LMFDB - Embedded newform 6025.2.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6025,2,Mod(1,6025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6025, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6025 = 5^{2} \cdot 241 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6025.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:	$48.1098672178$
Analytic rank:	$1$
Dimension:	$46$
Twist minimal:	no (minimal twist has level 1205)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.68673 q^{2} +2.58072 q^{3} +5.21850 q^{4} -6.93370 q^{6} +1.24945 q^{7} -8.64724 q^{8} +3.66013 q^{9} +O(q^{10})$	\(q-2.68673 q^{2} +2.58072 q^{3} +5.21850 q^{4} -6.93370 q^{6} +1.24945 q^{7} -8.64724 q^{8} +3.66013 q^{9} -5.71181 q^{11} +13.4675 q^{12} -3.28322 q^{13} -3.35693 q^{14} +12.7958 q^{16} +5.29090 q^{17} -9.83378 q^{18} +6.14996 q^{19} +3.22448 q^{21} +15.3461 q^{22} +2.22338 q^{23} -22.3161 q^{24} +8.82111 q^{26} +1.70362 q^{27} +6.52026 q^{28} -5.09304 q^{29} -4.50984 q^{31} -17.0842 q^{32} -14.7406 q^{33} -14.2152 q^{34} +19.1004 q^{36} -4.88794 q^{37} -16.5233 q^{38} -8.47308 q^{39} -4.77605 q^{41} -8.66331 q^{42} +2.72100 q^{43} -29.8071 q^{44} -5.97362 q^{46} -2.52165 q^{47} +33.0223 q^{48} -5.43887 q^{49} +13.6544 q^{51} -17.1335 q^{52} -7.67961 q^{53} -4.57716 q^{54} -10.8043 q^{56} +15.8714 q^{57} +13.6836 q^{58} -8.39323 q^{59} -12.2037 q^{61} +12.1167 q^{62} +4.57315 q^{63} +20.3092 q^{64} +39.6039 q^{66} +15.2352 q^{67} +27.6106 q^{68} +5.73793 q^{69} -6.27039 q^{71} -31.6500 q^{72} -5.27275 q^{73} +13.1325 q^{74} +32.0936 q^{76} -7.13662 q^{77} +22.7648 q^{78} +9.38460 q^{79} -6.58383 q^{81} +12.8319 q^{82} -2.13252 q^{83} +16.8270 q^{84} -7.31059 q^{86} -13.1437 q^{87} +49.3913 q^{88} +8.11125 q^{89} -4.10222 q^{91} +11.6027 q^{92} -11.6386 q^{93} +6.77498 q^{94} -44.0897 q^{96} -10.0464 q^{97} +14.6128 q^{98} -20.9060 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 46 q + 34 q^{4} - 4 q^{6} + 34 q^{9}+O(q^{10}) $ 46 * q + 34 * q^4 - 4 * q^6 + 34 * q^9	$ 46 q + 34 q^{4} - 4 q^{6} + 34 q^{9} - 64 q^{11} - 66 q^{14} + 22 q^{16} - 14 q^{21} - 50 q^{24} - 60 q^{26} - 36 q^{29} - 36 q^{31} + 12 q^{34} - 34 q^{36} - 88 q^{39} - 76 q^{41} - 100 q^{44} - 12 q^{46} + 22 q^{49} - 112 q^{51} - 26 q^{54} - 120 q^{56} - 84 q^{59} - 78 q^{61} + 28 q^{64} - 2 q^{66} - 24 q^{69} - 172 q^{71} - 16 q^{74} - 18 q^{76} - 54 q^{79} - 42 q^{81} + 44 q^{84} - 80 q^{86} - 86 q^{89} - 88 q^{91} + 4 q^{94} - 122 q^{96} - 148 q^{99}+O(q^{100}) $ 46 * q + 34 * q^4 - 4 * q^6 + 34 * q^9 - 64 * q^11 - 66 * q^14 + 22 * q^16 - 14 * q^21 - 50 * q^24 - 60 * q^26 - 36 * q^29 - 36 * q^31 + 12 * q^34 - 34 * q^36 - 88 * q^39 - 76 * q^41 - 100 * q^44 - 12 * q^46 + 22 * q^49 - 112 * q^51 - 26 * q^54 - 120 * q^56 - 84 * q^59 - 78 * q^61 + 28 * q^64 - 2 * q^66 - 24 * q^69 - 172 * q^71 - 16 * q^74 - 18 * q^76 - 54 * q^79 - 42 * q^81 + 44 * q^84 - 80 * q^86 - 86 * q^89 - 88 * q^91 + 4 * q^94 - 122 * q^96 - 148 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	−2.68673	−1.89980	−0.949901	−	0.312550i	$-0.898817\pi$
$2$	−2.68673	−1.89980	−0.949901	+	0.312550i	$0.898817\pi$
$3$	2.58072	1.48998	0.744991	−	0.667075i	$-0.232454\pi$
$3$	2.58072	1.48998	0.744991	+	0.667075i	$0.232454\pi$
$4$	5.21850	2.60925
$5$	0	0
$5$	0	0
$6$	−6.93370	−2.83067
$7$	1.24945	0.472248	0.236124	−	0.971723i	$-0.424123\pi$
$7$	1.24945	0.472248	0.236124	+	0.971723i	$0.424123\pi$
$8$	−8.64724	−3.05726
$9$	3.66013	1.22004
$10$	0	0
$11$	−5.71181	−1.72217	−0.861087	−	0.508457i	$-0.830216\pi$
$11$	−5.71181	−1.72217	−0.861087	+	0.508457i	$0.830216\pi$
$12$	13.4675	3.88774
$13$	−3.28322	−0.910601	−0.455300	−	0.890338i	$-0.650468\pi$
$13$	−3.28322	−0.910601	−0.455300	+	0.890338i	$0.650468\pi$
$14$	−3.35693	−0.897178
$15$	0	0
$16$	12.7958	3.19894
$17$	5.29090	1.28323	0.641616	−	0.767026i	$-0.278264\pi$
$17$	5.29090	1.28323	0.641616	+	0.767026i	$0.278264\pi$
$18$	−9.83378	−2.31784
$19$	6.14996	1.41090	0.705449	−	0.708760i	$-0.250745\pi$
$19$	6.14996	1.41090	0.705449	+	0.708760i	$0.250745\pi$
$20$	0	0
$21$	3.22448	0.703640
$22$	15.3461	3.27179
$23$	2.22338	0.463607	0.231804	−	0.972763i	$-0.425537\pi$
$23$	2.22338	0.463607	0.231804	+	0.972763i	$0.425537\pi$
$24$	−22.3161	−4.55526
$25$	0	0
$26$	8.82111	1.72996
$27$	1.70362	0.327862
$28$	6.52026	1.23221
$29$	−5.09304	−0.945753	−0.472877	−	0.881129i	$-0.656784\pi$
$29$	−5.09304	−0.945753	−0.472877	+	0.881129i	$0.656784\pi$
$30$	0	0
$31$	−4.50984	−0.809991	−0.404996	−	0.914319i	$-0.632727\pi$
$31$	−4.50984	−0.809991	−0.404996	+	0.914319i	$0.632727\pi$
$32$	−17.0842	−3.02010
$33$	−14.7406	−2.56601
$34$	−14.2152	−2.43789
$35$	0	0
$36$	19.1004	3.18340
$37$	−4.88794	−0.803572	−0.401786	−	0.915734i	$-0.631610\pi$
$37$	−4.88794	−0.803572	−0.401786	+	0.915734i	$0.631610\pi$
$38$	−16.5233	−2.68043
$39$	−8.47308	−1.35678
$40$	0	0
$41$	−4.77605	−0.745894	−0.372947	−	0.927853i	$-0.621653\pi$
$41$	−4.77605	−0.745894	−0.372947	+	0.927853i	$0.621653\pi$
$42$	−8.66331	−1.33678
$43$	2.72100	0.414949	0.207474	−	0.978240i	$-0.433476\pi$
$43$	2.72100	0.414949	0.207474	+	0.978240i	$0.433476\pi$
$44$	−29.8071	−4.49359
$45$	0	0
$46$	−5.97362	−0.880762
$47$	−2.52165	−0.367820	−0.183910	−	0.982943i	$-0.558876\pi$
$47$	−2.52165	−0.367820	−0.183910	+	0.982943i	$0.558876\pi$
$48$	33.0223	4.76636
$49$	−5.43887	−0.776982
$50$	0	0
$51$	13.6544	1.91199
$52$	−17.1335	−2.37599
$53$	−7.67961	−1.05488	−0.527438	−	0.849594i	$-0.676847\pi$
$53$	−7.67961	−1.05488	−0.527438	+	0.849594i	$0.676847\pi$
$54$	−4.57716	−0.622873
$55$	0	0
$56$	−10.8043	−1.44378
$57$	15.8714	2.10221
$58$	13.6836	1.79675
$59$	−8.39323	−1.09271	−0.546353	−	0.837555i	$-0.683984\pi$
$59$	−8.39323	−1.09271	−0.546353	+	0.837555i	$0.683984\pi$
$60$	0	0
$61$	−12.2037	−1.56252	−0.781260	−	0.624205i	$-0.785423\pi$
$61$	−12.2037	−1.56252	−0.781260	+	0.624205i	$0.785423\pi$
$62$	12.1167	1.53882
$63$	4.57315	0.576163
$64$	20.3092	2.53865
$65$	0	0
$66$	39.6039	4.87491
$67$	15.2352	1.86128	0.930641	−	0.365934i	$-0.119250\pi$
$67$	15.2352	1.86128	0.930641	+	0.365934i	$0.119250\pi$
$68$	27.6106	3.34828
$69$	5.73793	0.690766
$70$	0	0
$71$	−6.27039	−0.744159	−0.372079	−	0.928201i	$-0.621355\pi$
$71$	−6.27039	−0.744159	−0.372079	+	0.928201i	$0.621355\pi$
$72$	−31.6500	−3.72999
$73$	−5.27275	−0.617129	−0.308564	−	0.951203i	$-0.599849\pi$
$73$	−5.27275	−0.617129	−0.308564	+	0.951203i	$0.599849\pi$
$74$	13.1325	1.52663
$75$	0	0
$76$	32.0936	3.68139
$77$	−7.13662	−0.813293
$78$	22.7648	2.57761
$79$	9.38460	1.05585	0.527925	−	0.849291i	$-0.322970\pi$
$79$	9.38460	1.05585	0.527925	+	0.849291i	$0.322970\pi$
$80$	0	0
$81$	−6.58383	−0.731536
$82$	12.8319	1.41705
$83$	−2.13252	−0.234074	−0.117037	−	0.993128i	$-0.537340\pi$
$83$	−2.13252	−0.234074	−0.117037	+	0.993128i	$0.537340\pi$
$84$	16.8270	1.83597
$85$	0	0
$86$	−7.31059	−0.788321
$87$	−13.1437	−1.40915
$88$	49.3913	5.26514
$89$	8.11125	0.859791	0.429895	−	0.902879i	$-0.358550\pi$
$89$	8.11125	0.859791	0.429895	+	0.902879i	$0.358550\pi$
$90$	0	0
$91$	−4.10222	−0.430029
$92$	11.6027	1.20967
$93$	−11.6386	−1.20687
$94$	6.77498	0.698786
$95$	0	0
$96$	−44.0897	−4.49989
$97$	−10.0464	−1.02006	−0.510031	−	0.860156i	$-0.670366\pi$
$97$	−10.0464	−1.02006	−0.510031	+	0.860156i	$0.670366\pi$
$98$	14.6128	1.47611
$99$	−20.9060	−2.10113
$100$	0	0
$101$	−16.4859	−1.64041	−0.820207	−	0.572067i	$-0.806142\pi$
$101$	−16.4859	−1.64041	−0.820207	+	0.572067i	$0.806142\pi$
$102$	−36.6855	−3.63241
$103$	8.12695	0.800772	0.400386	−	0.916347i	$-0.368876\pi$
$103$	8.12695	0.800772	0.400386	+	0.916347i	$0.368876\pi$
$104$	28.3908	2.78394
$105$	0	0
$106$	20.6330	2.00406
$107$	15.5719	1.50539	0.752697	−	0.658368i	$-0.228753\pi$
$107$	15.5719	1.50539	0.752697	+	0.658368i	$0.228753\pi$
$108$	8.89035	0.855474
$109$	−17.5584	−1.68179	−0.840893	−	0.541202i	$-0.817970\pi$
$109$	−17.5584	−1.68179	−0.840893	+	0.541202i	$0.817970\pi$
$110$	0	0
$111$	−12.6144	−1.19731
$112$	15.9877	1.51069
$113$	−4.82918	−0.454291	−0.227146	−	0.973861i	$-0.572939\pi$
$113$	−4.82918	−0.454291	−0.227146	+	0.973861i	$0.572939\pi$
$114$	−42.6420	−3.99379
$115$	0	0
$116$	−26.5780	−2.46771
$117$	−12.0170	−1.11097
$118$	22.5503	2.07592
$119$	6.61072	0.606004
$120$	0	0
$121$	21.6247	1.96588
$122$	32.7880	2.96848
$123$	−12.3257	−1.11137
$124$	−23.5346	−2.11347
$125$	0	0
$126$	−12.2868	−1.09460
$127$	12.2103	1.08349	0.541746	−	0.840542i	$-0.317763\pi$
$127$	12.2103	1.08349	0.541746	+	0.840542i	$0.317763\pi$
$128$	−20.3967	−1.80283
$129$	7.02215	0.618266
$130$	0	0
$131$	−10.1109	−0.883392	−0.441696	−	0.897165i	$-0.645623\pi$
$131$	−10.1109	−0.883392	−0.441696	+	0.897165i	$0.645623\pi$
$132$	−76.9238	−6.69536
$133$	7.68407	0.666294
$134$	−40.9330	−3.53607
$135$	0	0
$136$	−45.7517	−3.92318
$137$	−13.2155	−1.12908	−0.564539	−	0.825406i	$-0.690946\pi$
$137$	−13.2155	−1.12908	−0.564539	+	0.825406i	$0.690946\pi$
$138$	−15.4163	−1.31232
$139$	15.9583	1.35357	0.676784	−	0.736182i	$-0.263373\pi$
$139$	15.9583	1.35357	0.676784	+	0.736182i	$0.263373\pi$
$140$	0	0
$141$	−6.50768	−0.548045
$142$	16.8468	1.41375
$143$	18.7531	1.56821
$144$	46.8342	3.90285
$145$	0	0
$146$	14.1664	1.17242
$147$	−14.0362	−1.15769
$148$	−25.5077	−2.09672
$149$	4.94740	0.405307	0.202653	−	0.979251i	$-0.435044\pi$
$149$	4.94740	0.405307	0.202653	+	0.979251i	$0.435044\pi$
$150$	0	0
$151$	−9.19121	−0.747970	−0.373985	−	0.927435i	$-0.622009\pi$
$151$	−9.19121	−0.747970	−0.373985	+	0.927435i	$0.622009\pi$
$152$	−53.1802	−4.31348
$153$	19.3654	1.56560
$154$	19.1741	1.54510
$155$	0	0
$156$	−44.2168	−3.54017
$157$	7.17472	0.572605	0.286302	−	0.958139i	$-0.407574\pi$
$157$	7.17472	0.572605	0.286302	+	0.958139i	$0.407574\pi$
$158$	−25.2138	−2.00591
$159$	−19.8190	−1.57174
$160$	0	0
$161$	2.77801	0.218937
$162$	17.6889	1.38977
$163$	14.9501	1.17098	0.585490	−	0.810680i	$-0.300902\pi$
$163$	14.9501	1.17098	0.585490	+	0.810680i	$0.300902\pi$
$164$	−24.9238	−1.94622
$165$	0	0
$166$	5.72950	0.444695
$167$	4.27656	0.330930	0.165465	−	0.986216i	$-0.447088\pi$
$167$	4.27656	0.330930	0.165465	+	0.986216i	$0.447088\pi$
$168$	−27.8829	−2.15121
$169$	−2.22048	−0.170807
$170$	0	0
$171$	22.5097	1.72136
$172$	14.1996	1.08271
$173$	−8.86646	−0.674105	−0.337052	−	0.941486i	$-0.609430\pi$
$173$	−8.86646	−0.674105	−0.337052	+	0.941486i	$0.609430\pi$
$174$	35.3136	2.67712
$175$	0	0
$176$	−73.0869	−5.50913
$177$	−21.6606	−1.62811
$178$	−21.7927	−1.63343
$179$	19.6157	1.46615	0.733075	−	0.680148i	$-0.238084\pi$
$179$	19.6157	1.46615	0.733075	+	0.680148i	$0.238084\pi$
$180$	0	0
$181$	15.9326	1.18426	0.592132	−	0.805841i	$-0.298286\pi$
$181$	15.9326	1.18426	0.592132	+	0.805841i	$0.298286\pi$
$182$	11.0215	0.816971
$183$	−31.4943	−2.32813
$184$	−19.2261	−1.41737
$185$	0	0
$186$	31.2699	2.29282
$187$	−30.2206	−2.20995
$188$	−13.1592	−0.959735
$189$	2.12859	0.154832
$190$	0	0
$191$	−13.9904	−1.01231	−0.506157	−	0.862442i	$-0.668934\pi$
$191$	−13.9904	−1.01231	−0.506157	+	0.862442i	$0.668934\pi$
$192$	52.4124	3.78254
$193$	5.85035	0.421117	0.210559	−	0.977581i	$-0.432472\pi$
$193$	5.85035	0.421117	0.210559	+	0.977581i	$0.432472\pi$
$194$	26.9920	1.93792
$195$	0	0
$196$	−28.3828	−2.02734
$197$	0.760772	0.0542028	0.0271014	−	0.999633i	$-0.491372\pi$
$197$	0.760772	0.0542028	0.0271014	+	0.999633i	$0.491372\pi$
$198$	56.1686	3.99173
$199$	−13.2393	−0.938511	−0.469255	−	0.883063i	$-0.655478\pi$
$199$	−13.2393	−0.938511	−0.469255	+	0.883063i	$0.655478\pi$
$200$	0	0
$201$	39.3180	2.77328
$202$	44.2932	3.11646
$203$	−6.36350	−0.446630
$204$	71.2553	4.98887
$205$	0	0
$206$	−21.8349	−1.52131
$207$	8.13787	0.565621
$208$	−42.0113	−2.91296
$209$	−35.1274	−2.42981
$210$	0	0
$211$	10.5091	0.723480	0.361740	−	0.932279i	$-0.382183\pi$
$211$	10.5091	0.723480	0.361740	+	0.932279i	$0.382183\pi$
$212$	−40.0761	−2.75244
$213$	−16.1821	−1.10878
$214$	−41.8375	−2.85995
$215$	0	0
$216$	−14.7316	−1.00236
$217$	−5.63482	−0.382516
$218$	47.1745	3.19506
$219$	−13.6075	−0.919510
$220$	0	0
$221$	−17.3712	−1.16851
$222$	33.8915	2.27465
$223$	−11.7804	−0.788876	−0.394438	−	0.918923i	$-0.629061\pi$
$223$	−11.7804	−0.788876	−0.394438	+	0.918923i	$0.629061\pi$
$224$	−21.3459	−1.42623
$225$	0	0
$226$	12.9747	0.863063
$227$	−9.33932	−0.619873	−0.309936	−	0.950757i	$-0.600308\pi$
$227$	−9.33932	−0.619873	−0.309936	+	0.950757i	$0.600308\pi$
$228$	82.8247	5.48520
$229$	−16.5911	−1.09637	−0.548186	−	0.836357i	$-0.684681\pi$
$229$	−16.5911	−1.09637	−0.548186	+	0.836357i	$0.684681\pi$
$230$	0	0
$231$	−18.4176	−1.21179
$232$	44.0407	2.89141
$233$	2.57802	0.168892	0.0844460	−	0.996428i	$-0.473088\pi$
$233$	2.57802	0.168892	0.0844460	+	0.996428i	$0.473088\pi$
$234$	32.2864	2.11063
$235$	0	0
$236$	−43.8001	−2.85114
$237$	24.2190	1.57320
$238$	−17.7612	−1.15129
$239$	−26.0214	−1.68318	−0.841592	−	0.540113i	$-0.818381\pi$
$239$	−26.0214	−1.68318	−0.841592	+	0.540113i	$0.818381\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	−58.0998	−3.73479
$243$	−22.1019	−1.41784
$244$	−63.6849	−4.07701
$245$	0	0
$246$	33.1157	2.11138
$247$	−20.1917	−1.28477
$248$	38.9977	2.47635
$249$	−5.50344	−0.348767
$250$	0	0
$251$	27.0049	1.70453	0.852267	−	0.523107i	$-0.175227\pi$
$251$	27.0049	1.70453	0.852267	+	0.523107i	$0.175227\pi$
$252$	23.8650	1.50335
$253$	−12.6995	−0.798413
$254$	−32.8059	−2.05842
$255$	0	0
$256$	14.1821	0.886381
$257$	−27.5253	−1.71698	−0.858490	−	0.512830i	$-0.828597\pi$
$257$	−27.5253	−1.71698	−0.858490	+	0.512830i	$0.828597\pi$
$258$	−18.8666	−1.17458
$259$	−6.10723	−0.379485
$260$	0	0
$261$	−18.6412	−1.15386
$262$	27.1652	1.67827
$263$	7.57437	0.467056	0.233528	−	0.972350i	$-0.424973\pi$
$263$	7.57437	0.467056	0.233528	+	0.972350i	$0.424973\pi$
$264$	127.465	7.84495
$265$	0	0
$266$	−20.6450	−1.26583
$267$	20.9329	1.28107
$268$	79.5052	4.85655
$269$	−8.18678	−0.499157	−0.249578	−	0.968355i	$-0.580292\pi$
$269$	−8.18678	−0.499157	−0.249578	+	0.968355i	$0.580292\pi$
$270$	0	0
$271$	−3.63408	−0.220755	−0.110377	−	0.993890i	$-0.535206\pi$
$271$	−3.63408	−0.220755	−0.110377	+	0.993890i	$0.535206\pi$
$272$	67.7012	4.10499
$273$	−10.5867	−0.640735
$274$	35.5065	2.14503
$275$	0	0
$276$	29.9434	1.80238
$277$	−2.42885	−0.145935	−0.0729677	−	0.997334i	$-0.523247\pi$
$277$	−2.42885	−0.145935	−0.0729677	+	0.997334i	$0.523247\pi$
$278$	−42.8757	−2.57151
$279$	−16.5066	−0.988225
$280$	0	0
$281$	−16.3717	−0.976655	−0.488328	−	0.872660i	$-0.662393\pi$
$281$	−16.3717	−0.976655	−0.488328	+	0.872660i	$0.662393\pi$
$282$	17.4844	1.04118
$283$	6.08266	0.361576	0.180788	−	0.983522i	$-0.442135\pi$
$283$	6.08266	0.361576	0.180788	+	0.983522i	$0.442135\pi$
$284$	−32.7221	−1.94170
$285$	0	0
$286$	−50.3845	−2.97930
$287$	−5.96744	−0.352247
$288$	−62.5306	−3.68465
$289$	10.9937	0.646686
$290$	0	0
$291$	−25.9271	−1.51987
$292$	−27.5159	−1.61024
$293$	10.4470	0.610322	0.305161	−	0.952301i	$-0.401290\pi$
$293$	10.4470	0.610322	0.305161	+	0.952301i	$0.401290\pi$
$294$	37.7115	2.19938
$295$	0	0
$296$	42.2671	2.45673
$297$	−9.73075	−0.564635
$298$	−13.2923	−0.770003
$299$	−7.29985	−0.422161
$300$	0	0
$301$	3.39976	0.195959
$302$	24.6943	1.42100
$303$	−42.5457	−2.44418
$304$	78.6935	4.51338
$305$	0	0
$306$	−52.0296	−2.97433
$307$	17.7832	1.01494	0.507470	−	0.861669i	$-0.330581\pi$
$307$	17.7832	1.01494	0.507470	+	0.861669i	$0.330581\pi$
$308$	−37.2425	−2.12209
$309$	20.9734	1.19314
$310$	0	0
$311$	−5.30634	−0.300895	−0.150447	−	0.988618i	$-0.548071\pi$
$311$	−5.30634	−0.300895	−0.150447	+	0.988618i	$0.548071\pi$
$312$	73.2687	4.14802
$313$	−25.9853	−1.46878	−0.734389	−	0.678729i	$-0.762531\pi$
$313$	−25.9853	−1.46878	−0.734389	+	0.678729i	$0.762531\pi$
$314$	−19.2765	−1.08784
$315$	0	0
$316$	48.9735	2.75498
$317$	0.511802	0.0287457	0.0143728	−	0.999897i	$-0.495425\pi$
$317$	0.511802	0.0287457	0.0143728	+	0.999897i	$0.495425\pi$
$318$	53.2481	2.98601
$319$	29.0904	1.62875
$320$	0	0
$321$	40.1868	2.24301
$322$	−7.46374	−0.415938
$323$	32.5389	1.81051
$324$	−34.3577	−1.90876
$325$	0	0
$326$	−40.1668	−2.22463
$327$	−45.3133	−2.50583
$328$	41.2997	2.28039
$329$	−3.15067	−0.173702
$330$	0	0
$331$	−4.37953	−0.240721	−0.120360	−	0.992730i	$-0.538405\pi$
$331$	−4.37953	−0.240721	−0.120360	+	0.992730i	$0.538405\pi$
$332$	−11.1286	−0.610759
$333$	−17.8905	−0.980393
$334$	−11.4899	−0.628702
$335$	0	0
$336$	41.2597	2.25090
$337$	−9.09028	−0.495179	−0.247589	−	0.968865i	$-0.579638\pi$
$337$	−9.09028	−0.495179	−0.247589	+	0.968865i	$0.579638\pi$
$338$	5.96584	0.324499
$339$	−12.4628	−0.676885
$340$	0	0
$341$	25.7593	1.39495
$342$	−60.4774	−3.27024
$343$	−15.5418	−0.839176
$344$	−23.5291	−1.26861
$345$	0	0
$346$	23.8218	1.28067
$347$	0.771121	0.0413959	0.0206980	−	0.999786i	$-0.493411\pi$
$347$	0.771121	0.0413959	0.0206980	+	0.999786i	$0.493411\pi$
$348$	−68.5905	−3.67684
$349$	30.0795	1.61012	0.805060	−	0.593194i	$-0.202133\pi$
$349$	30.0795	1.61012	0.805060	+	0.593194i	$0.202133\pi$
$350$	0	0
$351$	−5.59336	−0.298551
$352$	97.5819	5.20113
$353$	27.7365	1.47627	0.738133	−	0.674655i	$-0.235707\pi$
$353$	27.7365	1.47627	0.738133	+	0.674655i	$0.235707\pi$
$354$	58.1961	3.09309
$355$	0	0
$356$	42.3286	2.24341
$357$	17.0604	0.902934
$358$	−52.7022	−2.78540
$359$	6.19082	0.326739	0.163370	−	0.986565i	$-0.447764\pi$
$359$	6.19082	0.326739	0.163370	+	0.986565i	$0.447764\pi$
$360$	0	0
$361$	18.8221	0.990635
$362$	−42.8067	−2.24987
$363$	55.8075	2.92913
$364$	−21.4074	−1.12205
$365$	0	0
$366$	84.6166	4.42298
$367$	−14.6701	−0.765772	−0.382886	−	0.923796i	$-0.625070\pi$
$367$	−14.6701	−0.765772	−0.382886	+	0.923796i	$0.625070\pi$
$368$	28.4499	1.48305
$369$	−17.4810	−0.910024
$370$	0	0
$371$	−9.59529	−0.498163
$372$	−60.7363	−3.14903
$373$	−7.45878	−0.386201	−0.193100	−	0.981179i	$-0.561854\pi$
$373$	−7.45878	−0.386201	−0.193100	+	0.981179i	$0.561854\pi$
$374$	81.1946	4.19847
$375$	0	0
$376$	21.8053	1.12452
$377$	16.7216	0.861204
$378$	−5.71894	−0.294150
$379$	−32.3225	−1.66030	−0.830148	−	0.557543i	$-0.811744\pi$
$379$	−32.3225	−1.66030	−0.830148	+	0.557543i	$0.811744\pi$
$380$	0	0
$381$	31.5115	1.61438
$382$	37.5885	1.92320
$383$	4.30576	0.220014	0.110007	−	0.993931i	$-0.464913\pi$
$383$	4.30576	0.220014	0.110007	+	0.993931i	$0.464913\pi$
$384$	−52.6383	−2.68619
$385$	0	0
$386$	−15.7183	−0.800040
$387$	9.95923	0.506256
$388$	−52.4274	−2.66160
$389$	16.1415	0.818404	0.409202	−	0.912444i	$-0.365807\pi$
$389$	16.1415	0.818404	0.409202	+	0.912444i	$0.365807\pi$
$390$	0	0
$391$	11.7637	0.594916
$392$	47.0312	2.37544
$393$	−26.0934	−1.31624
$394$	−2.04399	−0.102975
$395$	0	0
$396$	−109.098	−5.48237
$397$	−31.3345	−1.57263	−0.786316	−	0.617825i	$-0.788014\pi$
$397$	−31.3345	−1.57263	−0.786316	+	0.617825i	$0.788014\pi$
$398$	35.5704	1.78299
$399$	19.8305	0.992765
$400$	0	0
$401$	−27.1821	−1.35741	−0.678705	−	0.734411i	$-0.737458\pi$
$401$	−27.1821	−1.35741	−0.678705	+	0.734411i	$0.737458\pi$
$402$	−105.637	−5.26868
$403$	14.8068	0.737578
$404$	−86.0320	−4.28025
$405$	0	0
$406$	17.0970	0.848509
$407$	27.9189	1.38389
$408$	−118.072	−5.84546
$409$	36.8301	1.82113	0.910566	−	0.413363i	$-0.135646\pi$
$409$	36.8301	1.82113	0.910566	+	0.413363i	$0.135646\pi$
$410$	0	0
$411$	−34.1056	−1.68230
$412$	42.4105	2.08942
$413$	−10.4869	−0.516028
$414$	−21.8642	−1.07457
$415$	0	0
$416$	56.0913	2.75010
$417$	41.1840	2.01679
$418$	94.3778	4.61617
$419$	−13.5764	−0.663249	−0.331624	−	0.943411i	$-0.607597\pi$
$419$	−13.5764	−0.663249	−0.331624	+	0.943411i	$0.607597\pi$
$420$	0	0
$421$	−12.1607	−0.592678	−0.296339	−	0.955083i	$-0.595766\pi$
$421$	−12.1607	−0.592678	−0.296339	+	0.955083i	$0.595766\pi$
$422$	−28.2352	−1.37447
$423$	−9.22957	−0.448757
$424$	66.4074	3.22503
$425$	0	0
$426$	43.4770	2.10647
$427$	−15.2479	−0.737897
$428$	81.2620	3.92795
$429$	48.3966	2.33661
$430$	0	0
$431$	−15.1822	−0.731301	−0.365650	−	0.930752i	$-0.619153\pi$
$431$	−15.1822	−0.731301	−0.365650	+	0.930752i	$0.619153\pi$
$432$	21.7991	1.04881
$433$	−28.7849	−1.38331	−0.691657	−	0.722226i	$-0.743119\pi$
$433$	−28.7849	−1.38331	−0.691657	+	0.722226i	$0.743119\pi$
$434$	15.1392	0.726706
$435$	0	0
$436$	−91.6283	−4.38820
$437$	13.6737	0.654103
$438$	36.5597	1.74689
$439$	39.1528	1.86866	0.934331	−	0.356406i	$-0.115998\pi$
$439$	39.1528	1.86866	0.934331	+	0.356406i	$0.115998\pi$
$440$	0	0
$441$	−19.9070	−0.947953
$442$	46.6716	2.21994
$443$	−26.4163	−1.25508	−0.627538	−	0.778586i	$-0.715937\pi$
$443$	−26.4163	−1.25508	−0.627538	+	0.778586i	$0.715937\pi$
$444$	−65.8283	−3.12407
$445$	0	0
$446$	31.6508	1.49871
$447$	12.7679	0.603899
$448$	25.3753	1.19887
$449$	−9.28281	−0.438083	−0.219041	−	0.975716i	$-0.570293\pi$
$449$	−9.28281	−0.438083	−0.219041	+	0.975716i	$0.570293\pi$
$450$	0	0
$451$	27.2799	1.28456
$452$	−25.2011	−1.18536
$453$	−23.7200	−1.11446
$454$	25.0922	1.17764
$455$	0	0
$456$	−137.243	−6.42701
$457$	8.44046	0.394828	0.197414	−	0.980320i	$-0.436746\pi$
$457$	8.44046	0.394828	0.197414	+	0.980320i	$0.436746\pi$
$458$	44.5758	2.08289
$459$	9.01369	0.420723
$460$	0	0
$461$	−2.99442	−0.139464	−0.0697319	−	0.997566i	$-0.522214\pi$
$461$	−2.99442	−0.139464	−0.0697319	+	0.997566i	$0.522214\pi$
$462$	49.4832	2.30216
$463$	10.3910	0.482910	0.241455	−	0.970412i	$-0.422375\pi$
$463$	10.3910	0.482910	0.241455	+	0.970412i	$0.422375\pi$
$464$	−65.1693	−3.02541
$465$	0	0
$466$	−6.92645	−0.320861
$467$	23.5923	1.09172	0.545861	−	0.837876i	$-0.316203\pi$
$467$	23.5923	1.09172	0.545861	+	0.837876i	$0.316203\pi$
$468$	−62.7108	−2.89881
$469$	19.0357	0.878986
$470$	0	0
$471$	18.5160	0.853170
$472$	72.5782	3.34068
$473$	−15.5418	−0.714614
$474$	−65.0700	−2.98876
$475$	0	0
$476$	34.4981	1.58122
$477$	−28.1084	−1.28700
$478$	69.9124	3.19772
$479$	−5.49906	−0.251259	−0.125629	−	0.992077i	$-0.540095\pi$
$479$	−5.49906	−0.251259	−0.125629	+	0.992077i	$0.540095\pi$
$480$	0	0
$481$	16.0482	0.731733
$482$	−2.68673	−0.122377
$483$	7.16926	0.326213
$484$	112.849	5.12949
$485$	0	0
$486$	59.3818	2.69361
$487$	−9.89752	−0.448499	−0.224250	−	0.974532i	$-0.571993\pi$
$487$	−9.89752	−0.448499	−0.224250	+	0.974532i	$0.571993\pi$
$488$	105.528	4.77703
$489$	38.5820	1.74474
$490$	0	0
$491$	−22.6134	−1.02053	−0.510265	−	0.860017i	$-0.670453\pi$
$491$	−22.6134	−1.02053	−0.510265	+	0.860017i	$0.670453\pi$
$492$	−64.3215	−2.89984
$493$	−26.9468	−1.21362
$494$	54.2495	2.44080
$495$	0	0
$496$	−57.7068	−2.59111
$497$	−7.83454	−0.351427
$498$	14.7863	0.662588
$499$	−25.9120	−1.15998	−0.579990	−	0.814624i	$-0.696944\pi$
$499$	−25.9120	−1.15998	−0.579990	+	0.814624i	$0.696944\pi$
$500$	0	0
$501$	11.0366	0.493080
$502$	−72.5548	−3.23828
$503$	1.28878	0.0574637	0.0287319	−	0.999587i	$-0.490853\pi$
$503$	1.28878	0.0574637	0.0287319	+	0.999587i	$0.490853\pi$
$504$	−39.5451	−1.76148
$505$	0	0
$506$	34.1202	1.51683
$507$	−5.73046	−0.254499
$508$	63.7197	2.82710
$509$	8.97659	0.397880	0.198940	−	0.980012i	$-0.436250\pi$
$509$	8.97659	0.397880	0.198940	+	0.980012i	$0.436250\pi$
$510$	0	0
$511$	−6.58804	−0.291438
$512$	2.69004	0.118884
$513$	10.4772	0.462580
$514$	73.9530	3.26192
$515$	0	0
$516$	36.6451	1.61321
$517$	14.4032	0.633451
$518$	16.4085	0.720947
$519$	−22.8819	−1.00440
$520$	0	0
$521$	−8.94427	−0.391856	−0.195928	−	0.980618i	$-0.562772\pi$
$521$	−8.94427	−0.391856	−0.195928	+	0.980618i	$0.562772\pi$
$522$	50.0838	2.19211
$523$	−29.1659	−1.27534	−0.637669	−	0.770311i	$-0.720101\pi$
$523$	−29.1659	−1.27534	−0.637669	+	0.770311i	$0.720101\pi$
$524$	−52.7636	−2.30499
$525$	0	0
$526$	−20.3503	−0.887313
$527$	−23.8611	−1.03941
$528$	−188.617	−8.20851
$529$	−18.0566	−0.785068
$530$	0	0
$531$	−30.7203	−1.33315
$532$	40.0994	1.73853
$533$	15.6808	0.679212
$534$	−56.2410	−2.43378
$535$	0	0
$536$	−131.743	−5.69042
$537$	50.6228	2.18454
$538$	21.9956	0.948300
$539$	31.0658	1.33810
$540$	0	0
$541$	33.3854	1.43535	0.717676	−	0.696378i	$-0.245206\pi$
$541$	33.3854	1.43535	0.717676	+	0.696378i	$0.245206\pi$
$542$	9.76379	0.419391
$543$	41.1177	1.76453
$544$	−90.3911	−3.87549
$545$	0	0
$546$	28.4435	1.21727
$547$	−2.45635	−0.105026	−0.0525129	−	0.998620i	$-0.516723\pi$
$547$	−2.45635	−0.105026	−0.0525129	+	0.998620i	$0.516723\pi$
$548$	−68.9652	−2.94605
$549$	−44.6671	−1.90634
$550$	0	0
$551$	−31.3220	−1.33436
$552$	−49.6173	−2.11185
$553$	11.7256	0.498623
$554$	6.52565	0.277248
$555$	0	0
$556$	83.2785	3.53180
$557$	−24.1979	−1.02530	−0.512649	−	0.858598i	$-0.671336\pi$
$557$	−24.1979	−1.02530	−0.512649	+	0.858598i	$0.671336\pi$
$558$	44.3488	1.87743
$559$	−8.93364	−0.377853
$560$	0	0
$561$	−77.9911	−3.29279
$562$	43.9864	1.85545
$563$	16.2348	0.684214	0.342107	−	0.939661i	$-0.388859\pi$
$563$	16.2348	0.684214	0.342107	+	0.939661i	$0.388859\pi$
$564$	−33.9603	−1.42999
$565$	0	0
$566$	−16.3424	−0.686924
$567$	−8.22616	−0.345466
$568$	54.2216	2.27509
$569$	21.7141	0.910303	0.455152	−	0.890414i	$-0.349585\pi$
$569$	21.7141	0.910303	0.455152	+	0.890414i	$0.349585\pi$
$570$	0	0
$571$	−12.1684	−0.509232	−0.254616	−	0.967042i	$-0.581949\pi$
$571$	−12.1684	−0.509232	−0.254616	+	0.967042i	$0.581949\pi$
$572$	97.8631	4.09186
$573$	−36.1055	−1.50833
$574$	16.0329	0.669199
$575$	0	0
$576$	74.3343	3.09726
$577$	−37.8843	−1.57714	−0.788572	−	0.614942i	$-0.789179\pi$
$577$	−37.8843	−1.57714	−0.788572	+	0.614942i	$0.789179\pi$
$578$	−29.5370	−1.22858
$579$	15.0981	0.627457
$580$	0	0
$581$	−2.66448	−0.110541
$582$	69.6590	2.88746
$583$	43.8644	1.81668
$584$	45.5947	1.88672
$585$	0	0
$586$	−28.0683	−1.15949
$587$	1.81908	0.0750813	0.0375406	−	0.999295i	$-0.488048\pi$
$587$	1.81908	0.0750813	0.0375406	+	0.999295i	$0.488048\pi$
$588$	−73.2481	−3.02070
$589$	−27.7354	−1.14282
$590$	0	0
$591$	1.96334	0.0807612
$592$	−62.5449	−2.57058
$593$	43.3283	1.77928	0.889640	−	0.456663i	$-0.150955\pi$
$593$	43.3283	1.77928	0.889640	+	0.456663i	$0.150955\pi$
$594$	26.1439	1.07270
$595$	0	0
$596$	25.8180	1.05755
$597$	−34.1670	−1.39836
$598$	19.6127	0.802023
$599$	−17.5241	−0.716017	−0.358008	−	0.933718i	$-0.616544\pi$
$599$	−17.5241	−0.716017	−0.358008	+	0.933718i	$0.616544\pi$
$600$	0	0
$601$	25.6494	1.04626	0.523129	−	0.852253i	$-0.324764\pi$
$601$	25.6494	1.04626	0.523129	+	0.852253i	$0.324764\pi$
$602$	−9.13421	−0.372283
$603$	55.7630	2.27085
$604$	−47.9644	−1.95164
$605$	0	0
$606$	114.309	4.64347
$607$	7.39927	0.300327	0.150163	−	0.988661i	$-0.452020\pi$
$607$	7.39927	0.300327	0.150163	+	0.988661i	$0.452020\pi$
$608$	−105.068	−4.26105
$609$	−16.4224	−0.665470
$610$	0	0
$611$	8.27912	0.334937
$612$	101.058	4.08505
$613$	−22.5538	−0.910938	−0.455469	−	0.890252i	$-0.650528\pi$
$613$	−22.5538	−0.910938	−0.455469	+	0.890252i	$0.650528\pi$
$614$	−47.7786	−1.92819
$615$	0	0
$616$	61.7120	2.48645
$617$	27.0999	1.09100	0.545500	−	0.838111i	$-0.316340\pi$
$617$	27.0999	1.09100	0.545500	+	0.838111i	$0.316340\pi$
$618$	−56.3498	−2.26672
$619$	−23.3911	−0.940168	−0.470084	−	0.882622i	$-0.655776\pi$
$619$	−23.3911	−0.940168	−0.470084	+	0.882622i	$0.655776\pi$
$620$	0	0
$621$	3.78780	0.151999
$622$	14.2567	0.571641
$623$	10.1346	0.406034
$624$	−108.419	−4.34025
$625$	0	0
$626$	69.8155	2.79039
$627$	−90.6541	−3.62038
$628$	37.4413	1.49407
$629$	−25.8616	−1.03117
$630$	0	0
$631$	−14.0498	−0.559314	−0.279657	−	0.960100i	$-0.590221\pi$
$631$	−14.0498	−0.559314	−0.279657	+	0.960100i	$0.590221\pi$
$632$	−81.1508	−3.22801
$633$	27.1212	1.07797
$634$	−1.37507	−0.0546111
$635$	0	0
$636$	−103.425	−4.10108
$637$	17.8570	0.707520
$638$	−78.1581	−3.09431
$639$	−22.9505	−0.907906
$640$	0	0
$641$	15.4541	0.610400	0.305200	−	0.952288i	$-0.401277\pi$
$641$	15.4541	0.610400	0.305200	+	0.952288i	$0.401277\pi$
$642$	−107.971	−4.26127
$643$	−41.9982	−1.65625	−0.828124	−	0.560545i	$-0.810592\pi$
$643$	−41.9982	−1.65625	−0.828124	+	0.560545i	$0.810592\pi$
$644$	14.4970	0.571263
$645$	0	0
$646$	−87.4231	−3.43961
$647$	11.2851	0.443664	0.221832	−	0.975085i	$-0.428796\pi$
$647$	11.2851	0.443664	0.221832	+	0.975085i	$0.428796\pi$
$648$	56.9319	2.23650
$649$	47.9405	1.88183
$650$	0	0
$651$	−14.5419	−0.569942
$652$	78.0170	3.05538
$653$	−27.4812	−1.07542	−0.537711	−	0.843129i	$-0.680711\pi$
$653$	−27.4812	−1.07542	−0.537711	+	0.843129i	$0.680711\pi$
$654$	121.744	4.76058
$655$	0	0
$656$	−61.1132	−2.38607
$657$	−19.2990	−0.752924
$658$	8.46500	0.330000
$659$	40.0611	1.56056	0.780279	−	0.625431i	$-0.215077\pi$
$659$	40.0611	1.56056	0.780279	+	0.625431i	$0.215077\pi$
$660$	0	0
$661$	49.2805	1.91679	0.958394	−	0.285449i	$-0.0921426\pi$
$661$	49.2805	1.91679	0.958394	+	0.285449i	$0.0921426\pi$
$662$	11.7666	0.457322
$663$	−44.8302	−1.74106
$664$	18.4404	0.715627
$665$	0	0
$666$	48.0669	1.86255
$667$	−11.3238	−0.438458
$668$	22.3172	0.863480
$669$	−30.4020	−1.17541
$670$	0	0
$671$	69.7050	2.69093
$672$	−55.0879	−2.12506
$673$	−4.32553	−0.166737	−0.0833685	−	0.996519i	$-0.526568\pi$
$673$	−4.32553	−0.166737	−0.0833685	+	0.996519i	$0.526568\pi$
$674$	24.4231	0.940742
$675$	0	0
$676$	−11.5876	−0.445677
$677$	−38.0959	−1.46415	−0.732073	−	0.681226i	$-0.761447\pi$
$677$	−38.0959	−1.46415	−0.732073	+	0.681226i	$0.761447\pi$
$678$	33.4841	1.28595
$679$	−12.5525	−0.481722
$680$	0	0
$681$	−24.1022	−0.923599
$682$	−69.2083	−2.65012
$683$	−35.9542	−1.37575	−0.687874	−	0.725830i	$-0.741456\pi$
$683$	−35.9542	−1.37575	−0.687874	+	0.725830i	$0.741456\pi$
$684$	117.467	4.49146
$685$	0	0
$686$	41.7564	1.59427
$687$	−42.8171	−1.63357
$688$	34.8173	1.32740
$689$	25.2138	0.960570
$690$	0	0
$691$	20.7606	0.789772	0.394886	−	0.918730i	$-0.370784\pi$
$691$	20.7606	0.789772	0.394886	+	0.918730i	$0.370784\pi$
$692$	−46.2697	−1.75891
$693$	−26.1210	−0.992253
$694$	−2.07179	−0.0786441
$695$	0	0
$696$	113.657	4.30815
$697$	−25.2696	−0.957156
$698$	−80.8155	−3.05891
$699$	6.65317	0.251646
$700$	0	0
$701$	3.55693	0.134343	0.0671716	−	0.997741i	$-0.478602\pi$
$701$	3.55693	0.134343	0.0671716	+	0.997741i	$0.478602\pi$
$702$	15.0278	0.567189
$703$	−30.0606	−1.13376
$704$	−116.002	−4.37199
$705$	0	0
$706$	−74.5205	−2.80462
$707$	−20.5984	−0.774681
$708$	−113.036	−4.24815
$709$	19.1731	0.720060	0.360030	−	0.932941i	$-0.382766\pi$
$709$	19.1731	0.720060	0.360030	+	0.932941i	$0.382766\pi$
$710$	0	0
$711$	34.3489	1.28818
$712$	−70.1399	−2.62860
$713$	−10.0271	−0.375518
$714$	−45.8367	−1.71540
$715$	0	0
$716$	102.365	3.82555
$717$	−67.1541	−2.50791
$718$	−16.6330	−0.620740
$719$	−17.5165	−0.653256	−0.326628	−	0.945153i	$-0.605912\pi$
$719$	−17.5165	−0.653256	−0.326628	+	0.945153i	$0.605912\pi$
$720$	0	0
$721$	10.1542	0.378163
$722$	−50.5697	−1.88201
$723$	2.58072	0.0959781
$724$	83.1445	3.09004
$725$	0	0
$726$	−149.939	−5.56477
$727$	−28.7088	−1.06475	−0.532376	−	0.846508i	$-0.678701\pi$
$727$	−28.7088	−1.06475	−0.532376	+	0.846508i	$0.678701\pi$
$728$	35.4728	1.31471
$729$	−37.2874	−1.38101
$730$	0	0
$731$	14.3966	0.532476
$732$	−164.353	−6.07467
$733$	−42.4695	−1.56865	−0.784323	−	0.620352i	$-0.786990\pi$
$733$	−42.4695	−1.56865	−0.784323	+	0.620352i	$0.786990\pi$
$734$	39.4145	1.45482
$735$	0	0
$736$	−37.9848	−1.40014
$737$	−87.0208	−3.20545
$738$	46.9666	1.72887
$739$	−38.8874	−1.43050	−0.715249	−	0.698870i	$-0.753687\pi$
$739$	−38.8874	−1.43050	−0.715249	+	0.698870i	$0.753687\pi$
$740$	0	0
$741$	−52.1091	−1.91428
$742$	25.7799	0.946411
$743$	1.03128	0.0378339	0.0189170	−	0.999821i	$-0.493978\pi$
$743$	1.03128	0.0378339	0.0189170	+	0.999821i	$0.493978\pi$
$744$	100.642	3.68972
$745$	0	0
$746$	20.0397	0.733706
$747$	−7.80531	−0.285581
$748$	−157.706	−5.76632
$749$	19.4563	0.710918
$750$	0	0
$751$	14.0086	0.511181	0.255590	−	0.966785i	$-0.417730\pi$
$751$	14.0086	0.511181	0.255590	+	0.966785i	$0.417730\pi$
$752$	−32.2664	−1.17664
$753$	69.6922	2.53972
$754$	−44.9262	−1.63612
$755$	0	0
$756$	11.1080	0.403996
$757$	38.1157	1.38534	0.692670	−	0.721255i	$-0.256434\pi$
$757$	38.1157	1.38534	0.692670	+	0.721255i	$0.256434\pi$
$758$	86.8418	3.15424
$759$	−32.7740	−1.18962
$760$	0	0
$761$	3.59136	0.130187	0.0650933	−	0.997879i	$-0.479266\pi$
$761$	3.59136	0.130187	0.0650933	+	0.997879i	$0.479266\pi$
$762$	−84.6628	−3.06701
$763$	−21.9383	−0.794219
$764$	−73.0092	−2.64138
$765$	0	0
$766$	−11.5684	−0.417984
$767$	27.5568	0.995018
$768$	36.6001	1.32069
$769$	−39.3482	−1.41893	−0.709466	−	0.704740i	$-0.751064\pi$
$769$	−39.3482	−1.41893	−0.709466	+	0.704740i	$0.751064\pi$
$770$	0	0
$771$	−71.0352	−2.55827
$772$	30.5301	1.09880
$773$	25.6925	0.924094	0.462047	−	0.886856i	$-0.347115\pi$
$773$	25.6925	0.924094	0.462047	+	0.886856i	$0.347115\pi$
$774$	−26.7577	−0.961786
$775$	0	0
$776$	86.8740	3.11859
$777$	−15.7611	−0.565425
$778$	−43.3677	−1.55481
$779$	−29.3726	−1.05238
$780$	0	0
$781$	35.8153	1.28157
$782$	−31.6059	−1.13022
$783$	−8.67660	−0.310077
$784$	−69.5945	−2.48552
$785$	0	0
$786$	70.1058	2.50059
$787$	3.98083	0.141901	0.0709506	−	0.997480i	$-0.477397\pi$
$787$	3.98083	0.141901	0.0709506	+	0.997480i	$0.477397\pi$
$788$	3.97009	0.141429
$789$	19.5473	0.695904
$790$	0	0
$791$	−6.03382	−0.214538
$792$	180.779	6.42370
$793$	40.0673	1.42283
$794$	84.1871	2.98769
$795$	0	0
$796$	−69.0894	−2.44881
$797$	39.8549	1.41173	0.705866	−	0.708345i	$-0.250558\pi$
$797$	39.8549	1.41173	0.705866	+	0.708345i	$0.250558\pi$
$798$	−53.2791	−1.88606
$799$	−13.3418	−0.471999
$800$	0	0
$801$	29.6882	1.04898
$802$	73.0309	2.57881
$803$	30.1169	1.06280
$804$	205.181	7.23617
$805$	0	0
$806$	−39.7818	−1.40125
$807$	−21.1278	−0.743734
$808$	142.558	5.01517
$809$	14.3004	0.502774	0.251387	−	0.967887i	$-0.419113\pi$
$809$	14.3004	0.502774	0.251387	+	0.967887i	$0.419113\pi$
$810$	0	0
$811$	52.5174	1.84414	0.922068	−	0.387028i	$-0.126498\pi$
$811$	52.5174	1.84414	0.922068	+	0.387028i	$0.126498\pi$
$812$	−33.2079	−1.16537
$813$	−9.37857	−0.328921
$814$	−75.0106	−2.62912
$815$	0	0
$816$	174.718	6.11635
$817$	16.7341	0.585451
$818$	−98.9525	−3.45979
$819$	−15.0147	−0.524655
$820$	0	0
$821$	−18.4325	−0.643300	−0.321650	−	0.946859i	$-0.604237\pi$
$821$	−18.4325	−0.643300	−0.321650	+	0.946859i	$0.604237\pi$
$822$	91.6324	3.19605
$823$	−38.9994	−1.35944	−0.679718	−	0.733474i	$-0.737898\pi$
$823$	−38.9994	−1.35944	−0.679718	+	0.733474i	$0.737898\pi$
$824$	−70.2757	−2.44817
$825$	0	0
$826$	28.1755	0.980351
$827$	18.2963	0.636226	0.318113	−	0.948053i	$-0.396951\pi$
$827$	18.2963	0.636226	0.318113	+	0.948053i	$0.396951\pi$
$828$	42.4675	1.47585
$829$	2.36565	0.0821624	0.0410812	−	0.999156i	$-0.486920\pi$
$829$	2.36565	0.0821624	0.0410812	+	0.999156i	$0.486920\pi$
$830$	0	0
$831$	−6.26818	−0.217441
$832$	−66.6795	−2.31169
$833$	−28.7766	−0.997049
$834$	−110.650	−3.83150
$835$	0	0
$836$	−183.312	−6.33999
$837$	−7.68306	−0.265565
$838$	36.4760	1.26004
$839$	−19.8535	−0.685420	−0.342710	−	0.939441i	$-0.611345\pi$
$839$	−19.8535	−0.685420	−0.342710	+	0.939441i	$0.611345\pi$
$840$	0	0
$841$	−3.06096	−0.105550
$842$	32.6726	1.12597
$843$	−42.2509	−1.45520
$844$	54.8420	1.88774
$845$	0	0
$846$	24.7973	0.852550
$847$	27.0190	0.928385
$848$	−98.2665	−3.37448
$849$	15.6977	0.538742
$850$	0	0
$851$	−10.8677	−0.372542
$852$	−84.4466	−2.89309
$853$	17.5358	0.600415	0.300208	−	0.953874i	$-0.402944\pi$
$853$	17.5358	0.600415	0.300208	+	0.953874i	$0.402944\pi$
$854$	40.9669	1.40186
$855$	0	0
$856$	−134.654	−4.60238
$857$	−42.6899	−1.45826	−0.729130	−	0.684375i	$-0.760075\pi$
$857$	−42.6899	−1.45826	−0.729130	+	0.684375i	$0.760075\pi$
$858$	−130.028	−4.43910
$859$	−19.1579	−0.653659	−0.326830	−	0.945083i	$-0.605980\pi$
$859$	−19.1579	−0.653659	−0.326830	+	0.945083i	$0.605980\pi$
$860$	0	0
$861$	−15.4003	−0.524841
$862$	40.7904	1.38933
$863$	12.8156	0.436248	0.218124	−	0.975921i	$-0.430006\pi$
$863$	12.8156	0.436248	0.218124	+	0.975921i	$0.430006\pi$
$864$	−29.1051	−0.990175
$865$	0	0
$866$	77.3372	2.62803
$867$	28.3716	0.963550
$868$	−29.4053	−0.998082
$869$	−53.6030	−1.81836
$870$	0	0
$871$	−50.0206	−1.69488
$872$	151.831	5.14166
$873$	−36.7713	−1.24452
$874$	−36.7376	−1.24267
$875$	0	0
$876$	−71.0108	−2.39923
$877$	29.6224	1.00028	0.500139	−	0.865945i	$-0.333282\pi$
$877$	29.6224	1.00028	0.500139	+	0.865945i	$0.333282\pi$
$878$	−105.193	−3.55009
$879$	26.9609	0.909368
$880$	0	0
$881$	27.5400	0.927845	0.463922	−	0.885876i	$-0.346442\pi$
$881$	27.5400	0.927845	0.463922	+	0.885876i	$0.346442\pi$
$882$	53.4847	1.80092
$883$	9.69519	0.326269	0.163134	−	0.986604i	$-0.447840\pi$
$883$	9.69519	0.326269	0.163134	+	0.986604i	$0.447840\pi$
$884$	−90.6516	−3.04894
$885$	0	0
$886$	70.9733	2.38440
$887$	6.60583	0.221802	0.110901	−	0.993831i	$-0.464626\pi$
$887$	6.60583	0.221802	0.110901	+	0.993831i	$0.464626\pi$
$888$	109.080	3.66048
$889$	15.2562	0.511677
$890$	0	0
$891$	37.6055	1.25983
$892$	−61.4762	−2.05838
$893$	−15.5080	−0.518957
$894$	−34.3038	−1.14729
$895$	0	0
$896$	−25.4847	−0.851384
$897$	−18.8389	−0.629012
$898$	24.9404	0.832271
$899$	22.9688	0.766052
$900$	0	0
$901$	−40.6321	−1.35365
$902$	−73.2936	−2.44041
$903$	8.77383	0.291975
$904$	41.7591	1.38889
$905$	0	0
$906$	63.7291	2.11726
$907$	18.6330	0.618700	0.309350	−	0.950948i	$-0.399888\pi$
$907$	18.6330	0.618700	0.309350	+	0.950948i	$0.399888\pi$
$908$	−48.7373	−1.61740
$909$	−60.3408	−2.00138
$910$	0	0
$911$	−10.4778	−0.347146	−0.173573	−	0.984821i	$-0.555531\pi$
$911$	−10.4778	−0.347146	−0.173573	+	0.984821i	$0.555531\pi$
$912$	203.086	6.72485
$913$	12.1805	0.403117
$914$	−22.6772	−0.750096
$915$	0	0
$916$	−86.5807	−2.86071
$917$	−12.6330	−0.417180
$918$	−24.2173	−0.799291
$919$	33.4620	1.10381	0.551905	−	0.833907i	$-0.313901\pi$
$919$	33.4620	1.10381	0.551905	+	0.833907i	$0.313901\pi$
$920$	0	0
$921$	45.8935	1.51224
$922$	8.04518	0.264954
$923$	20.5871	0.677631
$924$	−96.1125	−3.16187
$925$	0	0
$926$	−27.9177	−0.917433
$927$	29.7457	0.976978
$928$	87.0107	2.85627
$929$	38.0410	1.24809	0.624043	−	0.781390i	$-0.285489\pi$
$929$	38.0410	1.24809	0.624043	+	0.781390i	$0.285489\pi$
$930$	0	0
$931$	−33.4489	−1.09624
$932$	13.4534	0.440681
$933$	−13.6942	−0.448328
$934$	−63.3861	−2.07406
$935$	0	0
$936$	103.914	3.39653
$937$	27.2333	0.889673	0.444836	−	0.895612i	$-0.353262\pi$
$937$	27.2333	0.889673	0.444836	+	0.895612i	$0.353262\pi$
$938$	−51.1437	−1.66990
$939$	−67.0609	−2.18845
$940$	0	0
$941$	−39.6482	−1.29249	−0.646247	−	0.763129i	$-0.723662\pi$
$941$	−39.6482	−1.29249	−0.646247	+	0.763129i	$0.723662\pi$
$942$	−49.7473	−1.62086
$943$	−10.6190	−0.345802
$944$	−107.398	−3.49550
$945$	0	0
$946$	41.7567	1.35763
$947$	−3.86951	−0.125742	−0.0628711	−	0.998022i	$-0.520026\pi$
$947$	−3.86951	−0.125742	−0.0628711	+	0.998022i	$0.520026\pi$
$948$	126.387	4.10486
$949$	17.3116	0.561958
$950$	0	0
$951$	1.32082	0.0428305
$952$	−57.1645	−1.85271
$953$	20.5864	0.666859	0.333430	−	0.942775i	$-0.391794\pi$
$953$	20.5864	0.666859	0.333430	+	0.942775i	$0.391794\pi$
$954$	75.5196	2.44504
$955$	0	0
$956$	−135.793	−4.39185
$957$	75.0744	2.42681
$958$	14.7745	0.477342
$959$	−16.5121	−0.533204
$960$	0	0
$961$	−10.6613	−0.343914
$962$	−43.1170	−1.39015
$963$	56.9952	1.83665
$964$	5.21850	0.168077
$965$	0	0
$966$	−19.2619	−0.619740
$967$	−52.3145	−1.68232	−0.841161	−	0.540785i	$-0.818127\pi$
$967$	−52.3145	−1.68232	−0.841161	+	0.540785i	$0.818127\pi$
$968$	−186.994	−6.01022
$969$	83.9738	2.69763
$970$	0	0
$971$	29.3275	0.941165	0.470582	−	0.882356i	$-0.344044\pi$
$971$	29.3275	0.941165	0.470582	+	0.882356i	$0.344044\pi$
$972$	−115.339	−3.69949
$973$	19.9391	0.639219
$974$	26.5919	0.852061
$975$	0	0
$976$	−156.155	−4.99841
$977$	−43.0111	−1.37605	−0.688024	−	0.725688i	$-0.741522\pi$
$977$	−43.0111	−1.37605	−0.688024	+	0.725688i	$0.741522\pi$
$978$	−103.659	−3.31466
$979$	−46.3299	−1.48071
$980$	0	0
$981$	−64.2659	−2.05185
$982$	60.7561	1.93880
$983$	47.7953	1.52443	0.762216	−	0.647322i	$-0.224111\pi$
$983$	47.7953	1.52443	0.762216	+	0.647322i	$0.224111\pi$
$984$	106.583	3.39774
$985$	0	0
$986$	72.3986	2.30564
$987$	−8.13102	−0.258813
$988$	−105.370	−3.35228
$989$	6.04983	0.192373
$990$	0	0
$991$	−8.79367	−0.279340	−0.139670	−	0.990198i	$-0.544604\pi$
$991$	−8.79367	−0.279340	−0.139670	+	0.990198i	$0.544604\pi$
$992$	77.0472	2.44625
$993$	−11.3023	−0.358669
$994$	21.0493	0.667642
$995$	0	0
$996$	−28.7197	−0.910020
$997$	25.3984	0.804375	0.402188	−	0.915557i	$-0.368250\pi$
$997$	25.3984	0.804375	0.402188	+	0.915557i	$0.368250\pi$
$998$	69.6185	2.20373
$999$	−8.32719	−0.263461

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.p.1.1		46
5.2	odd	4		1205.2.b.c.724.1	✓	46
5.3	odd	4		1205.2.b.c.724.46	yes	46
5.4	even	2	inner	6025.2.a.p.1.46		46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1205.2.b.c.724.1	✓	46	5.2	odd	4
1205.2.b.c.724.46	yes	46	5.3	odd	4
6025.2.a.p.1.1		46	1.1	even	1	trivial
6025.2.a.p.1.46		46	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 6025 = 5^{2} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6025.a (trivial)

\(f(q)\)	\(=\)	\(q-2.68673 q^{2} +2.58072 q^{3} +5.21850 q^{4} -6.93370 q^{6} +1.24945 q^{7} -8.64724 q^{8} +3.66013 q^{9} +O(q^{10})\)	\(q-2.68673 q^{2} +2.58072 q^{3} +5.21850 q^{4} -6.93370 q^{6} +1.24945 q^{7} -8.64724 q^{8} +3.66013 q^{9} -5.71181 q^{11} +13.4675 q^{12} -3.28322 q^{13} -3.35693 q^{14} +12.7958 q^{16} +5.29090 q^{17} -9.83378 q^{18} +6.14996 q^{19} +3.22448 q^{21} +15.3461 q^{22} +2.22338 q^{23} -22.3161 q^{24} +8.82111 q^{26} +1.70362 q^{27} +6.52026 q^{28} -5.09304 q^{29} -4.50984 q^{31} -17.0842 q^{32} -14.7406 q^{33} -14.2152 q^{34} +19.1004 q^{36} -4.88794 q^{37} -16.5233 q^{38} -8.47308 q^{39} -4.77605 q^{41} -8.66331 q^{42} +2.72100 q^{43} -29.8071 q^{44} -5.97362 q^{46} -2.52165 q^{47} +33.0223 q^{48} -5.43887 q^{49} +13.6544 q^{51} -17.1335 q^{52} -7.67961 q^{53} -4.57716 q^{54} -10.8043 q^{56} +15.8714 q^{57} +13.6836 q^{58} -8.39323 q^{59} -12.2037 q^{61} +12.1167 q^{62} +4.57315 q^{63} +20.3092 q^{64} +39.6039 q^{66} +15.2352 q^{67} +27.6106 q^{68} +5.73793 q^{69} -6.27039 q^{71} -31.6500 q^{72} -5.27275 q^{73} +13.1325 q^{74} +32.0936 q^{76} -7.13662 q^{77} +22.7648 q^{78} +9.38460 q^{79} -6.58383 q^{81} +12.8319 q^{82} -2.13252 q^{83} +16.8270 q^{84} -7.31059 q^{86} -13.1437 q^{87} +49.3913 q^{88} +8.11125 q^{89} -4.10222 q^{91} +11.6027 q^{92} -11.6386 q^{93} +6.77498 q^{94} -44.0897 q^{96} -10.0464 q^{97} +14.6128 q^{98} -20.9060 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q + 34 q^{4} - 4 q^{6} + 34 q^{9}+O(q^{10}) \) 46 * q + 34 * q^4 - 4 * q^6 + 34 * q^9	\( 46 q + 34 q^{4} - 4 q^{6} + 34 q^{9} - 64 q^{11} - 66 q^{14} + 22 q^{16} - 14 q^{21} - 50 q^{24} - 60 q^{26} - 36 q^{29} - 36 q^{31} + 12 q^{34} - 34 q^{36} - 88 q^{39} - 76 q^{41} - 100 q^{44} - 12 q^{46} + 22 q^{49} - 112 q^{51} - 26 q^{54} - 120 q^{56} - 84 q^{59} - 78 q^{61} + 28 q^{64} - 2 q^{66} - 24 q^{69} - 172 q^{71} - 16 q^{74} - 18 q^{76} - 54 q^{79} - 42 q^{81} + 44 q^{84} - 80 q^{86} - 86 q^{89} - 88 q^{91} + 4 q^{94} - 122 q^{96} - 148 q^{99}+O(q^{100}) \) 46 * q + 34 * q^4 - 4 * q^6 + 34 * q^9 - 64 * q^11 - 66 * q^14 + 22 * q^16 - 14 * q^21 - 50 * q^24 - 60 * q^26 - 36 * q^29 - 36 * q^31 + 12 * q^34 - 34 * q^36 - 88 * q^39 - 76 * q^41 - 100 * q^44 - 12 * q^46 + 22 * q^49 - 112 * q^51 - 26 * q^54 - 120 * q^56 - 84 * q^59 - 78 * q^61 + 28 * q^64 - 2 * q^66 - 24 * q^69 - 172 * q^71 - 16 * q^74 - 18 * q^76 - 54 * q^79 - 42 * q^81 + 44 * q^84 - 80 * q^86 - 86 * q^89 - 88 * q^91 + 4 * q^94 - 122 * q^96 - 148 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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