LMFDB - Embedded newform 4275.2.a.v.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4275,2,Mod(1,4275)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4275, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4275.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4275 = 3^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4275.a (trivial)

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:	yes
Analytic conductor:	$34.1360468641$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1425)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	4275.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.618034 q^{2} -1.61803 q^{4} +2.23607 q^{7} +2.23607 q^{8} +4.00000 q^{11} -6.47214 q^{13} -1.38197 q^{14} +1.85410 q^{16} +1.23607 q^{17} -1.00000 q^{19} -2.47214 q^{22} +3.23607 q^{23} +4.00000 q^{26} -3.61803 q^{28} -7.47214 q^{29} -10.4721 q^{31} -5.61803 q^{32} -0.763932 q^{34} -2.76393 q^{37} +0.618034 q^{38} +5.00000 q^{41} -4.00000 q^{43} -6.47214 q^{44} -2.00000 q^{46} +0.472136 q^{47} -2.00000 q^{49} +10.4721 q^{52} -5.00000 q^{53} +5.00000 q^{56} +4.61803 q^{58} +14.7082 q^{59} +7.94427 q^{61} +6.47214 q^{62} -0.236068 q^{64} +10.9443 q^{67} -2.00000 q^{68} -9.18034 q^{71} -3.47214 q^{73} +1.70820 q^{74} +1.61803 q^{76} +8.94427 q^{77} -2.29180 q^{79} -3.09017 q^{82} -6.94427 q^{83} +2.47214 q^{86} +8.94427 q^{88} +9.94427 q^{89} -14.4721 q^{91} -5.23607 q^{92} -0.291796 q^{94} +1.70820 q^{97} +1.23607 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - q^{4} + 8 q^{11} - 4 q^{13} - 5 q^{14} - 3 q^{16} - 2 q^{17} - 2 q^{19} + 4 q^{22} + 2 q^{23} + 8 q^{26} - 5 q^{28} - 6 q^{29} - 12 q^{31} - 9 q^{32} - 6 q^{34} - 10 q^{37} - q^{38} + 10 q^{41}+ \cdots - 2 q^{98}+O(q^{100}) $ 2 * q + q^2 - q^4 + 8 * q^11 - 4 * q^13 - 5 * q^14 - 3 * q^16 - 2 * q^17 - 2 * q^19 + 4 * q^22 + 2 * q^23 + 8 * q^26 - 5 * q^28 - 6 * q^29 - 12 * q^31 - 9 * q^32 - 6 * q^34 - 10 * q^37 - q^38 + 10 * q^41 - 8 * q^43 - 4 * q^44 - 4 * q^46 - 8 * q^47 - 4 * q^49 + 12 * q^52 - 10 * q^53 + 10 * q^56 + 7 * q^58 + 16 * q^59 - 2 * q^61 + 4 * q^62 + 4 * q^64 + 4 * q^67 - 4 * q^68 + 4 * q^71 + 2 * q^73 - 10 * q^74 + q^76 - 18 * q^79 + 5 * q^82 + 4 * q^83 - 4 * q^86 + 2 * q^89 - 20 * q^91 - 6 * q^92 - 14 * q^94 - 10 * 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 $

$2$	−0.618034	−0.437016	−0.218508	−	0.975835i	$-0.570119\pi$
$2$	−0.618034	−0.437016	−0.218508	+	0.975835i	$0.570119\pi$
$3$	0	0
$3$	0	0
$4$	−1.61803	−0.809017
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.23607	0.845154	0.422577	−	0.906327i	$-0.361126\pi$
$7$	2.23607	0.845154	0.422577	+	0.906327i	$0.361126\pi$
$8$	2.23607	0.790569
$9$	0	0
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	−6.47214	−1.79505	−0.897524	−	0.440966i	$-0.854636\pi$
$13$	−6.47214	−1.79505	−0.897524	+	0.440966i	$0.854636\pi$
$14$	−1.38197	−0.369346
$15$	0	0
$16$	1.85410	0.463525
$17$	1.23607	0.299791	0.149895	−	0.988702i	$-0.452106\pi$
$17$	1.23607	0.299791	0.149895	+	0.988702i	$0.452106\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	−2.47214	−0.527061
$23$	3.23607	0.674767	0.337383	−	0.941367i	$-0.390458\pi$
$23$	3.23607	0.674767	0.337383	+	0.941367i	$0.390458\pi$
$24$	0	0
$25$	0	0
$26$	4.00000	0.784465
$27$	0	0
$28$	−3.61803	−0.683744
$29$	−7.47214	−1.38754	−0.693770	−	0.720196i	$-0.744052\pi$
$29$	−7.47214	−1.38754	−0.693770	+	0.720196i	$0.744052\pi$
$30$	0	0
$31$	−10.4721	−1.88085	−0.940426	−	0.340000i	$-0.889573\pi$
$31$	−10.4721	−1.88085	−0.940426	+	0.340000i	$0.889573\pi$
$32$	−5.61803	−0.993137
$33$	0	0
$34$	−0.763932	−0.131013
$35$	0	0
$36$	0	0
$37$	−2.76393	−0.454388	−0.227194	−	0.973850i	$-0.572955\pi$
$37$	−2.76393	−0.454388	−0.227194	+	0.973850i	$0.572955\pi$
$38$	0.618034	0.100258
$39$	0	0
$40$	0	0
$41$	5.00000	0.780869	0.390434	−	0.920631i	$-0.372325\pi$
$41$	5.00000	0.780869	0.390434	+	0.920631i	$0.372325\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	−6.47214	−0.975711
$45$	0	0
$46$	−2.00000	−0.294884
$47$	0.472136	0.0688681	0.0344341	−	0.999407i	$-0.489037\pi$
$47$	0.472136	0.0688681	0.0344341	+	0.999407i	$0.489037\pi$
$48$	0	0
$49$	−2.00000	−0.285714
$50$	0	0
$51$	0	0
$52$	10.4721	1.45222
$53$	−5.00000	−0.686803	−0.343401	−	0.939189i	$-0.611579\pi$
$53$	−5.00000	−0.686803	−0.343401	+	0.939189i	$0.611579\pi$
$54$	0	0
$55$	0	0
$56$	5.00000	0.668153
$57$	0	0
$58$	4.61803	0.606378
$59$	14.7082	1.91485	0.957423	−	0.288690i	$-0.0932198\pi$
$59$	14.7082	1.91485	0.957423	+	0.288690i	$0.0932198\pi$
$60$	0	0
$61$	7.94427	1.01716	0.508580	−	0.861015i	$-0.330171\pi$
$61$	7.94427	1.01716	0.508580	+	0.861015i	$0.330171\pi$
$62$	6.47214	0.821962
$63$	0	0
$64$	−0.236068	−0.0295085
$65$	0	0
$66$	0	0
$67$	10.9443	1.33706	0.668528	−	0.743687i	$-0.266925\pi$
$67$	10.9443	1.33706	0.668528	+	0.743687i	$0.266925\pi$
$68$	−2.00000	−0.242536
$69$	0	0
$70$	0	0
$71$	−9.18034	−1.08951	−0.544753	−	0.838597i	$-0.683377\pi$
$71$	−9.18034	−1.08951	−0.544753	+	0.838597i	$0.683377\pi$
$72$	0	0
$73$	−3.47214	−0.406383	−0.203191	−	0.979139i	$-0.565131\pi$
$73$	−3.47214	−0.406383	−0.203191	+	0.979139i	$0.565131\pi$
$74$	1.70820	0.198575
$75$	0	0
$76$	1.61803	0.185601
$77$	8.94427	1.01929
$78$	0	0
$79$	−2.29180	−0.257847	−0.128924	−	0.991655i	$-0.541152\pi$
$79$	−2.29180	−0.257847	−0.128924	+	0.991655i	$0.541152\pi$
$80$	0	0
$81$	0	0
$82$	−3.09017	−0.341252
$83$	−6.94427	−0.762233	−0.381116	−	0.924527i	$-0.624460\pi$
$83$	−6.94427	−0.762233	−0.381116	+	0.924527i	$0.624460\pi$
$84$	0	0
$85$	0	0
$86$	2.47214	0.266577
$87$	0	0
$88$	8.94427	0.953463
$89$	9.94427	1.05409	0.527045	−	0.849837i	$-0.323300\pi$
$89$	9.94427	1.05409	0.527045	+	0.849837i	$0.323300\pi$
$90$	0	0
$91$	−14.4721	−1.51709
$92$	−5.23607	−0.545898
$93$	0	0
$94$	−0.291796	−0.0300965
$95$	0	0
$96$	0	0
$97$	1.70820	0.173442	0.0867209	−	0.996233i	$-0.472361\pi$
$97$	1.70820	0.173442	0.0867209	+	0.996233i	$0.472361\pi$
$98$	1.23607	0.124862
$99$	0	0
$100$	0	0
$101$	14.1803	1.41100	0.705498	−	0.708712i	$-0.250723\pi$
$101$	14.1803	1.41100	0.705498	+	0.708712i	$0.250723\pi$
$102$	0	0
$103$	−16.9443	−1.66957	−0.834784	−	0.550577i	$-0.814408\pi$
$103$	−16.9443	−1.66957	−0.834784	+	0.550577i	$0.814408\pi$
$104$	−14.4721	−1.41911
$105$	0	0
$106$	3.09017	0.300144
$107$	−16.2361	−1.56960	−0.784800	−	0.619749i	$-0.787234\pi$
$107$	−16.2361	−1.56960	−0.784800	+	0.619749i	$0.787234\pi$
$108$	0	0
$109$	−7.70820	−0.738312	−0.369156	−	0.929367i	$-0.620353\pi$
$109$	−7.70820	−0.738312	−0.369156	+	0.929367i	$0.620353\pi$
$110$	0	0
$111$	0	0
$112$	4.14590	0.391751
$113$	−12.4721	−1.17328	−0.586640	−	0.809848i	$-0.699550\pi$
$113$	−12.4721	−1.17328	−0.586640	+	0.809848i	$0.699550\pi$
$114$	0	0
$115$	0	0
$116$	12.0902	1.12254
$117$	0	0
$118$	−9.09017	−0.836818
$119$	2.76393	0.253369
$120$	0	0
$121$	5.00000	0.454545
$122$	−4.90983	−0.444515
$123$	0	0
$124$	16.9443	1.52164
$125$	0	0
$126$	0	0
$127$	−10.7639	−0.955145	−0.477572	−	0.878592i	$-0.658483\pi$
$127$	−10.7639	−0.955145	−0.477572	+	0.878592i	$0.658483\pi$
$128$	11.3820	1.00603
$129$	0	0
$130$	0	0
$131$	6.94427	0.606724	0.303362	−	0.952875i	$-0.401891\pi$
$131$	6.94427	0.606724	0.303362	+	0.952875i	$0.401891\pi$
$132$	0	0
$133$	−2.23607	−0.193892
$134$	−6.76393	−0.584315
$135$	0	0
$136$	2.76393	0.237005
$137$	4.29180	0.366673	0.183336	−	0.983050i	$-0.441310\pi$
$137$	4.29180	0.366673	0.183336	+	0.983050i	$0.441310\pi$
$138$	0	0
$139$	−15.7639	−1.33708	−0.668540	−	0.743677i	$-0.733080\pi$
$139$	−15.7639	−1.33708	−0.668540	+	0.743677i	$0.733080\pi$
$140$	0	0
$141$	0	0
$142$	5.67376	0.476132
$143$	−25.8885	−2.16491
$144$	0	0
$145$	0	0
$146$	2.14590	0.177596
$147$	0	0
$148$	4.47214	0.367607
$149$	19.4164	1.59065	0.795327	−	0.606181i	$-0.207299\pi$
$149$	19.4164	1.59065	0.795327	+	0.606181i	$0.207299\pi$
$150$	0	0
$151$	11.7082	0.952800	0.476400	−	0.879229i	$-0.341941\pi$
$151$	11.7082	0.952800	0.476400	+	0.879229i	$0.341941\pi$
$152$	−2.23607	−0.181369
$153$	0	0
$154$	−5.52786	−0.445448
$155$	0	0
$156$	0	0
$157$	−7.00000	−0.558661	−0.279330	−	0.960195i	$-0.590112\pi$
$157$	−7.00000	−0.558661	−0.279330	+	0.960195i	$0.590112\pi$
$158$	1.41641	0.112683
$159$	0	0
$160$	0	0
$161$	7.23607	0.570282
$162$	0	0
$163$	11.1803	0.875712	0.437856	−	0.899045i	$-0.355738\pi$
$163$	11.1803	0.875712	0.437856	+	0.899045i	$0.355738\pi$
$164$	−8.09017	−0.631736
$165$	0	0
$166$	4.29180	0.333108
$167$	−24.5967	−1.90335	−0.951677	−	0.307102i	$-0.900641\pi$
$167$	−24.5967	−1.90335	−0.951677	+	0.307102i	$0.900641\pi$
$168$	0	0
$169$	28.8885	2.22220
$170$	0	0
$171$	0	0
$172$	6.47214	0.493496
$173$	−13.4721	−1.02427	−0.512134	−	0.858906i	$-0.671145\pi$
$173$	−13.4721	−1.02427	−0.512134	+	0.858906i	$0.671145\pi$
$174$	0	0
$175$	0	0
$176$	7.41641	0.559033
$177$	0	0
$178$	−6.14590	−0.460655
$179$	−22.1246	−1.65367	−0.826836	−	0.562444i	$-0.809861\pi$
$179$	−22.1246	−1.65367	−0.826836	+	0.562444i	$0.809861\pi$
$180$	0	0
$181$	10.4721	0.778388	0.389194	−	0.921156i	$-0.372754\pi$
$181$	10.4721	0.778388	0.389194	+	0.921156i	$0.372754\pi$
$182$	8.94427	0.662994
$183$	0	0
$184$	7.23607	0.533450
$185$	0	0
$186$	0	0
$187$	4.94427	0.361561
$188$	−0.763932	−0.0557155
$189$	0	0
$190$	0	0
$191$	−7.41641	−0.536632	−0.268316	−	0.963331i	$-0.586467\pi$
$191$	−7.41641	−0.536632	−0.268316	+	0.963331i	$0.586467\pi$
$192$	0	0
$193$	3.23607	0.232937	0.116469	−	0.993194i	$-0.462843\pi$
$193$	3.23607	0.232937	0.116469	+	0.993194i	$0.462843\pi$
$194$	−1.05573	−0.0757969
$195$	0	0
$196$	3.23607	0.231148
$197$	−22.6525	−1.61392	−0.806961	−	0.590605i	$-0.798889\pi$
$197$	−22.6525	−1.61392	−0.806961	+	0.590605i	$0.798889\pi$
$198$	0	0
$199$	1.76393	0.125042	0.0625209	−	0.998044i	$-0.480086\pi$
$199$	1.76393	0.125042	0.0625209	+	0.998044i	$0.480086\pi$
$200$	0	0
$201$	0	0
$202$	−8.76393	−0.616628
$203$	−16.7082	−1.17269
$204$	0	0
$205$	0	0
$206$	10.4721	0.729628
$207$	0	0
$208$	−12.0000	−0.832050
$209$	−4.00000	−0.276686
$210$	0	0
$211$	5.81966	0.400642	0.200321	−	0.979730i	$-0.435802\pi$
$211$	5.81966	0.400642	0.200321	+	0.979730i	$0.435802\pi$
$212$	8.09017	0.555635
$213$	0	0
$214$	10.0344	0.685940
$215$	0	0
$216$	0	0
$217$	−23.4164	−1.58961
$218$	4.76393	0.322654
$219$	0	0
$220$	0	0
$221$	−8.00000	−0.538138
$222$	0	0
$223$	−9.23607	−0.618493	−0.309246	−	0.950982i	$-0.600077\pi$
$223$	−9.23607	−0.618493	−0.309246	+	0.950982i	$0.600077\pi$
$224$	−12.5623	−0.839354
$225$	0	0
$226$	7.70820	0.512742
$227$	28.1246	1.86670	0.933348	−	0.358973i	$-0.116873\pi$
$227$	28.1246	1.86670	0.933348	+	0.358973i	$0.116873\pi$
$228$	0	0
$229$	−4.47214	−0.295527	−0.147764	−	0.989023i	$-0.547207\pi$
$229$	−4.47214	−0.295527	−0.147764	+	0.989023i	$0.547207\pi$
$230$	0	0
$231$	0	0
$232$	−16.7082	−1.09695
$233$	2.94427	0.192886	0.0964428	−	0.995339i	$-0.469254\pi$
$233$	2.94427	0.192886	0.0964428	+	0.995339i	$0.469254\pi$
$234$	0	0
$235$	0	0
$236$	−23.7984	−1.54914
$237$	0	0
$238$	−1.70820	−0.110726
$239$	−0.291796	−0.0188747	−0.00943736	−	0.999955i	$-0.503004\pi$
$239$	−0.291796	−0.0188747	−0.00943736	+	0.999955i	$0.503004\pi$
$240$	0	0
$241$	−16.1803	−1.04227	−0.521134	−	0.853475i	$-0.674491\pi$
$241$	−16.1803	−1.04227	−0.521134	+	0.853475i	$0.674491\pi$
$242$	−3.09017	−0.198644
$243$	0	0
$244$	−12.8541	−0.822900
$245$	0	0
$246$	0	0
$247$	6.47214	0.411812
$248$	−23.4164	−1.48694
$249$	0	0
$250$	0	0
$251$	−3.70820	−0.234060	−0.117030	−	0.993128i	$-0.537337\pi$
$251$	−3.70820	−0.234060	−0.117030	+	0.993128i	$0.537337\pi$
$252$	0	0
$253$	12.9443	0.813799
$254$	6.65248	0.417413
$255$	0	0
$256$	−6.56231	−0.410144
$257$	−13.0000	−0.810918	−0.405459	−	0.914113i	$-0.632888\pi$
$257$	−13.0000	−0.810918	−0.405459	+	0.914113i	$0.632888\pi$
$258$	0	0
$259$	−6.18034	−0.384028
$260$	0	0
$261$	0	0
$262$	−4.29180	−0.265148
$263$	−10.2918	−0.634619	−0.317310	−	0.948322i	$-0.602779\pi$
$263$	−10.2918	−0.634619	−0.317310	+	0.948322i	$0.602779\pi$
$264$	0	0
$265$	0	0
$266$	1.38197	0.0847338
$267$	0	0
$268$	−17.7082	−1.08170
$269$	−20.4721	−1.24821	−0.624104	−	0.781341i	$-0.714536\pi$
$269$	−20.4721	−1.24821	−0.624104	+	0.781341i	$0.714536\pi$
$270$	0	0
$271$	3.18034	0.193192	0.0965959	−	0.995324i	$-0.469205\pi$
$271$	3.18034	0.193192	0.0965959	+	0.995324i	$0.469205\pi$
$272$	2.29180	0.138961
$273$	0	0
$274$	−2.65248	−0.160242
$275$	0	0
$276$	0	0
$277$	−5.58359	−0.335486	−0.167743	−	0.985831i	$-0.553648\pi$
$277$	−5.58359	−0.335486	−0.167743	+	0.985831i	$0.553648\pi$
$278$	9.74265	0.584325
$279$	0	0
$280$	0	0
$281$	−7.88854	−0.470591	−0.235296	−	0.971924i	$-0.575606\pi$
$281$	−7.88854	−0.470591	−0.235296	+	0.971924i	$0.575606\pi$
$282$	0	0
$283$	−9.88854	−0.587813	−0.293906	−	0.955834i	$-0.594955\pi$
$283$	−9.88854	−0.587813	−0.293906	+	0.955834i	$0.594955\pi$
$284$	14.8541	0.881429
$285$	0	0
$286$	16.0000	0.946100
$287$	11.1803	0.659955
$288$	0	0
$289$	−15.4721	−0.910126
$290$	0	0
$291$	0	0
$292$	5.61803	0.328771
$293$	2.00000	0.116841	0.0584206	−	0.998292i	$-0.481394\pi$
$293$	2.00000	0.116841	0.0584206	+	0.998292i	$0.481394\pi$
$294$	0	0
$295$	0	0
$296$	−6.18034	−0.359225
$297$	0	0
$298$	−12.0000	−0.695141
$299$	−20.9443	−1.21124
$300$	0	0
$301$	−8.94427	−0.515539
$302$	−7.23607	−0.416389
$303$	0	0
$304$	−1.85410	−0.106340
$305$	0	0
$306$	0	0
$307$	3.05573	0.174400	0.0871998	−	0.996191i	$-0.472208\pi$
$307$	3.05573	0.174400	0.0871998	+	0.996191i	$0.472208\pi$
$308$	−14.4721	−0.824626
$309$	0	0
$310$	0	0
$311$	−4.00000	−0.226819	−0.113410	−	0.993548i	$-0.536177\pi$
$311$	−4.00000	−0.226819	−0.113410	+	0.993548i	$0.536177\pi$
$312$	0	0
$313$	30.9443	1.74907	0.874537	−	0.484959i	$-0.161166\pi$
$313$	30.9443	1.74907	0.874537	+	0.484959i	$0.161166\pi$
$314$	4.32624	0.244144
$315$	0	0
$316$	3.70820	0.208603
$317$	−7.94427	−0.446195	−0.223097	−	0.974796i	$-0.571617\pi$
$317$	−7.94427	−0.446195	−0.223097	+	0.974796i	$0.571617\pi$
$318$	0	0
$319$	−29.8885	−1.67344
$320$	0	0
$321$	0	0
$322$	−4.47214	−0.249222
$323$	−1.23607	−0.0687767
$324$	0	0
$325$	0	0
$326$	−6.90983	−0.382700
$327$	0	0
$328$	11.1803	0.617331
$329$	1.05573	0.0582042
$330$	0	0
$331$	−26.3607	−1.44891	−0.724457	−	0.689320i	$-0.757909\pi$
$331$	−26.3607	−1.44891	−0.724457	+	0.689320i	$0.757909\pi$
$332$	11.2361	0.616659
$333$	0	0
$334$	15.2016	0.831796
$335$	0	0
$336$	0	0
$337$	8.47214	0.461507	0.230753	−	0.973012i	$-0.425881\pi$
$337$	8.47214	0.461507	0.230753	+	0.973012i	$0.425881\pi$
$338$	−17.8541	−0.971135
$339$	0	0
$340$	0	0
$341$	−41.8885	−2.26839
$342$	0	0
$343$	−20.1246	−1.08663
$344$	−8.94427	−0.482243
$345$	0	0
$346$	8.32624	0.447621
$347$	−1.70820	−0.0917012	−0.0458506	−	0.998948i	$-0.514600\pi$
$347$	−1.70820	−0.0917012	−0.0458506	+	0.998948i	$0.514600\pi$
$348$	0	0
$349$	−17.8328	−0.954569	−0.477284	−	0.878749i	$-0.658379\pi$
$349$	−17.8328	−0.954569	−0.477284	+	0.878749i	$0.658379\pi$
$350$	0	0
$351$	0	0
$352$	−22.4721	−1.19777
$353$	−26.4721	−1.40897	−0.704485	−	0.709719i	$-0.748822\pi$
$353$	−26.4721	−1.40897	−0.704485	+	0.709719i	$0.748822\pi$
$354$	0	0
$355$	0	0
$356$	−16.0902	−0.852777
$357$	0	0
$358$	13.6738	0.722681
$359$	25.2361	1.33191	0.665954	−	0.745992i	$-0.268025\pi$
$359$	25.2361	1.33191	0.665954	+	0.745992i	$0.268025\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−6.47214	−0.340168
$363$	0	0
$364$	23.4164	1.22735
$365$	0	0
$366$	0	0
$367$	−33.3050	−1.73850	−0.869252	−	0.494369i	$-0.835399\pi$
$367$	−33.3050	−1.73850	−0.869252	+	0.494369i	$0.835399\pi$
$368$	6.00000	0.312772
$369$	0	0
$370$	0	0
$371$	−11.1803	−0.580454
$372$	0	0
$373$	−12.1803	−0.630674	−0.315337	−	0.948980i	$-0.602118\pi$
$373$	−12.1803	−0.630674	−0.315337	+	0.948980i	$0.602118\pi$
$374$	−3.05573	−0.158008
$375$	0	0
$376$	1.05573	0.0544450
$377$	48.3607	2.49070
$378$	0	0
$379$	6.76393	0.347440	0.173720	−	0.984795i	$-0.444421\pi$
$379$	6.76393	0.347440	0.173720	+	0.984795i	$0.444421\pi$
$380$	0	0
$381$	0	0
$382$	4.58359	0.234517
$383$	16.1246	0.823929	0.411965	−	0.911200i	$-0.364843\pi$
$383$	16.1246	0.823929	0.411965	+	0.911200i	$0.364843\pi$
$384$	0	0
$385$	0	0
$386$	−2.00000	−0.101797
$387$	0	0
$388$	−2.76393	−0.140317
$389$	−8.00000	−0.405616	−0.202808	−	0.979219i	$-0.565007\pi$
$389$	−8.00000	−0.405616	−0.202808	+	0.979219i	$0.565007\pi$
$390$	0	0
$391$	4.00000	0.202289
$392$	−4.47214	−0.225877
$393$	0	0
$394$	14.0000	0.705310
$395$	0	0
$396$	0	0
$397$	−22.3607	−1.12225	−0.561125	−	0.827731i	$-0.689631\pi$
$397$	−22.3607	−1.12225	−0.561125	+	0.827731i	$0.689631\pi$
$398$	−1.09017	−0.0546453
$399$	0	0
$400$	0	0
$401$	20.8328	1.04034	0.520171	−	0.854062i	$-0.325868\pi$
$401$	20.8328	1.04034	0.520171	+	0.854062i	$0.325868\pi$
$402$	0	0
$403$	67.7771	3.37622
$404$	−22.9443	−1.14152
$405$	0	0
$406$	10.3262	0.512483
$407$	−11.0557	−0.548012
$408$	0	0
$409$	8.76393	0.433349	0.216674	−	0.976244i	$-0.430479\pi$
$409$	8.76393	0.433349	0.216674	+	0.976244i	$0.430479\pi$
$410$	0	0
$411$	0	0
$412$	27.4164	1.35071
$413$	32.8885	1.61834
$414$	0	0
$415$	0	0
$416$	36.3607	1.78273
$417$	0	0
$418$	2.47214	0.120916
$419$	−39.3050	−1.92017	−0.960086	−	0.279704i	$-0.909764\pi$
$419$	−39.3050	−1.92017	−0.960086	+	0.279704i	$0.909764\pi$
$420$	0	0
$421$	11.5279	0.561834	0.280917	−	0.959732i	$-0.409361\pi$
$421$	11.5279	0.561834	0.280917	+	0.959732i	$0.409361\pi$
$422$	−3.59675	−0.175087
$423$	0	0
$424$	−11.1803	−0.542965
$425$	0	0
$426$	0	0
$427$	17.7639	0.859657
$428$	26.2705	1.26983
$429$	0	0
$430$	0	0
$431$	5.18034	0.249528	0.124764	−	0.992186i	$-0.460183\pi$
$431$	5.18034	0.249528	0.124764	+	0.992186i	$0.460183\pi$
$432$	0	0
$433$	19.5279	0.938449	0.469225	−	0.883079i	$-0.344533\pi$
$433$	19.5279	0.938449	0.469225	+	0.883079i	$0.344533\pi$
$434$	14.4721	0.694685
$435$	0	0
$436$	12.4721	0.597307
$437$	−3.23607	−0.154802
$438$	0	0
$439$	−16.7639	−0.800099	−0.400049	−	0.916494i	$-0.631007\pi$
$439$	−16.7639	−0.800099	−0.400049	+	0.916494i	$0.631007\pi$
$440$	0	0
$441$	0	0
$442$	4.94427	0.235175
$443$	13.4164	0.637433	0.318716	−	0.947850i	$-0.396748\pi$
$443$	13.4164	0.637433	0.318716	+	0.947850i	$0.396748\pi$
$444$	0	0
$445$	0	0
$446$	5.70820	0.270291
$447$	0	0
$448$	−0.527864	−0.0249392
$449$	25.4721	1.20210	0.601052	−	0.799210i	$-0.294748\pi$
$449$	25.4721	1.20210	0.601052	+	0.799210i	$0.294748\pi$
$450$	0	0
$451$	20.0000	0.941763
$452$	20.1803	0.949203
$453$	0	0
$454$	−17.3820	−0.815776
$455$	0	0
$456$	0	0
$457$	−8.88854	−0.415789	−0.207894	−	0.978151i	$-0.566661\pi$
$457$	−8.88854	−0.415789	−0.207894	+	0.978151i	$0.566661\pi$
$458$	2.76393	0.129150
$459$	0	0
$460$	0	0
$461$	1.81966	0.0847500	0.0423750	−	0.999102i	$-0.486508\pi$
$461$	1.81966	0.0847500	0.0423750	+	0.999102i	$0.486508\pi$
$462$	0	0
$463$	−8.58359	−0.398913	−0.199457	−	0.979907i	$-0.563918\pi$
$463$	−8.58359	−0.398913	−0.199457	+	0.979907i	$0.563918\pi$
$464$	−13.8541	−0.643161
$465$	0	0
$466$	−1.81966	−0.0842941
$467$	−26.1803	−1.21148	−0.605741	−	0.795662i	$-0.707123\pi$
$467$	−26.1803	−1.21148	−0.605741	+	0.795662i	$0.707123\pi$
$468$	0	0
$469$	24.4721	1.13002
$470$	0	0
$471$	0	0
$472$	32.8885	1.51382
$473$	−16.0000	−0.735681
$474$	0	0
$475$	0	0
$476$	−4.47214	−0.204980
$477$	0	0
$478$	0.180340	0.00824855
$479$	38.7639	1.77117	0.885585	−	0.464478i	$-0.153758\pi$
$479$	38.7639	1.77117	0.885585	+	0.464478i	$0.153758\pi$
$480$	0	0
$481$	17.8885	0.815647
$482$	10.0000	0.455488
$483$	0	0
$484$	−8.09017	−0.367735
$485$	0	0
$486$	0	0
$487$	−24.0000	−1.08754	−0.543772	−	0.839233i	$-0.683004\pi$
$487$	−24.0000	−1.08754	−0.543772	+	0.839233i	$0.683004\pi$
$488$	17.7639	0.804135
$489$	0	0
$490$	0	0
$491$	8.29180	0.374204	0.187102	−	0.982341i	$-0.440091\pi$
$491$	8.29180	0.374204	0.187102	+	0.982341i	$0.440091\pi$
$492$	0	0
$493$	−9.23607	−0.415972
$494$	−4.00000	−0.179969
$495$	0	0
$496$	−19.4164	−0.871822
$497$	−20.5279	−0.920801
$498$	0	0
$499$	−31.1803	−1.39582	−0.697912	−	0.716184i	$-0.745887\pi$
$499$	−31.1803	−1.39582	−0.697912	+	0.716184i	$0.745887\pi$
$500$	0	0
$501$	0	0
$502$	2.29180	0.102288
$503$	−9.52786	−0.424826	−0.212413	−	0.977180i	$-0.568132\pi$
$503$	−9.52786	−0.424826	−0.212413	+	0.977180i	$0.568132\pi$
$504$	0	0
$505$	0	0
$506$	−8.00000	−0.355643
$507$	0	0
$508$	17.4164	0.772728
$509$	9.00000	0.398918	0.199459	−	0.979906i	$-0.436082\pi$
$509$	9.00000	0.398918	0.199459	+	0.979906i	$0.436082\pi$
$510$	0	0
$511$	−7.76393	−0.343456
$512$	−18.7082	−0.826794
$513$	0	0
$514$	8.03444	0.354384
$515$	0	0
$516$	0	0
$517$	1.88854	0.0830581
$518$	3.81966	0.167826
$519$	0	0
$520$	0	0
$521$	35.8328	1.56986	0.784932	−	0.619582i	$-0.212698\pi$
$521$	35.8328	1.56986	0.784932	+	0.619582i	$0.212698\pi$
$522$	0	0
$523$	−41.1246	−1.79825	−0.899127	−	0.437688i	$-0.855797\pi$
$523$	−41.1246	−1.79825	−0.899127	+	0.437688i	$0.855797\pi$
$524$	−11.2361	−0.490850
$525$	0	0
$526$	6.36068	0.277339
$527$	−12.9443	−0.563861
$528$	0	0
$529$	−12.5279	−0.544690
$530$	0	0
$531$	0	0
$532$	3.61803	0.156862
$533$	−32.3607	−1.40170
$534$	0	0
$535$	0	0
$536$	24.4721	1.05704
$537$	0	0
$538$	12.6525	0.545487
$539$	−8.00000	−0.344584
$540$	0	0
$541$	24.8328	1.06765	0.533823	−	0.845596i	$-0.320755\pi$
$541$	24.8328	1.06765	0.533823	+	0.845596i	$0.320755\pi$
$542$	−1.96556	−0.0844280
$543$	0	0
$544$	−6.94427	−0.297733
$545$	0	0
$546$	0	0
$547$	7.70820	0.329579	0.164790	−	0.986329i	$-0.447306\pi$
$547$	7.70820	0.329579	0.164790	+	0.986329i	$0.447306\pi$
$548$	−6.94427	−0.296645
$549$	0	0
$550$	0	0
$551$	7.47214	0.318324
$552$	0	0
$553$	−5.12461	−0.217921
$554$	3.45085	0.146613
$555$	0	0
$556$	25.5066	1.08172
$557$	−33.8885	−1.43590	−0.717952	−	0.696093i	$-0.754920\pi$
$557$	−33.8885	−1.43590	−0.717952	+	0.696093i	$0.754920\pi$
$558$	0	0
$559$	25.8885	1.09497
$560$	0	0
$561$	0	0
$562$	4.87539	0.205656
$563$	23.6525	0.996833	0.498417	−	0.866938i	$-0.333915\pi$
$563$	23.6525	0.996833	0.498417	+	0.866938i	$0.333915\pi$
$564$	0	0
$565$	0	0
$566$	6.11146	0.256884
$567$	0	0
$568$	−20.5279	−0.861330
$569$	20.8885	0.875693	0.437847	−	0.899050i	$-0.355741\pi$
$569$	20.8885	0.875693	0.437847	+	0.899050i	$0.355741\pi$
$570$	0	0
$571$	−15.6525	−0.655036	−0.327518	−	0.944845i	$-0.606212\pi$
$571$	−15.6525	−0.655036	−0.327518	+	0.944845i	$0.606212\pi$
$572$	41.8885	1.75145
$573$	0	0
$574$	−6.90983	−0.288411
$575$	0	0
$576$	0	0
$577$	1.41641	0.0589658	0.0294829	−	0.999565i	$-0.490614\pi$
$577$	1.41641	0.0589658	0.0294829	+	0.999565i	$0.490614\pi$
$578$	9.56231	0.397739
$579$	0	0
$580$	0	0
$581$	−15.5279	−0.644204
$582$	0	0
$583$	−20.0000	−0.828315
$584$	−7.76393	−0.321274
$585$	0	0
$586$	−1.23607	−0.0510615
$587$	12.6525	0.522224	0.261112	−	0.965309i	$-0.415911\pi$
$587$	12.6525	0.522224	0.261112	+	0.965309i	$0.415911\pi$
$588$	0	0
$589$	10.4721	0.431497
$590$	0	0
$591$	0	0
$592$	−5.12461	−0.210620
$593$	−13.8885	−0.570334	−0.285167	−	0.958478i	$-0.592049\pi$
$593$	−13.8885	−0.570334	−0.285167	+	0.958478i	$0.592049\pi$
$594$	0	0
$595$	0	0
$596$	−31.4164	−1.28687
$597$	0	0
$598$	12.9443	0.529331
$599$	20.0000	0.817178	0.408589	−	0.912719i	$-0.366021\pi$
$599$	20.0000	0.817178	0.408589	+	0.912719i	$0.366021\pi$
$600$	0	0
$601$	−35.7771	−1.45938	−0.729689	−	0.683779i	$-0.760335\pi$
$601$	−35.7771	−1.45938	−0.729689	+	0.683779i	$0.760335\pi$
$602$	5.52786	0.225299
$603$	0	0
$604$	−18.9443	−0.770831
$605$	0	0
$606$	0	0
$607$	19.1246	0.776244	0.388122	−	0.921608i	$-0.373124\pi$
$607$	19.1246	0.776244	0.388122	+	0.921608i	$0.373124\pi$
$608$	5.61803	0.227841
$609$	0	0
$610$	0	0
$611$	−3.05573	−0.123622
$612$	0	0
$613$	−23.0000	−0.928961	−0.464481	−	0.885583i	$-0.653759\pi$
$613$	−23.0000	−0.928961	−0.464481	+	0.885583i	$0.653759\pi$
$614$	−1.88854	−0.0762154
$615$	0	0
$616$	20.0000	0.805823
$617$	22.0000	0.885687	0.442843	−	0.896599i	$-0.353970\pi$
$617$	22.0000	0.885687	0.442843	+	0.896599i	$0.353970\pi$
$618$	0	0
$619$	30.7082	1.23427	0.617133	−	0.786858i	$-0.288294\pi$
$619$	30.7082	1.23427	0.617133	+	0.786858i	$0.288294\pi$
$620$	0	0
$621$	0	0
$622$	2.47214	0.0991236
$623$	22.2361	0.890869
$624$	0	0
$625$	0	0
$626$	−19.1246	−0.764373
$627$	0	0
$628$	11.3262	0.451966
$629$	−3.41641	−0.136221
$630$	0	0
$631$	−43.4164	−1.72838	−0.864190	−	0.503166i	$-0.832169\pi$
$631$	−43.4164	−1.72838	−0.864190	+	0.503166i	$0.832169\pi$
$632$	−5.12461	−0.203846
$633$	0	0
$634$	4.90983	0.194994
$635$	0	0
$636$	0	0
$637$	12.9443	0.512871
$638$	18.4721	0.731319
$639$	0	0
$640$	0	0
$641$	−6.94427	−0.274282	−0.137141	−	0.990552i	$-0.543791\pi$
$641$	−6.94427	−0.274282	−0.137141	+	0.990552i	$0.543791\pi$
$642$	0	0
$643$	17.1803	0.677526	0.338763	−	0.940872i	$-0.389991\pi$
$643$	17.1803	0.677526	0.338763	+	0.940872i	$0.389991\pi$
$644$	−11.7082	−0.461368
$645$	0	0
$646$	0.763932	0.0300565
$647$	17.3050	0.680328	0.340164	−	0.940366i	$-0.389517\pi$
$647$	17.3050	0.680328	0.340164	+	0.940366i	$0.389517\pi$
$648$	0	0
$649$	58.8328	2.30939
$650$	0	0
$651$	0	0
$652$	−18.0902	−0.708466
$653$	25.8885	1.01310	0.506549	−	0.862211i	$-0.330921\pi$
$653$	25.8885	1.01310	0.506549	+	0.862211i	$0.330921\pi$
$654$	0	0
$655$	0	0
$656$	9.27051	0.361953
$657$	0	0
$658$	−0.652476	−0.0254362
$659$	−24.9443	−0.971691	−0.485845	−	0.874045i	$-0.661488\pi$
$659$	−24.9443	−0.971691	−0.485845	+	0.874045i	$0.661488\pi$
$660$	0	0
$661$	−23.1246	−0.899443	−0.449722	−	0.893169i	$-0.648477\pi$
$661$	−23.1246	−0.899443	−0.449722	+	0.893169i	$0.648477\pi$
$662$	16.2918	0.633199
$663$	0	0
$664$	−15.5279	−0.602598
$665$	0	0
$666$	0	0
$667$	−24.1803	−0.936266
$668$	39.7984	1.53985
$669$	0	0
$670$	0	0
$671$	31.7771	1.22674
$672$	0	0
$673$	34.2492	1.32021	0.660105	−	0.751173i	$-0.270512\pi$
$673$	34.2492	1.32021	0.660105	+	0.751173i	$0.270512\pi$
$674$	−5.23607	−0.201686
$675$	0	0
$676$	−46.7426	−1.79779
$677$	5.83282	0.224173	0.112087	−	0.993698i	$-0.464247\pi$
$677$	5.83282	0.224173	0.112087	+	0.993698i	$0.464247\pi$
$678$	0	0
$679$	3.81966	0.146585
$680$	0	0
$681$	0	0
$682$	25.8885	0.991324
$683$	8.34752	0.319409	0.159705	−	0.987165i	$-0.448946\pi$
$683$	8.34752	0.319409	0.159705	+	0.987165i	$0.448946\pi$
$684$	0	0
$685$	0	0
$686$	12.4377	0.474873
$687$	0	0
$688$	−7.41641	−0.282748
$689$	32.3607	1.23284
$690$	0	0
$691$	−7.05573	−0.268413	−0.134206	−	0.990953i	$-0.542848\pi$
$691$	−7.05573	−0.268413	−0.134206	+	0.990953i	$0.542848\pi$
$692$	21.7984	0.828650
$693$	0	0
$694$	1.05573	0.0400749
$695$	0	0
$696$	0	0
$697$	6.18034	0.234097
$698$	11.0213	0.417162
$699$	0	0
$700$	0	0
$701$	−17.3475	−0.655207	−0.327603	−	0.944815i	$-0.606241\pi$
$701$	−17.3475	−0.655207	−0.327603	+	0.944815i	$0.606241\pi$
$702$	0	0
$703$	2.76393	0.104244
$704$	−0.944272	−0.0355886
$705$	0	0
$706$	16.3607	0.615742
$707$	31.7082	1.19251
$708$	0	0
$709$	−40.4164	−1.51787	−0.758935	−	0.651166i	$-0.774280\pi$
$709$	−40.4164	−1.51787	−0.758935	+	0.651166i	$0.774280\pi$
$710$	0	0
$711$	0	0
$712$	22.2361	0.833332
$713$	−33.8885	−1.26914
$714$	0	0
$715$	0	0
$716$	35.7984	1.33785
$717$	0	0
$718$	−15.5967	−0.582065
$719$	37.5279	1.39955	0.699777	−	0.714362i	$-0.253283\pi$
$719$	37.5279	1.39955	0.699777	+	0.714362i	$0.253283\pi$
$720$	0	0
$721$	−37.8885	−1.41104
$722$	−0.618034	−0.0230008
$723$	0	0
$724$	−16.9443	−0.629729
$725$	0	0
$726$	0	0
$727$	15.7639	0.584652	0.292326	−	0.956319i	$-0.405571\pi$
$727$	15.7639	0.584652	0.292326	+	0.956319i	$0.405571\pi$
$728$	−32.3607	−1.19937
$729$	0	0
$730$	0	0
$731$	−4.94427	−0.182871
$732$	0	0
$733$	12.4164	0.458610	0.229305	−	0.973355i	$-0.426355\pi$
$733$	12.4164	0.458610	0.229305	+	0.973355i	$0.426355\pi$
$734$	20.5836	0.759754
$735$	0	0
$736$	−18.1803	−0.670136
$737$	43.7771	1.61255
$738$	0	0
$739$	−4.59675	−0.169094	−0.0845470	−	0.996419i	$-0.526944\pi$
$739$	−4.59675	−0.169094	−0.0845470	+	0.996419i	$0.526944\pi$
$740$	0	0
$741$	0	0
$742$	6.90983	0.253668
$743$	42.1246	1.54540	0.772701	−	0.634770i	$-0.218905\pi$
$743$	42.1246	1.54540	0.772701	+	0.634770i	$0.218905\pi$
$744$	0	0
$745$	0	0
$746$	7.52786	0.275615
$747$	0	0
$748$	−8.00000	−0.292509
$749$	−36.3050	−1.32655
$750$	0	0
$751$	−20.4721	−0.747039	−0.373519	−	0.927622i	$-0.621849\pi$
$751$	−20.4721	−0.747039	−0.373519	+	0.927622i	$0.621849\pi$
$752$	0.875388	0.0319221
$753$	0	0
$754$	−29.8885	−1.08848
$755$	0	0
$756$	0	0
$757$	−23.0000	−0.835949	−0.417975	−	0.908459i	$-0.637260\pi$
$757$	−23.0000	−0.835949	−0.417975	+	0.908459i	$0.637260\pi$
$758$	−4.18034	−0.151837
$759$	0	0
$760$	0	0
$761$	25.1246	0.910766	0.455383	−	0.890296i	$-0.349502\pi$
$761$	25.1246	0.910766	0.455383	+	0.890296i	$0.349502\pi$
$762$	0	0
$763$	−17.2361	−0.623988
$764$	12.0000	0.434145
$765$	0	0
$766$	−9.96556	−0.360070
$767$	−95.1935	−3.43724
$768$	0	0
$769$	−5.58359	−0.201349	−0.100675	−	0.994919i	$-0.532100\pi$
$769$	−5.58359	−0.201349	−0.100675	+	0.994919i	$0.532100\pi$
$770$	0	0
$771$	0	0
$772$	−5.23607	−0.188450
$773$	−3.94427	−0.141866	−0.0709328	−	0.997481i	$-0.522598\pi$
$773$	−3.94427	−0.141866	−0.0709328	+	0.997481i	$0.522598\pi$
$774$	0	0
$775$	0	0
$776$	3.81966	0.137118
$777$	0	0
$778$	4.94427	0.177261
$779$	−5.00000	−0.179144
$780$	0	0
$781$	−36.7214	−1.31399
$782$	−2.47214	−0.0884034
$783$	0	0
$784$	−3.70820	−0.132436
$785$	0	0
$786$	0	0
$787$	19.3475	0.689665	0.344832	−	0.938664i	$-0.387936\pi$
$787$	19.3475	0.689665	0.344832	+	0.938664i	$0.387936\pi$
$788$	36.6525	1.30569
$789$	0	0
$790$	0	0
$791$	−27.8885	−0.991602
$792$	0	0
$793$	−51.4164	−1.82585
$794$	13.8197	0.490441
$795$	0	0
$796$	−2.85410	−0.101161
$797$	−31.3607	−1.11085	−0.555426	−	0.831566i	$-0.687445\pi$
$797$	−31.3607	−1.11085	−0.555426	+	0.831566i	$0.687445\pi$
$798$	0	0
$799$	0.583592	0.0206460
$800$	0	0
$801$	0	0
$802$	−12.8754	−0.454646
$803$	−13.8885	−0.490116
$804$	0	0
$805$	0	0
$806$	−41.8885	−1.47546
$807$	0	0
$808$	31.7082	1.11549
$809$	16.8328	0.591810	0.295905	−	0.955217i	$-0.404379\pi$
$809$	16.8328	0.591810	0.295905	+	0.955217i	$0.404379\pi$
$810$	0	0
$811$	46.7214	1.64061	0.820304	−	0.571927i	$-0.193804\pi$
$811$	46.7214	1.64061	0.820304	+	0.571927i	$0.193804\pi$
$812$	27.0344	0.948723
$813$	0	0
$814$	6.83282	0.239490
$815$	0	0
$816$	0	0
$817$	4.00000	0.139942
$818$	−5.41641	−0.189380
$819$	0	0
$820$	0	0
$821$	−5.12461	−0.178850	−0.0894251	−	0.995994i	$-0.528503\pi$
$821$	−5.12461	−0.178850	−0.0894251	+	0.995994i	$0.528503\pi$
$822$	0	0
$823$	33.1803	1.15659	0.578297	−	0.815826i	$-0.303718\pi$
$823$	33.1803	1.15659	0.578297	+	0.815826i	$0.303718\pi$
$824$	−37.8885	−1.31991
$825$	0	0
$826$	−20.3262	−0.707240
$827$	−36.9443	−1.28468	−0.642339	−	0.766421i	$-0.722036\pi$
$827$	−36.9443	−1.28468	−0.642339	+	0.766421i	$0.722036\pi$
$828$	0	0
$829$	55.6656	1.93335	0.966674	−	0.256012i	$-0.0824086\pi$
$829$	55.6656	1.93335	0.966674	+	0.256012i	$0.0824086\pi$
$830$	0	0
$831$	0	0
$832$	1.52786	0.0529692
$833$	−2.47214	−0.0856544
$834$	0	0
$835$	0	0
$836$	6.47214	0.223844
$837$	0	0
$838$	24.2918	0.839146
$839$	38.2361	1.32006	0.660028	−	0.751241i	$-0.270544\pi$
$839$	38.2361	1.32006	0.660028	+	0.751241i	$0.270544\pi$
$840$	0	0
$841$	26.8328	0.925270
$842$	−7.12461	−0.245530
$843$	0	0
$844$	−9.41641	−0.324126
$845$	0	0
$846$	0	0
$847$	11.1803	0.384161
$848$	−9.27051	−0.318351
$849$	0	0
$850$	0	0
$851$	−8.94427	−0.306606
$852$	0	0
$853$	36.8885	1.26304	0.631520	−	0.775360i	$-0.282431\pi$
$853$	36.8885	1.26304	0.631520	+	0.775360i	$0.282431\pi$
$854$	−10.9787	−0.375684
$855$	0	0
$856$	−36.3050	−1.24088
$857$	32.7771	1.11964	0.559822	−	0.828613i	$-0.310870\pi$
$857$	32.7771	1.11964	0.559822	+	0.828613i	$0.310870\pi$
$858$	0	0
$859$	−21.2918	−0.726467	−0.363233	−	0.931698i	$-0.618327\pi$
$859$	−21.2918	−0.726467	−0.363233	+	0.931698i	$0.618327\pi$
$860$	0	0
$861$	0	0
$862$	−3.20163	−0.109048
$863$	34.8197	1.18528	0.592638	−	0.805469i	$-0.298087\pi$
$863$	34.8197	1.18528	0.592638	+	0.805469i	$0.298087\pi$
$864$	0	0
$865$	0	0
$866$	−12.0689	−0.410117
$867$	0	0
$868$	37.8885	1.28602
$869$	−9.16718	−0.310975
$870$	0	0
$871$	−70.8328	−2.40008
$872$	−17.2361	−0.583687
$873$	0	0
$874$	2.00000	0.0676510
$875$	0	0
$876$	0	0
$877$	44.9443	1.51766	0.758830	−	0.651289i	$-0.225771\pi$
$877$	44.9443	1.51766	0.758830	+	0.651289i	$0.225771\pi$
$878$	10.3607	0.349656
$879$	0	0
$880$	0	0
$881$	0.944272	0.0318133	0.0159067	−	0.999873i	$-0.494937\pi$
$881$	0.944272	0.0318133	0.0159067	+	0.999873i	$0.494937\pi$
$882$	0	0
$883$	−45.6525	−1.53633	−0.768164	−	0.640253i	$-0.778829\pi$
$883$	−45.6525	−1.53633	−0.768164	+	0.640253i	$0.778829\pi$
$884$	12.9443	0.435363
$885$	0	0
$886$	−8.29180	−0.278568
$887$	−18.8328	−0.632344	−0.316172	−	0.948702i	$-0.602398\pi$
$887$	−18.8328	−0.632344	−0.316172	+	0.948702i	$0.602398\pi$
$888$	0	0
$889$	−24.0689	−0.807244
$890$	0	0
$891$	0	0
$892$	14.9443	0.500371
$893$	−0.472136	−0.0157994
$894$	0	0
$895$	0	0
$896$	25.4508	0.850253
$897$	0	0
$898$	−15.7426	−0.525339
$899$	78.2492	2.60976
$900$	0	0
$901$	−6.18034	−0.205897
$902$	−12.3607	−0.411566
$903$	0	0
$904$	−27.8885	−0.927559
$905$	0	0
$906$	0	0
$907$	29.8197	0.990146	0.495073	−	0.868852i	$-0.335141\pi$
$907$	29.8197	0.990146	0.495073	+	0.868852i	$0.335141\pi$
$908$	−45.5066	−1.51019
$909$	0	0
$910$	0	0
$911$	−11.7639	−0.389756	−0.194878	−	0.980827i	$-0.562431\pi$
$911$	−11.7639	−0.389756	−0.194878	+	0.980827i	$0.562431\pi$
$912$	0	0
$913$	−27.7771	−0.919287
$914$	5.49342	0.181706
$915$	0	0
$916$	7.23607	0.239086
$917$	15.5279	0.512775
$918$	0	0
$919$	41.2918	1.36209	0.681045	−	0.732241i	$-0.261526\pi$
$919$	41.2918	1.36209	0.681045	+	0.732241i	$0.261526\pi$
$920$	0	0
$921$	0	0
$922$	−1.12461	−0.0370371
$923$	59.4164	1.95571
$924$	0	0
$925$	0	0
$926$	5.30495	0.174332
$927$	0	0
$928$	41.9787	1.37802
$929$	−47.5967	−1.56160	−0.780799	−	0.624782i	$-0.785188\pi$
$929$	−47.5967	−1.56160	−0.780799	+	0.624782i	$0.785188\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	−4.76393	−0.156048
$933$	0	0
$934$	16.1803	0.529437
$935$	0	0
$936$	0	0
$937$	−39.9443	−1.30492	−0.652461	−	0.757822i	$-0.726263\pi$
$937$	−39.9443	−1.30492	−0.652461	+	0.757822i	$0.726263\pi$
$938$	−15.1246	−0.493836
$939$	0	0
$940$	0	0
$941$	−36.4721	−1.18896	−0.594479	−	0.804111i	$-0.702642\pi$
$941$	−36.4721	−1.18896	−0.594479	+	0.804111i	$0.702642\pi$
$942$	0	0
$943$	16.1803	0.526904
$944$	27.2705	0.887579
$945$	0	0
$946$	9.88854	0.321504
$947$	−36.9443	−1.20053	−0.600264	−	0.799802i	$-0.704938\pi$
$947$	−36.9443	−1.20053	−0.600264	+	0.799802i	$0.704938\pi$
$948$	0	0
$949$	22.4721	0.729476
$950$	0	0
$951$	0	0
$952$	6.18034	0.200306
$953$	53.2492	1.72491	0.862456	−	0.506132i	$-0.168925\pi$
$953$	53.2492	1.72491	0.862456	+	0.506132i	$0.168925\pi$
$954$	0	0
$955$	0	0
$956$	0.472136	0.0152700
$957$	0	0
$958$	−23.9574	−0.774029
$959$	9.59675	0.309895
$960$	0	0
$961$	78.6656	2.53760
$962$	−11.0557	−0.356451
$963$	0	0
$964$	26.1803	0.843212
$965$	0	0
$966$	0	0
$967$	−22.5967	−0.726662	−0.363331	−	0.931660i	$-0.618361\pi$
$967$	−22.5967	−0.726662	−0.363331	+	0.931660i	$0.618361\pi$
$968$	11.1803	0.359350
$969$	0	0
$970$	0	0
$971$	−21.0689	−0.676133	−0.338066	−	0.941122i	$-0.609773\pi$
$971$	−21.0689	−0.676133	−0.338066	+	0.941122i	$0.609773\pi$
$972$	0	0
$973$	−35.2492	−1.13004
$974$	14.8328	0.475274
$975$	0	0
$976$	14.7295	0.471479
$977$	5.05573	0.161747	0.0808735	−	0.996724i	$-0.474229\pi$
$977$	5.05573	0.161747	0.0808735	+	0.996724i	$0.474229\pi$
$978$	0	0
$979$	39.7771	1.27128
$980$	0	0
$981$	0	0
$982$	−5.12461	−0.163533
$983$	57.3050	1.82774	0.913872	−	0.406002i	$-0.133078\pi$
$983$	57.3050	1.82774	0.913872	+	0.406002i	$0.133078\pi$
$984$	0	0
$985$	0	0
$986$	5.70820	0.181786
$987$	0	0
$988$	−10.4721	−0.333163
$989$	−12.9443	−0.411604
$990$	0	0
$991$	−37.0557	−1.17711	−0.588557	−	0.808456i	$-0.700304\pi$
$991$	−37.0557	−1.17711	−0.588557	+	0.808456i	$0.700304\pi$
$992$	58.8328	1.86794
$993$	0	0
$994$	12.6869	0.402405
$995$	0	0
$996$	0	0
$997$	−12.4721	−0.394997	−0.197498	−	0.980303i	$-0.563282\pi$
$997$	−12.4721	−0.394997	−0.197498	+	0.980303i	$0.563282\pi$
$998$	19.2705	0.609997
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4275.2.a.v.1.1		2
3.2	odd	2		1425.2.a.n.1.2	✓	2
5.4	even	2		4275.2.a.s.1.2		2
15.2	even	4		1425.2.c.m.799.3		4
15.8	even	4		1425.2.c.m.799.2		4
15.14	odd	2		1425.2.a.q.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1425.2.a.n.1.2	✓	2	3.2	odd	2
1425.2.a.q.1.1	yes	2	15.14	odd	2
1425.2.c.m.799.2		4	15.8	even	4
1425.2.c.m.799.3		4	15.2	even	4
4275.2.a.s.1.2		2	5.4	even	2
4275.2.a.v.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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4275 = 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4275.a (trivial)

\(f(q)\)	\(=\)	\(q-0.618034 q^{2} -1.61803 q^{4} +2.23607 q^{7} +2.23607 q^{8} +4.00000 q^{11} -6.47214 q^{13} -1.38197 q^{14} +1.85410 q^{16} +1.23607 q^{17} -1.00000 q^{19} -2.47214 q^{22} +3.23607 q^{23} +4.00000 q^{26} -3.61803 q^{28} -7.47214 q^{29} -10.4721 q^{31} -5.61803 q^{32} -0.763932 q^{34} -2.76393 q^{37} +0.618034 q^{38} +5.00000 q^{41} -4.00000 q^{43} -6.47214 q^{44} -2.00000 q^{46} +0.472136 q^{47} -2.00000 q^{49} +10.4721 q^{52} -5.00000 q^{53} +5.00000 q^{56} +4.61803 q^{58} +14.7082 q^{59} +7.94427 q^{61} +6.47214 q^{62} -0.236068 q^{64} +10.9443 q^{67} -2.00000 q^{68} -9.18034 q^{71} -3.47214 q^{73} +1.70820 q^{74} +1.61803 q^{76} +8.94427 q^{77} -2.29180 q^{79} -3.09017 q^{82} -6.94427 q^{83} +2.47214 q^{86} +8.94427 q^{88} +9.94427 q^{89} -14.4721 q^{91} -5.23607 q^{92} -0.291796 q^{94} +1.70820 q^{97} +1.23607 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{4} + 8 q^{11} - 4 q^{13} - 5 q^{14} - 3 q^{16} - 2 q^{17} - 2 q^{19} + 4 q^{22} + 2 q^{23} + 8 q^{26} - 5 q^{28} - 6 q^{29} - 12 q^{31} - 9 q^{32} - 6 q^{34} - 10 q^{37} - q^{38} + 10 q^{41}+ \cdots - 2 q^{98}+O(q^{100}) \) 2 * q + q^2 - q^4 + 8 * q^11 - 4 * q^13 - 5 * q^14 - 3 * q^16 - 2 * q^17 - 2 * q^19 + 4 * q^22 + 2 * q^23 + 8 * q^26 - 5 * q^28 - 6 * q^29 - 12 * q^31 - 9 * q^32 - 6 * q^34 - 10 * q^37 - q^38 + 10 * q^41 - 8 * q^43 - 4 * q^44 - 4 * q^46 - 8 * q^47 - 4 * q^49 + 12 * q^52 - 10 * q^53 + 10 * q^56 + 7 * q^58 + 16 * q^59 - 2 * q^61 + 4 * q^62 + 4 * q^64 + 4 * q^67 - 4 * q^68 + 4 * q^71 + 2 * q^73 - 10 * q^74 + q^76 - 18 * q^79 + 5 * q^82 + 4 * q^83 - 4 * q^86 + 2 * q^89 - 20 * q^91 - 6 * q^92 - 14 * q^94 - 10 * q^97 - 2 * q^98

Introduction

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