08850
$815$	0	0
$816$	0	0
$817$	−46.8328	−1.63847
$818$	6.00000	0.209785
$819$	0	0
$820$	0	0
$821$	21.5279	0.751328	0.375664	−	0.926756i	$-0.377415\pi$
$821$	21.5279	0.751328	0.375664	+	0.926756i	$0.377415\pi$
$822$	0	0
$823$	−32.2492	−1.12414	−0.562069	−	0.827091i	$-0.689994\pi$
$823$	−32.2492	−1.12414	−0.562069	+	0.827091i	$0.689994\pi$
$824$	−0.472136	−0.0164476
$825$	0	0
$826$	0.583592	0.0203058
$827$	6.11146	0.212516	0.106258	−	0.994339i	$-0.466113\pi$
$827$	6.11146	0.212516	0.106258	+	0.994339i	$0.466113\pi$
$828$	0	0
$829$	−46.7214	−1.62270	−0.811350	−	0.584561i	$-0.801267\pi$
$829$	−46.7214	−1.62270	−0.811350	+	0.584561i	$0.801267\pi$
$830$	0	0
$831$	0	0
$832$	1.00000	0.0346688
$833$	15.1246	0.524037
$834$	0	0
$835$	0	0
$836$	−32.3607	−1.11922
$837$	0	0
$838$	21.5967	0.746047
$839$	−19.7771	−0.682781	−0.341390	−	0.939922i	$-0.610898\pi$
$839$	−19.7771	−0.682781	−0.341390	+	0.939922i	$0.610898\pi$
$840$	0	0
$841$	65.2492	2.24997
$842$	−0.291796	−0.0100560
$843$	0	0
$844$	−21.8885	−0.753435
$845$	0	0
$846$	0	0
$847$	−11.1246	−0.382246
$848$	−8.47214	−0.290934
$849$	0	0
$850$	0	0
$851$	−50.2492	−1.72252
$852$	0	0
$853$	14.5836	0.499333	0.249666	−	0.968332i	$-0.419679\pi$
$853$	14.5836	0.499333	0.249666	+	0.968332i	$0.419679\pi$
$854$	−8.58359	−0.293724
$855$	0	0
$856$	10.4721	0.357930
$857$	−22.1803	−0.757666	−0.378833	−	0.925465i	$-0.623675\pi$
$857$	−22.1803	−0.757666	−0.378833	+	0.925465i	$0.623675\pi$
$858$	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
$861$	0	0
$862$	−38.8328	−1.32265
$863$	−13.5279	−0.460494	−0.230247	−	0.973132i	$-0.573953\pi$
$863$	−13.5279	−0.460494	−0.230247	+	0.973132i	$0.573953\pi$
$864$	0	0
$865$	0	0
$866$	−18.4721	−0.627709
$867$	0	0
$868$	4.94427	0.167820
$869$	−17.8885	−0.606827
$870$	0	0
$871$	0	0
$872$	−15.7082	−0.531947
$873$	0	0
$874$	52.3607	1.77113
$875$	0	0
$876$	0	0
$877$	15.5279	0.524339	0.262169	−	0.965022i	$-0.415562\pi$
$877$	15.5279	0.524339	0.262169	+	0.965022i	$0.415562\pi$
$878$	12.9443	0.436848
$879$	0	0
$880$	0	0
$881$	10.3607	0.349060	0.174530	−	0.984652i	$-0.444159\pi$
$881$	10.3607	0.349060	0.174530	+	0.984652i	$0.444159\pi$
$882$	0	0
$883$	−9.88854	−0.332776	−0.166388	−	0.986060i	$-0.553210\pi$
$883$	−9.88854	−0.332776	−0.166388	+	0.986060i	$0.553210\pi$
$884$	−2.76393	−0.0929611
$885$	0	0
$886$	13.5279	0.454477
$887$	−12.1803	−0.408976	−0.204488	−	0.978869i	$-0.565553\pi$
$887$	−12.1803	−0.408976	−0.204488	+	0.978869i	$0.565553\pi$
$888$	0	0
$889$	6.69505	0.224545
$890$	0	0
$891$	0	0
$892$	21.2361	0.711036
$893$	−35.7771	−1.19723
$894$	0	0
$895$	0	0
$896$	−1.23607	−0.0412941
$897$	0	0
$898$	8.47214	0.282719
$899$	38.8328	1.29515
$900$	0	0
$901$	23.4164	0.780114
$902$	−55.7771	−1.85717
$903$	0	0
$904$	12.6525	0.420815
$905$	0	0
$906$	0	0
$907$	54.4721	1.80872	0.904359	−	0.426773i	$-0.140350\pi$
$907$	54.4721	1.80872	0.904359	+	0.426773i	$0.140350\pi$
$908$	−24.9443	−0.827805
$909$	0	0
$910$	0	0
$911$	28.9443	0.958967	0.479483	−	0.877551i	$-0.340824\pi$
$911$	28.9443	0.958967	0.479483	+	0.877551i	$0.340824\pi$
$912$	0	0
$913$	−57.8885	−1.91583
$914$	9.70820	0.321119
$915$	0	0
$916$	−3.70820	−0.122523
$917$	−4.58359	−0.151364
$918$	0	0
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	0	0
$921$	0	0
$922$	−6.11146	−0.201270
$923$	−6.47214	−0.213033
$924$	0	0
$925$	0	0
$926$	−35.7082	−1.17344
$927$	0	0
$928$	−9.70820	−0.318687
$929$	−23.5279	−0.771924	−0.385962	−	0.922515i	$-0.626130\pi$
$929$	−23.5279	−0.771924	−0.385962	+	0.922515i	$0.626130\pi$
$930$	0	0
$931$	39.5967	1.29773
$932$	2.18034	0.0714194
$933$	0	0
$934$	2.47214	0.0808908
$935$	0	0
$936$	0	0
$937$	−51.7771	−1.69148	−0.845742	−	0.533592i	$-0.820842\pi$
$937$	−51.7771	−1.69148	−0.845742	+	0.533592i	$0.820842\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	46.4721	1.51495	0.757474	−	0.652865i	$-0.226433\pi$
$941$	46.4721	1.51495	0.757474	+	0.652865i	$0.226433\pi$
$942$	0	0
$943$	90.2492	2.93892
$944$	−0.472136	−0.0153667
$945$	0	0
$946$	28.9443	0.941059
$947$	12.9443	0.420632	0.210316	−	0.977633i	$-0.432551\pi$
$947$	12.9443	0.420632	0.210316	+	0.977633i	$0.432551\pi$
$948$	0	0
$949$	8.76393	0.284489
$950$	0	0
$951$	0	0
$952$	3.41641	0.110726
$953$	11.1246	0.360362	0.180181	−	0.983634i	$-0.442332\pi$
$953$	11.1246	0.360362	0.180181	+	0.983634i	$0.442332\pi$
$954$	0	0
$955$	0	0
$956$	−13.8885	−0.449188
$957$	0	0
$958$	36.0000	1.16311
$959$	19.6393	0.634187
$960$	0	0
$961$	−15.0000	−0.483871
$962$	6.94427	0.223892
$963$	0	0
$964$	12.4721	0.401700
$965$	0	0
$966$	0	0
$967$	10.5410	0.338976	0.169488	−	0.985532i	$-0.445789\pi$
$967$	10.5410	0.338976	0.169488	+	0.985532i	$0.445789\pi$
$968$	9.00000	0.289271
$969$	0	0
$970$	0	0
$971$	−18.7639	−0.602163	−0.301082	−	0.953598i	$-0.597348\pi$
$971$	−18.7639	−0.602163	−0.301082	+	0.953598i	$0.597348\pi$
$972$	0	0
$973$	−3.05573	−0.0979621
$974$	25.5967	0.820173
$975$	0	0
$976$	6.94427	0.222281
$977$	−34.3607	−1.09930	−0.549648	−	0.835397i	$-0.685238\pi$
$977$	−34.3607	−1.09930	−0.549648	+	0.835397i	$0.685238\pi$
$978$	0	0
$979$	37.8885	1.21092
$980$	0	0
$981$	0	0
$982$	−6.18034	−0.197223
$983$	−19.4164	−0.619287	−0.309644	−	0.950853i	$-0.600210\pi$
$983$	−19.4164	−0.619287	−0.309644	+	0.950853i	$0.600210\pi$
$984$	0	0
$985$	0	0
$986$	26.8328	0.854531
$987$	0	0
$988$	−7.23607	−0.230210
$989$	−46.8328	−1.48920
$990$	0	0
$991$	7.05573	0.224133	0.112066	−	0.993701i	$-0.464253\pi$
$991$	7.05573	0.224133	0.112066	+	0.993701i	$0.464253\pi$
$992$	−4.00000	−0.127000
$993$	0	0
$994$	8.00000	0.253745
$995$	0	0
$996$	0	0
$997$	8.11146	0.256892	0.128446	−	0.991716i	$-0.459001\pi$
$997$	8.11146	0.256892	0.128446	+	0.991716i	$0.459001\pi$
$998$	1.12461	0.0355990
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5850.2.a.cm.1.1		2
3.2	odd	2		1950.2.a.be.1.1		2
5.2	odd	4		1170.2.e.e.469.4		4
5.3	odd	4		1170.2.e.e.469.1		4
5.4	even	2		5850.2.a.cf.1.2		2
15.2	even	4		390.2.e.e.79.1	✓	4
15.8	even	4		390.2.e.e.79.4	yes	4
15.14	odd	2		1950.2.a.bf.1.2		2
60.23	odd	4		3120.2.l.k.1249.2		4
60.47	odd	4		3120.2.l.k.1249.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.e.e.79.1	✓	4	15.2	even	4
390.2.e.e.79.4	yes	4	15.8	even	4
1170.2.e.e.469.1		4	5.3	odd	4
1170.2.e.e.469.4		4	5.2	odd	4
1950.2.a.be.1.1		2	3.2	odd	2
1950.2.a.bf.1.2		2	15.14	odd	2
3120.2.l.k.1249.2		4	60.23	odd	4
3120.2.l.k.1249.3		4	60.47	odd	4
5850.2.a.cf.1.2		2	5.4	even	2
5850.2.a.cm.1.1		2	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5850.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.23607 q^{7} +1.00000 q^{8} +4.47214 q^{11} +1.00000 q^{13} -1.23607 q^{14} +1.00000 q^{16} -2.76393 q^{17} -7.23607 q^{19} +4.47214 q^{22} -7.23607 q^{23} +1.00000 q^{26} -1.23607 q^{28} -9.70820 q^{29} -4.00000 q^{31} +1.00000 q^{32} -2.76393 q^{34} +6.94427 q^{37} -7.23607 q^{38} -12.4721 q^{41} +6.47214 q^{43} +4.47214 q^{44} -7.23607 q^{46} +4.94427 q^{47} -5.47214 q^{49} +1.00000 q^{52} -8.47214 q^{53} -1.23607 q^{56} -9.70820 q^{58} -0.472136 q^{59} +6.94427 q^{61} -4.00000 q^{62} +1.00000 q^{64} -2.76393 q^{68} -6.47214 q^{71} +8.76393 q^{73} +6.94427 q^{74} -7.23607 q^{76} -5.52786 q^{77} -4.00000 q^{79} -12.4721 q^{82} -12.9443 q^{83} +6.47214 q^{86} +4.47214 q^{88} +8.47214 q^{89} -1.23607 q^{91} -7.23607 q^{92} +4.94427 q^{94} +9.70820 q^{97} -5.47214 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{7} + 2 q^{8} + 2 q^{13} + 2 q^{14} + 2 q^{16} - 10 q^{17} - 10 q^{19} - 10 q^{23} + 2 q^{26} + 2 q^{28} - 6 q^{29} - 8 q^{31} + 2 q^{32} - 10 q^{34} - 4 q^{37} - 10 q^{38}+ \cdots - 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^7 + 2 * q^8 + 2 * q^13 + 2 * q^14 + 2 * q^16 - 10 * q^17 - 10 * q^19 - 10 * q^23 + 2 * q^26 + 2 * q^28 - 6 * q^29 - 8 * q^31 + 2 * q^32 - 10 * q^34 - 4 * q^37 - 10 * q^38 - 16 * q^41 + 4 * q^43 - 10 * q^46 - 8 * q^47 - 2 * q^49 + 2 * q^52 - 8 * q^53 + 2 * q^56 - 6 * q^58 + 8 * q^59 - 4 * q^61 - 8 * q^62 + 2 * q^64 - 10 * q^68 - 4 * q^71 + 22 * q^73 - 4 * q^74 - 10 * q^76 - 20 * q^77 - 8 * q^79 - 16 * q^82 - 8 * q^83 + 4 * q^86 + 8 * q^89 + 2 * q^91 - 10 * q^92 - 8 * q^94 + 6 * q^97 - 2 * q^98

Introduction

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