LMFDB - Embedded newform 6025.2.a.k.1.6

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:	$0$
Dimension:	$25$
Twist minimal:	no (minimal twist has level 1205)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
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)

$f(q)$	$=$	$q-1.67265 q^{2} +2.12506 q^{3} +0.797742 q^{4} -3.55448 q^{6} +2.13130 q^{7} +2.01095 q^{8} +1.51590 q^{9} +O(q^{10})$	\(q-1.67265 q^{2} +2.12506 q^{3} +0.797742 q^{4} -3.55448 q^{6} +2.13130 q^{7} +2.01095 q^{8} +1.51590 q^{9} +3.16246 q^{11} +1.69525 q^{12} +1.26789 q^{13} -3.56491 q^{14} -4.95909 q^{16} -4.19249 q^{17} -2.53555 q^{18} +4.52893 q^{19} +4.52915 q^{21} -5.28967 q^{22} -4.86838 q^{23} +4.27340 q^{24} -2.12074 q^{26} -3.15382 q^{27} +1.70023 q^{28} -1.42636 q^{29} -0.656493 q^{31} +4.27290 q^{32} +6.72042 q^{33} +7.01254 q^{34} +1.20929 q^{36} -9.27717 q^{37} -7.57529 q^{38} +2.69436 q^{39} +11.0268 q^{41} -7.57567 q^{42} +7.88686 q^{43} +2.52283 q^{44} +8.14308 q^{46} +5.41293 q^{47} -10.5384 q^{48} -2.45755 q^{49} -8.90930 q^{51} +1.01145 q^{52} +7.24242 q^{53} +5.27522 q^{54} +4.28594 q^{56} +9.62426 q^{57} +2.38580 q^{58} +7.89385 q^{59} -0.114744 q^{61} +1.09808 q^{62} +3.23083 q^{63} +2.77114 q^{64} -11.2409 q^{66} -2.32822 q^{67} -3.34452 q^{68} -10.3456 q^{69} +8.84974 q^{71} +3.04839 q^{72} +7.70217 q^{73} +15.5174 q^{74} +3.61292 q^{76} +6.74015 q^{77} -4.50670 q^{78} +14.8781 q^{79} -11.2497 q^{81} -18.4440 q^{82} -9.17652 q^{83} +3.61310 q^{84} -13.1919 q^{86} -3.03111 q^{87} +6.35955 q^{88} +11.5016 q^{89} +2.70227 q^{91} -3.88372 q^{92} -1.39509 q^{93} -9.05391 q^{94} +9.08018 q^{96} -1.39764 q^{97} +4.11061 q^{98} +4.79395 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q + 4 q^{2} - 9 q^{3} + 36 q^{4} + 7 q^{6} - 7 q^{7} + 15 q^{8} + 36 q^{9}+O(q^{10}) $ 25 * q + 4 * q^2 - 9 * q^3 + 36 * q^4 + 7 * q^6 - 7 * q^7 + 15 * q^8 + 36 * q^9	$ 25 q + 4 q^{2} - 9 q^{3} + 36 q^{4} + 7 q^{6} - 7 q^{7} + 15 q^{8} + 36 q^{9} + 10 q^{11} - 22 q^{12} - 10 q^{13} + 13 q^{14} + 54 q^{16} - q^{17} + 13 q^{18} + 50 q^{19} + 9 q^{21} - 11 q^{22} + 31 q^{23} + 22 q^{24} + 8 q^{26} - 42 q^{27} - 14 q^{28} + 4 q^{29} + 34 q^{31} + 44 q^{32} - 28 q^{33} + 33 q^{34} + 83 q^{36} - 14 q^{37} + 10 q^{38} + 23 q^{39} + 11 q^{41} - 23 q^{42} - 49 q^{43} + 20 q^{44} + 27 q^{46} + 28 q^{47} - 30 q^{48} + 66 q^{49} + 49 q^{51} - 39 q^{52} + 16 q^{53} + 5 q^{54} + 51 q^{56} - 10 q^{57} + 8 q^{58} + 30 q^{59} + 35 q^{61} + 18 q^{62} + 73 q^{64} - 13 q^{66} - 37 q^{67} - 11 q^{68} - 4 q^{69} + 12 q^{71} + 90 q^{72} - 36 q^{73} - 12 q^{74} + 57 q^{76} + 31 q^{77} + 9 q^{78} + 16 q^{79} + 65 q^{81} + 11 q^{82} - 43 q^{83} - 62 q^{84} - 9 q^{86} + 22 q^{87} - 20 q^{88} + 38 q^{89} + 86 q^{91} + 119 q^{92} - 10 q^{93} - 18 q^{94} - 34 q^{96} - 17 q^{97} + 32 q^{98} + 26 q^{99}+O(q^{100}) $ 25 * q + 4 * q^2 - 9 * q^3 + 36 * q^4 + 7 * q^6 - 7 * q^7 + 15 * q^8 + 36 * q^9 + 10 * q^11 - 22 * q^12 - 10 * q^13 + 13 * q^14 + 54 * q^16 - q^17 + 13 * q^18 + 50 * q^19 + 9 * q^21 - 11 * q^22 + 31 * q^23 + 22 * q^24 + 8 * q^26 - 42 * q^27 - 14 * q^28 + 4 * q^29 + 34 * q^31 + 44 * q^32 - 28 * q^33 + 33 * q^34 + 83 * q^36 - 14 * q^37 + 10 * q^38 + 23 * q^39 + 11 * q^41 - 23 * q^42 - 49 * q^43 + 20 * q^44 + 27 * q^46 + 28 * q^47 - 30 * q^48 + 66 * q^49 + 49 * q^51 - 39 * q^52 + 16 * q^53 + 5 * q^54 + 51 * q^56 - 10 * q^57 + 8 * q^58 + 30 * q^59 + 35 * q^61 + 18 * q^62 + 73 * q^64 - 13 * q^66 - 37 * q^67 - 11 * q^68 - 4 * q^69 + 12 * q^71 + 90 * q^72 - 36 * q^73 - 12 * q^74 + 57 * q^76 + 31 * q^77 + 9 * q^78 + 16 * q^79 + 65 * q^81 + 11 * q^82 - 43 * q^83 - 62 * q^84 - 9 * q^86 + 22 * q^87 - 20 * q^88 + 38 * q^89 + 86 * q^91 + 119 * q^92 - 10 * q^93 - 18 * q^94 - 34 * q^96 - 17 * q^97 + 32 * q^98 + 26 * 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 $

$2$	−1.67265	−1.18274	−0.591369	−	0.806401i	$-0.701412\pi$
$2$	−1.67265	−1.18274	−0.591369	+	0.806401i	$0.701412\pi$
$3$	2.12506	1.22691	0.613453	−	0.789731i	$-0.289780\pi$
$3$	2.12506	1.22691	0.613453	+	0.789731i	$0.289780\pi$
$4$	0.797742	0.398871
$5$	0	0
$5$	0	0
$6$	−3.55448	−1.45111
$7$	2.13130	0.805557	0.402778	−	0.915298i	$-0.368045\pi$
$7$	2.13130	0.805557	0.402778	+	0.915298i	$0.368045\pi$
$8$	2.01095	0.710978
$9$	1.51590	0.505298
$10$	0	0
$11$	3.16246	0.953517	0.476758	−	0.879034i	$-0.341812\pi$
$11$	3.16246	0.953517	0.476758	+	0.879034i	$0.341812\pi$
$12$	1.69525	0.489377
$13$	1.26789	0.351651	0.175825	−	0.984421i	$-0.443741\pi$
$13$	1.26789	0.351651	0.175825	+	0.984421i	$0.443741\pi$
$14$	−3.56491	−0.952763
$15$	0	0
$16$	−4.95909	−1.23977
$17$	−4.19249	−1.01683	−0.508414	−	0.861113i	$-0.669768\pi$
$17$	−4.19249	−1.01683	−0.508414	+	0.861113i	$0.669768\pi$
$18$	−2.53555	−0.597636
$19$	4.52893	1.03901	0.519504	−	0.854468i	$-0.326117\pi$
$19$	4.52893	1.03901	0.519504	+	0.854468i	$0.326117\pi$
$20$	0	0
$21$	4.52915	0.988342
$22$	−5.28967	−1.12776
$23$	−4.86838	−1.01513	−0.507564	−	0.861614i	$-0.669454\pi$
$23$	−4.86838	−1.01513	−0.507564	+	0.861614i	$0.669454\pi$
$24$	4.27340	0.872304
$25$	0	0
$26$	−2.12074	−0.415911
$27$	−3.15382	−0.606952
$28$	1.70023	0.321313
$29$	−1.42636	−0.264869	−0.132434	−	0.991192i	$-0.542279\pi$
$29$	−1.42636	−0.264869	−0.132434	+	0.991192i	$0.542279\pi$
$30$	0	0
$31$	−0.656493	−0.117910	−0.0589548	−	0.998261i	$-0.518777\pi$
$31$	−0.656493	−0.117910	−0.0589548	+	0.998261i	$0.518777\pi$
$32$	4.27290	0.755349
$33$	6.72042	1.16988
$34$	7.01254	1.20264
$35$	0	0
$36$	1.20929	0.201549
$37$	−9.27717	−1.52516	−0.762579	−	0.646895i	$-0.776067\pi$
$37$	−9.27717	−1.52516	−0.762579	+	0.646895i	$0.776067\pi$
$38$	−7.57529	−1.22887
$39$	2.69436	0.431442
$40$	0	0
$41$	11.0268	1.72210	0.861051	−	0.508519i	$-0.169807\pi$
$41$	11.0268	1.72210	0.861051	+	0.508519i	$0.169807\pi$
$42$	−7.57567	−1.16895
$43$	7.88686	1.20273	0.601367	−	0.798973i	$-0.294623\pi$
$43$	7.88686	1.20273	0.601367	+	0.798973i	$0.294623\pi$
$44$	2.52283	0.380330
$45$	0	0
$46$	8.14308	1.20063
$47$	5.41293	0.789557	0.394778	−	0.918776i	$-0.370821\pi$
$47$	5.41293	0.789557	0.394778	+	0.918776i	$0.370821\pi$
$48$	−10.5384	−1.52108
$49$	−2.45755	−0.351079
$50$	0	0
$51$	−8.90930	−1.24755
$52$	1.01145	0.140263
$53$	7.24242	0.994823	0.497411	−	0.867515i	$-0.334284\pi$
$53$	7.24242	0.994823	0.497411	+	0.867515i	$0.334284\pi$
$54$	5.27522	0.717866
$55$	0	0
$56$	4.28594	0.572733
$57$	9.62426	1.27476
$58$	2.38580	0.313271
$59$	7.89385	1.02769	0.513846	−	0.857882i	$-0.328220\pi$
$59$	7.89385	1.02769	0.513846	+	0.857882i	$0.328220\pi$
$60$	0	0
$61$	−0.114744	−0.0146915	−0.00734574	−	0.999973i	$-0.502338\pi$
$61$	−0.114744	−0.0146915	−0.00734574	+	0.999973i	$0.502338\pi$
$62$	1.09808	0.139456
$63$	3.23083	0.407046
$64$	2.77114	0.346392
$65$	0	0
$66$	−11.2409	−1.38366
$67$	−2.32822	−0.284437	−0.142219	−	0.989835i	$-0.545424\pi$
$67$	−2.32822	−0.284437	−0.142219	+	0.989835i	$0.545424\pi$
$68$	−3.34452	−0.405583
$69$	−10.3456	−1.24547
$70$	0	0
$71$	8.84974	1.05027	0.525136	−	0.851018i	$-0.324015\pi$
$71$	8.84974	1.05027	0.525136	+	0.851018i	$0.324015\pi$
$72$	3.04839	0.359256
$73$	7.70217	0.901471	0.450735	−	0.892658i	$-0.351162\pi$
$73$	7.70217	0.901471	0.450735	+	0.892658i	$0.351162\pi$
$74$	15.5174	1.80386
$75$	0	0
$76$	3.61292	0.414430
$77$	6.74015	0.768112
$78$	−4.50670	−0.510283
$79$	14.8781	1.67392	0.836958	−	0.547267i	$-0.184332\pi$
$79$	14.8781	1.67392	0.836958	+	0.547267i	$0.184332\pi$
$80$	0	0
$81$	−11.2497	−1.24997
$82$	−18.4440	−2.03680
$83$	−9.17652	−1.00725	−0.503627	−	0.863921i	$-0.668001\pi$
$83$	−9.17652	−1.00725	−0.503627	+	0.863921i	$0.668001\pi$
$84$	3.61310	0.394221
$85$	0	0
$86$	−13.1919	−1.42252
$87$	−3.03111	−0.324969
$88$	6.35955	0.677930
$89$	11.5016	1.21916	0.609582	−	0.792723i	$-0.291337\pi$
$89$	11.5016	1.21916	0.609582	+	0.792723i	$0.291337\pi$
$90$	0	0
$91$	2.70227	0.283274
$92$	−3.88372	−0.404905
$93$	−1.39509	−0.144664
$94$	−9.05391	−0.933840
$95$	0	0
$96$	9.08018	0.926742
$97$	−1.39764	−0.141909	−0.0709543	−	0.997480i	$-0.522604\pi$
$97$	−1.39764	−0.141909	−0.0709543	+	0.997480i	$0.522604\pi$
$98$	4.11061	0.415234
$99$	4.79395	0.481810
$100$	0	0
$101$	6.88244	0.684828	0.342414	−	0.939549i	$-0.388755\pi$
$101$	6.88244	0.684828	0.342414	+	0.939549i	$0.388755\pi$
$102$	14.9021	1.47553
$103$	1.37117	0.135106	0.0675528	−	0.997716i	$-0.478481\pi$
$103$	1.37117	0.135106	0.0675528	+	0.997716i	$0.478481\pi$
$104$	2.54967	0.250016
$105$	0	0
$106$	−12.1140	−1.17662
$107$	−2.28232	−0.220640	−0.110320	−	0.993896i	$-0.535188\pi$
$107$	−2.28232	−0.220640	−0.110320	+	0.993896i	$0.535188\pi$
$108$	−2.51593	−0.242096
$109$	−2.69339	−0.257980	−0.128990	−	0.991646i	$-0.541174\pi$
$109$	−2.69339	−0.257980	−0.128990	+	0.991646i	$0.541174\pi$
$110$	0	0
$111$	−19.7146	−1.87123
$112$	−10.5693	−0.998707
$113$	16.5589	1.55773	0.778865	−	0.627191i	$-0.215796\pi$
$113$	16.5589	1.55773	0.778865	+	0.627191i	$0.215796\pi$
$114$	−16.0980	−1.50771
$115$	0	0
$116$	−1.13787	−0.105649
$117$	1.92199	0.177688
$118$	−13.2036	−1.21549
$119$	−8.93546	−0.819112
$120$	0	0
$121$	−0.998863	−0.0908057
$122$	0.191926	0.0173762
$123$	23.4327	2.11286
$124$	−0.523712	−0.0470307
$125$	0	0
$126$	−5.40403	−0.481430
$127$	14.6327	1.29844	0.649219	−	0.760601i	$-0.275096\pi$
$127$	14.6327	1.29844	0.649219	+	0.760601i	$0.275096\pi$
$128$	−13.1809	−1.16504
$129$	16.7601	1.47564
$130$	0	0
$131$	−9.91795	−0.866536	−0.433268	−	0.901265i	$-0.642640\pi$
$131$	−9.91795	−0.866536	−0.433268	+	0.901265i	$0.642640\pi$
$132$	5.36117	0.466630
$133$	9.65251	0.836979
$134$	3.89429	0.336415
$135$	0	0
$136$	−8.43089	−0.722942
$137$	−13.1186	−1.12079	−0.560397	−	0.828224i	$-0.689351\pi$
$137$	−13.1186	−1.12079	−0.560397	+	0.828224i	$0.689351\pi$
$138$	17.3046	1.47306
$139$	6.70586	0.568783	0.284392	−	0.958708i	$-0.408208\pi$
$139$	6.70586	0.568783	0.284392	+	0.958708i	$0.408208\pi$
$140$	0	0
$141$	11.5028	0.968712
$142$	−14.8025	−1.24220
$143$	4.00966	0.335305
$144$	−7.51746	−0.626455
$145$	0	0
$146$	−12.8830	−1.06620
$147$	−5.22245	−0.430740
$148$	−7.40079	−0.608341
$149$	7.22421	0.591830	0.295915	−	0.955214i	$-0.404375\pi$
$149$	7.22421	0.591830	0.295915	+	0.955214i	$0.404375\pi$
$150$	0	0
$151$	4.83007	0.393065	0.196533	−	0.980497i	$-0.437032\pi$
$151$	4.83007	0.393065	0.196533	+	0.980497i	$0.437032\pi$
$152$	9.10745	0.738712
$153$	−6.35537	−0.513801
$154$	−11.2739	−0.908476
$155$	0	0
$156$	2.14940	0.172090
$157$	−13.7188	−1.09488	−0.547438	−	0.836846i	$-0.684397\pi$
$157$	−13.7188	−1.09488	−0.547438	+	0.836846i	$0.684397\pi$
$158$	−24.8858	−1.97981
$159$	15.3906	1.22055
$160$	0	0
$161$	−10.3760	−0.817743
$162$	18.8168	1.47839
$163$	−19.0140	−1.48929	−0.744645	−	0.667461i	$-0.767381\pi$
$163$	−19.0140	−1.48929	−0.744645	+	0.667461i	$0.767381\pi$
$164$	8.79657	0.686897
$165$	0	0
$166$	15.3491	1.19132
$167$	5.86594	0.453920	0.226960	−	0.973904i	$-0.427121\pi$
$167$	5.86594	0.453920	0.226960	+	0.973904i	$0.427121\pi$
$168$	9.10790	0.702690
$169$	−11.3924	−0.876342
$170$	0	0
$171$	6.86538	0.525009
$172$	6.29168	0.479736
$173$	−9.48368	−0.721031	−0.360515	−	0.932753i	$-0.617399\pi$
$173$	−9.48368	−0.721031	−0.360515	+	0.932753i	$0.617399\pi$
$174$	5.06997	0.384354
$175$	0	0
$176$	−15.6829	−1.18214
$177$	16.7749	1.26088
$178$	−19.2380	−1.44195
$179$	7.02742	0.525254	0.262627	−	0.964897i	$-0.415411\pi$
$179$	7.02742	0.525254	0.262627	+	0.964897i	$0.415411\pi$
$180$	0	0
$181$	−7.90416	−0.587511	−0.293756	−	0.955881i	$-0.594905\pi$
$181$	−7.90416	−0.587511	−0.293756	+	0.955881i	$0.594905\pi$
$182$	−4.51993	−0.335040
$183$	−0.243838	−0.0180251
$184$	−9.79008	−0.721734
$185$	0	0
$186$	2.33349	0.171100
$187$	−13.2586	−0.969562
$188$	4.31812	0.314931
$189$	−6.72174	−0.488935
$190$	0	0
$191$	−15.6154	−1.12989	−0.564945	−	0.825129i	$-0.691103\pi$
$191$	−15.6154	−1.12989	−0.564945	+	0.825129i	$0.691103\pi$
$192$	5.88884	0.424991
$193$	3.69901	0.266261	0.133130	−	0.991099i	$-0.457497\pi$
$193$	3.69901	0.266261	0.133130	+	0.991099i	$0.457497\pi$
$194$	2.33775	0.167841
$195$	0	0
$196$	−1.96049	−0.140035
$197$	22.8950	1.63120	0.815601	−	0.578614i	$-0.196406\pi$
$197$	22.8950	1.63120	0.815601	+	0.578614i	$0.196406\pi$
$198$	−8.01858	−0.569856
$199$	20.4401	1.44896	0.724480	−	0.689296i	$-0.242080\pi$
$199$	20.4401	1.44896	0.724480	+	0.689296i	$0.242080\pi$
$200$	0	0
$201$	−4.94762	−0.348978
$202$	−11.5119	−0.809973
$203$	−3.04001	−0.213367
$204$	−7.10733	−0.497612
$205$	0	0
$206$	−2.29348	−0.159795
$207$	−7.37996	−0.512943
$208$	−6.28760	−0.435967
$209$	14.3225	0.990711
$210$	0	0
$211$	3.66994	0.252649	0.126325	−	0.991989i	$-0.459682\pi$
$211$	3.66994	0.252649	0.126325	+	0.991989i	$0.459682\pi$
$212$	5.77758	0.396806
$213$	18.8063	1.28858
$214$	3.81752	0.260960
$215$	0	0
$216$	−6.34217	−0.431530
$217$	−1.39918	−0.0949829
$218$	4.50509	0.305123
$219$	16.3676	1.10602
$220$	0	0
$221$	−5.31563	−0.357568
$222$	32.9755	2.21317
$223$	−25.7588	−1.72494	−0.862470	−	0.506109i	$-0.831083\pi$
$223$	−25.7588	−1.72494	−0.862470	+	0.506109i	$0.831083\pi$
$224$	9.10684	0.608477
$225$	0	0
$226$	−27.6972	−1.84239
$227$	−23.3939	−1.55271	−0.776355	−	0.630296i	$-0.782934\pi$
$227$	−23.3939	−1.55271	−0.776355	+	0.630296i	$0.782934\pi$
$228$	7.67768	0.508467
$229$	−13.9789	−0.923753	−0.461877	−	0.886944i	$-0.652824\pi$
$229$	−13.9789	−0.923753	−0.461877	+	0.886944i	$0.652824\pi$
$230$	0	0
$231$	14.3233	0.942401
$232$	−2.86834	−0.188316
$233$	21.1400	1.38493	0.692464	−	0.721452i	$-0.256525\pi$
$233$	21.1400	1.38493	0.692464	+	0.721452i	$0.256525\pi$
$234$	−3.21481	−0.210159
$235$	0	0
$236$	6.29726	0.409917
$237$	31.6169	2.05374
$238$	14.9459	0.968796
$239$	26.8848	1.73903	0.869517	−	0.493902i	$-0.164430\pi$
$239$	26.8848	1.73903	0.869517	+	0.493902i	$0.164430\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	1.67074	0.107399
$243$	−14.4450	−0.926646
$244$	−0.0915362	−0.00586000
$245$	0	0
$246$	−39.1946	−2.49896
$247$	5.74220	0.365367
$248$	−1.32017	−0.0838312
$249$	−19.5007	−1.23581
$250$	0	0
$251$	−0.416018	−0.0262588	−0.0131294	−	0.999914i	$-0.504179\pi$
$251$	−0.416018	−0.0262588	−0.0131294	+	0.999914i	$0.504179\pi$
$252$	2.57737	0.162359
$253$	−15.3961	−0.967942
$254$	−24.4753	−1.53571
$255$	0	0
$256$	16.5047	1.03155
$257$	−7.06231	−0.440535	−0.220267	−	0.975440i	$-0.570693\pi$
$257$	−7.06231	−0.440535	−0.220267	+	0.975440i	$0.570693\pi$
$258$	−28.0337	−1.74530
$259$	−19.7725	−1.22860
$260$	0	0
$261$	−2.16222	−0.133838
$262$	16.5892	1.02489
$263$	18.1126	1.11687	0.558436	−	0.829547i	$-0.311402\pi$
$263$	18.1126	1.11687	0.558436	+	0.829547i	$0.311402\pi$
$264$	13.5144	0.831756
$265$	0	0
$266$	−16.1452	−0.989928
$267$	24.4416	1.49580
$268$	−1.85732	−0.113454
$269$	11.8544	0.722778	0.361389	−	0.932415i	$-0.382303\pi$
$269$	11.8544	0.722778	0.361389	+	0.932415i	$0.382303\pi$
$270$	0	0
$271$	20.5759	1.24989	0.624947	−	0.780667i	$-0.285120\pi$
$271$	20.5759	1.24989	0.624947	+	0.780667i	$0.285120\pi$
$272$	20.7909	1.26064
$273$	5.74249	0.347551
$274$	21.9427	1.32561
$275$	0	0
$276$	−8.25314	−0.496781
$277$	27.3049	1.64059	0.820295	−	0.571941i	$-0.193809\pi$
$277$	27.3049	1.64059	0.820295	+	0.571941i	$0.193809\pi$
$278$	−11.2165	−0.672722
$279$	−0.995174	−0.0595795
$280$	0	0
$281$	16.6924	0.995784	0.497892	−	0.867239i	$-0.334108\pi$
$281$	16.6924	0.995784	0.497892	+	0.867239i	$0.334108\pi$
$282$	−19.2401	−1.14573
$283$	7.21998	0.429183	0.214592	−	0.976704i	$-0.431158\pi$
$283$	7.21998	0.429183	0.214592	+	0.976704i	$0.431158\pi$
$284$	7.05982	0.418923
$285$	0	0
$286$	−6.70674	−0.396578
$287$	23.5015	1.38725
$288$	6.47727	0.381677
$289$	0.576952	0.0339384
$290$	0	0
$291$	−2.97007	−0.174109
$292$	6.14435	0.359571
$293$	5.96161	0.348281	0.174140	−	0.984721i	$-0.444285\pi$
$293$	5.96161	0.348281	0.174140	+	0.984721i	$0.444285\pi$
$294$	8.73531	0.509454
$295$	0	0
$296$	−18.6559	−1.08435
$297$	−9.97381	−0.578739
$298$	−12.0835	−0.699981
$299$	−6.17260	−0.356970
$300$	0	0
$301$	16.8093	0.968870
$302$	−8.07899	−0.464894
$303$	14.6256	0.840220
$304$	−22.4594	−1.28813
$305$	0	0
$306$	10.6303	0.607693
$307$	−15.3240	−0.874588	−0.437294	−	0.899319i	$-0.644063\pi$
$307$	−15.3240	−0.874588	−0.437294	+	0.899319i	$0.644063\pi$
$308$	5.37691	0.306378
$309$	2.91383	0.165762
$310$	0	0
$311$	−3.77975	−0.214330	−0.107165	−	0.994241i	$-0.534177\pi$
$311$	−3.77975	−0.214330	−0.107165	+	0.994241i	$0.534177\pi$
$312$	5.41822	0.306746
$313$	13.6778	0.773116	0.386558	−	0.922265i	$-0.373664\pi$
$313$	13.6778	0.773116	0.386558	+	0.922265i	$0.373664\pi$
$314$	22.9466	1.29495
$315$	0	0
$316$	11.8689	0.667677
$317$	33.0055	1.85377	0.926887	−	0.375341i	$-0.122474\pi$
$317$	33.0055	1.85377	0.926887	+	0.375341i	$0.122474\pi$
$318$	−25.7430	−1.44360
$319$	−4.51081	−0.252557
$320$	0	0
$321$	−4.85008	−0.270705
$322$	17.3554	0.967177
$323$	−18.9875	−1.05649
$324$	−8.97440	−0.498578
$325$	0	0
$326$	31.8036	1.76144
$327$	−5.72363	−0.316518
$328$	22.1744	1.22438
$329$	11.5366	0.636033
$330$	0	0
$331$	31.1530	1.71232	0.856162	−	0.516708i	$-0.172843\pi$
$331$	31.1530	1.71232	0.856162	+	0.516708i	$0.172843\pi$
$332$	−7.32050	−0.401765
$333$	−14.0632	−0.770660
$334$	−9.81164	−0.536869
$335$	0	0
$336$	−22.4605	−1.22532
$337$	5.24807	0.285881	0.142940	−	0.989731i	$-0.454344\pi$
$337$	5.24807	0.285881	0.142940	+	0.989731i	$0.454344\pi$
$338$	19.0555	1.03648
$339$	35.1887	1.91119
$340$	0	0
$341$	−2.07613	−0.112429
$342$	−11.4833	−0.620948
$343$	−20.1569	−1.08837
$344$	15.8601	0.855118
$345$	0	0
$346$	15.8628	0.852791
$347$	−26.8277	−1.44019	−0.720093	−	0.693878i	$-0.755901\pi$
$347$	−26.8277	−1.44019	−0.720093	+	0.693878i	$0.755901\pi$
$348$	−2.41805	−0.129621
$349$	−2.87879	−0.154098	−0.0770491	−	0.997027i	$-0.524550\pi$
$349$	−2.87879	−0.154098	−0.0770491	+	0.997027i	$0.524550\pi$
$350$	0	0
$351$	−3.99871	−0.213435
$352$	13.5129	0.720238
$353$	−22.4843	−1.19672	−0.598360	−	0.801228i	$-0.704181\pi$
$353$	−22.4843	−1.19672	−0.598360	+	0.801228i	$0.704181\pi$
$354$	−28.0585	−1.49129
$355$	0	0
$356$	9.17529	0.486289
$357$	−18.9884	−1.00497
$358$	−11.7544	−0.621238
$359$	−23.9779	−1.26551	−0.632753	−	0.774353i	$-0.718075\pi$
$359$	−23.9779	−1.26551	−0.632753	+	0.774353i	$0.718075\pi$
$360$	0	0
$361$	1.51118	0.0795360
$362$	13.2209	0.694873
$363$	−2.12265	−0.111410
$364$	2.15571	0.112990
$365$	0	0
$366$	0.407855	0.0213189
$367$	−3.14019	−0.163917	−0.0819583	−	0.996636i	$-0.526117\pi$
$367$	−3.14019	−0.163917	−0.0819583	+	0.996636i	$0.526117\pi$
$368$	24.1428	1.25853
$369$	16.7155	0.870175
$370$	0	0
$371$	15.4358	0.801386
$372$	−1.11292	−0.0577023
$373$	−3.02847	−0.156808	−0.0784041	−	0.996922i	$-0.524982\pi$
$373$	−3.02847	−0.156808	−0.0784041	+	0.996922i	$0.524982\pi$
$374$	22.1769	1.14674
$375$	0	0
$376$	10.8851	0.561358
$377$	−1.80848	−0.0931413
$378$	11.2431	0.578282
$379$	5.96663	0.306485	0.153243	−	0.988189i	$-0.451028\pi$
$379$	5.96663	0.306485	0.153243	+	0.988189i	$0.451028\pi$
$380$	0	0
$381$	31.0953	1.59306
$382$	26.1190	1.33636
$383$	−0.185474	−0.00947728	−0.00473864	−	0.999989i	$-0.501508\pi$
$383$	−0.185474	−0.00947728	−0.00473864	+	0.999989i	$0.501508\pi$
$384$	−28.0103	−1.42940
$385$	0	0
$386$	−6.18714	−0.314917
$387$	11.9556	0.607740
$388$	−1.11496	−0.0566033
$389$	−10.3188	−0.523183	−0.261592	−	0.965179i	$-0.584247\pi$
$389$	−10.3188	−0.523183	−0.261592	+	0.965179i	$0.584247\pi$
$390$	0	0
$391$	20.4106	1.03221
$392$	−4.94201	−0.249609
$393$	−21.0763	−1.06316
$394$	−38.2952	−1.92929
$395$	0	0
$396$	3.82434	0.192180
$397$	−13.7263	−0.688904	−0.344452	−	0.938804i	$-0.611935\pi$
$397$	−13.7263	−0.688904	−0.344452	+	0.938804i	$0.611935\pi$
$398$	−34.1890	−1.71374
$399$	20.5122	1.02689
$400$	0	0
$401$	11.4875	0.573660	0.286830	−	0.957981i	$-0.407398\pi$
$401$	11.4875	0.573660	0.286830	+	0.957981i	$0.407398\pi$
$402$	8.27561	0.412750
$403$	−0.832363	−0.0414630
$404$	5.49041	0.273158
$405$	0	0
$406$	5.08486	0.252357
$407$	−29.3387	−1.45426
$408$	−17.9162	−0.886982
$409$	3.82387	0.189078	0.0945391	−	0.995521i	$-0.469862\pi$
$409$	3.82387	0.189078	0.0945391	+	0.995521i	$0.469862\pi$
$410$	0	0
$411$	−27.8778	−1.37511
$412$	1.09384	0.0538897
$413$	16.8242	0.827864
$414$	12.3441	0.606677
$415$	0	0
$416$	5.41759	0.265619
$417$	14.2504	0.697844
$418$	−23.9565	−1.17175
$419$	−29.4526	−1.43885	−0.719427	−	0.694568i	$-0.755596\pi$
$419$	−29.4526	−1.43885	−0.719427	+	0.694568i	$0.755596\pi$
$420$	0	0
$421$	2.50876	0.122270	0.0611348	−	0.998130i	$-0.480528\pi$
$421$	2.50876	0.122270	0.0611348	+	0.998130i	$0.480528\pi$
$422$	−6.13851	−0.298818
$423$	8.20543	0.398962
$424$	14.5641	0.707298
$425$	0	0
$426$	−31.4562	−1.52406
$427$	−0.244554	−0.0118348
$428$	−1.82071	−0.0880071
$429$	8.52078	0.411387
$430$	0	0
$431$	−8.33964	−0.401706	−0.200853	−	0.979621i	$-0.564371\pi$
$431$	−8.33964	−0.401706	−0.200853	+	0.979621i	$0.564371\pi$
$432$	15.6401	0.752483
$433$	−23.4028	−1.12467	−0.562334	−	0.826910i	$-0.690096\pi$
$433$	−23.4028	−1.12467	−0.562334	+	0.826910i	$0.690096\pi$
$434$	2.34034	0.112340
$435$	0	0
$436$	−2.14863	−0.102901
$437$	−22.0486	−1.05473
$438$	−27.3772	−1.30813
$439$	−1.42198	−0.0678674	−0.0339337	−	0.999424i	$-0.510804\pi$
$439$	−1.42198	−0.0678674	−0.0339337	+	0.999424i	$0.510804\pi$
$440$	0	0
$441$	−3.72539	−0.177399
$442$	8.89116	0.422910
$443$	−3.28199	−0.155932	−0.0779661	−	0.996956i	$-0.524843\pi$
$443$	−3.28199	−0.155932	−0.0779661	+	0.996956i	$0.524843\pi$
$444$	−15.7272	−0.746378
$445$	0	0
$446$	43.0854	2.04015
$447$	15.3519	0.726120
$448$	5.90613	0.279038
$449$	8.98263	0.423916	0.211958	−	0.977279i	$-0.432016\pi$
$449$	8.98263	0.423916	0.211958	+	0.977279i	$0.432016\pi$
$450$	0	0
$451$	34.8719	1.64205
$452$	13.2097	0.621334
$453$	10.2642	0.482254
$454$	39.1297	1.83645
$455$	0	0
$456$	19.3539	0.906330
$457$	−6.59620	−0.308557	−0.154279	−	0.988027i	$-0.549305\pi$
$457$	−6.59620	−0.308557	−0.154279	+	0.988027i	$0.549305\pi$
$458$	23.3818	1.09256
$459$	13.2223	0.617166
$460$	0	0
$461$	−2.29358	−0.106823	−0.0534113	−	0.998573i	$-0.517009\pi$
$461$	−2.29358	−0.106823	−0.0534113	+	0.998573i	$0.517009\pi$
$462$	−23.9577	−1.11461
$463$	36.1609	1.68054	0.840270	−	0.542169i	$-0.182397\pi$
$463$	36.1609	1.68054	0.840270	+	0.542169i	$0.182397\pi$
$464$	7.07346	0.328377
$465$	0	0
$466$	−35.3598	−1.63801
$467$	−36.4882	−1.68847	−0.844235	−	0.535973i	$-0.819945\pi$
$467$	−36.4882	−1.68847	−0.844235	+	0.535973i	$0.819945\pi$
$468$	1.53326	0.0708748
$469$	−4.96214	−0.229130
$470$	0	0
$471$	−29.1532	−1.34331
$472$	15.8741	0.730667
$473$	24.9418	1.14683
$474$	−52.8839	−2.42904
$475$	0	0
$476$	−7.12819	−0.326720
$477$	10.9787	0.502682
$478$	−44.9688	−2.05682
$479$	15.6687	0.715920	0.357960	−	0.933737i	$-0.383472\pi$
$479$	15.6687	0.715920	0.357960	+	0.933737i	$0.383472\pi$
$480$	0	0
$481$	−11.7625	−0.536323
$482$	−1.67265	−0.0761869
$483$	−22.0497	−1.00329
$484$	−0.796835	−0.0362198
$485$	0	0
$486$	24.1613	1.09598
$487$	−32.6230	−1.47829	−0.739145	−	0.673546i	$-0.764770\pi$
$487$	−32.6230	−1.47829	−0.739145	+	0.673546i	$0.764770\pi$
$488$	−0.230745	−0.0104453
$489$	−40.4059	−1.82722
$490$	0	0
$491$	35.7228	1.61215	0.806075	−	0.591814i	$-0.201588\pi$
$491$	35.7228	1.61215	0.806075	+	0.591814i	$0.201588\pi$
$492$	18.6933	0.842757
$493$	5.98001	0.269326
$494$	−9.60466	−0.432134
$495$	0	0
$496$	3.25561	0.146181
$497$	18.8615	0.846053
$498$	32.6177	1.46164
$499$	−33.1348	−1.48332	−0.741658	−	0.670778i	$-0.765960\pi$
$499$	−33.1348	−1.48332	−0.741658	+	0.670778i	$0.765960\pi$
$500$	0	0
$501$	12.4655	0.556918
$502$	0.695850	0.0310573
$503$	33.0054	1.47164	0.735818	−	0.677179i	$-0.236798\pi$
$503$	33.0054	1.47164	0.735818	+	0.677179i	$0.236798\pi$
$504$	6.49704	0.289401
$505$	0	0
$506$	25.7521	1.14482
$507$	−24.2097	−1.07519
$508$	11.6731	0.517910
$509$	7.73448	0.342825	0.171412	−	0.985199i	$-0.445167\pi$
$509$	7.73448	0.342825	0.171412	+	0.985199i	$0.445167\pi$
$510$	0	0
$511$	16.4157	0.726186
$512$	−1.24473	−0.0550097
$513$	−14.2834	−0.630628
$514$	11.8127	0.521037
$515$	0	0
$516$	13.3702	0.588591
$517$	17.1182	0.752856
$518$	33.0723	1.45311
$519$	−20.1534	−0.884637
$520$	0	0
$521$	−13.5496	−0.593617	−0.296809	−	0.954937i	$-0.595922\pi$
$521$	−13.5496	−0.593617	−0.296809	+	0.954937i	$0.595922\pi$
$522$	3.61662	0.158295
$523$	−6.18312	−0.270369	−0.135184	−	0.990820i	$-0.543163\pi$
$523$	−6.18312	−0.270369	−0.135184	+	0.990820i	$0.543163\pi$
$524$	−7.91197	−0.345636
$525$	0	0
$526$	−30.2960	−1.32097
$527$	2.75234	0.119894
$528$	−33.3272	−1.45038
$529$	0.701170	0.0304856
$530$	0	0
$531$	11.9662	0.519291
$532$	7.70022	0.333847
$533$	13.9808	0.605578
$534$	−40.8821	−1.76914
$535$	0	0
$536$	−4.68194	−0.202229
$537$	14.9337	0.644437
$538$	−19.8283	−0.854857
$539$	−7.77190	−0.334759
$540$	0	0
$541$	2.97219	0.127784	0.0638922	−	0.997957i	$-0.479649\pi$
$541$	2.97219	0.127784	0.0638922	+	0.997957i	$0.479649\pi$
$542$	−34.4161	−1.47830
$543$	−16.7968	−0.720821
$544$	−17.9141	−0.768060
$545$	0	0
$546$	−9.60514	−0.411062
$547$	−20.5142	−0.877124	−0.438562	−	0.898701i	$-0.644512\pi$
$547$	−20.5142	−0.877124	−0.438562	+	0.898701i	$0.644512\pi$
$548$	−10.4652	−0.447052
$549$	−0.173940	−0.00742358
$550$	0	0
$551$	−6.45989	−0.275201
$552$	−20.8045	−0.885500
$553$	31.7097	1.34843
$554$	−45.6713	−1.94039
$555$	0	0
$556$	5.34955	0.226871
$557$	−21.9225	−0.928888	−0.464444	−	0.885602i	$-0.653746\pi$
$557$	−21.9225	−0.928888	−0.464444	+	0.885602i	$0.653746\pi$
$558$	1.66457	0.0704670
$559$	9.99970	0.422942
$560$	0	0
$561$	−28.1753	−1.18956
$562$	−27.9204	−1.17775
$563$	10.6405	0.448443	0.224222	−	0.974538i	$-0.428016\pi$
$563$	10.6405	0.448443	0.224222	+	0.974538i	$0.428016\pi$
$564$	9.17628	0.386391
$565$	0	0
$566$	−12.0765	−0.507612
$567$	−23.9766	−1.00692
$568$	17.7964	0.746720
$569$	−4.17956	−0.175216	−0.0876082	−	0.996155i	$-0.527922\pi$
$569$	−4.17956	−0.175216	−0.0876082	+	0.996155i	$0.527922\pi$
$570$	0	0
$571$	15.9159	0.666061	0.333031	−	0.942916i	$-0.391929\pi$
$571$	15.9159	0.666061	0.333031	+	0.942916i	$0.391929\pi$
$572$	3.19868	0.133743
$573$	−33.1837	−1.38627
$574$	−39.3097	−1.64075
$575$	0	0
$576$	4.20075	0.175031
$577$	−33.6759	−1.40195	−0.700974	−	0.713187i	$-0.747251\pi$
$577$	−33.6759	−1.40195	−0.700974	+	0.713187i	$0.747251\pi$
$578$	−0.965037	−0.0401402
$579$	7.86064	0.326677
$580$	0	0
$581$	−19.5579	−0.811400
$582$	4.96787	0.205925
$583$	22.9038	0.948580
$584$	15.4887	0.640926
$585$	0	0
$586$	−9.97165	−0.411925
$587$	18.0797	0.746228	0.373114	−	0.927786i	$-0.378290\pi$
$587$	18.0797	0.746228	0.373114	+	0.927786i	$0.378290\pi$
$588$	−4.16617	−0.171810
$589$	−2.97321	−0.122509
$590$	0	0
$591$	48.6534	2.00133
$592$	46.0063	1.89085
$593$	28.4760	1.16937	0.584684	−	0.811261i	$-0.301219\pi$
$593$	28.4760	1.16937	0.584684	+	0.811261i	$0.301219\pi$
$594$	16.6827	0.684497
$595$	0	0
$596$	5.76306	0.236064
$597$	43.4365	1.77774
$598$	10.3246	0.422203
$599$	−45.4297	−1.85621	−0.928103	−	0.372324i	$-0.878561\pi$
$599$	−45.4297	−1.85621	−0.928103	+	0.372324i	$0.878561\pi$
$600$	0	0
$601$	21.6197	0.881887	0.440943	−	0.897535i	$-0.354644\pi$
$601$	21.6197	0.881887	0.440943	+	0.897535i	$0.354644\pi$
$602$	−28.1160	−1.14592
$603$	−3.52934	−0.143726
$604$	3.85315	0.156782
$605$	0	0
$606$	−24.4635	−0.993761
$607$	42.6283	1.73023	0.865114	−	0.501576i	$-0.167246\pi$
$607$	42.6283	1.73023	0.865114	+	0.501576i	$0.167246\pi$
$608$	19.3517	0.784813
$609$	−6.46021	−0.261781
$610$	0	0
$611$	6.86302	0.277648
$612$	−5.06995	−0.204941
$613$	−2.24542	−0.0906915	−0.0453457	−	0.998971i	$-0.514439\pi$
$613$	−2.24542	−0.0906915	−0.0453457	+	0.998971i	$0.514439\pi$
$614$	25.6317	1.03441
$615$	0	0
$616$	13.5541	0.546111
$617$	24.1415	0.971901	0.485951	−	0.873986i	$-0.338474\pi$
$617$	24.1415	0.971901	0.485951	+	0.873986i	$0.338474\pi$
$618$	−4.87380	−0.196053
$619$	−46.0057	−1.84912	−0.924562	−	0.381032i	$-0.875569\pi$
$619$	−46.0057	−1.84912	−0.924562	+	0.381032i	$0.875569\pi$
$620$	0	0
$621$	15.3540	0.616135
$622$	6.32219	0.253497
$623$	24.5133	0.982106
$624$	−13.3616	−0.534890
$625$	0	0
$626$	−22.8781	−0.914395
$627$	30.4363	1.21551
$628$	−10.9440	−0.436714
$629$	38.8944	1.55082
$630$	0	0
$631$	−26.2254	−1.04402	−0.522009	−	0.852940i	$-0.674817\pi$
$631$	−26.2254	−1.04402	−0.522009	+	0.852940i	$0.674817\pi$
$632$	29.9191	1.19012
$633$	7.79885	0.309977
$634$	−55.2065	−2.19253
$635$	0	0
$636$	12.2777	0.486844
$637$	−3.11591	−0.123457
$638$	7.54499	0.298709
$639$	13.4153	0.530700
$640$	0	0
$641$	10.5183	0.415447	0.207724	−	0.978188i	$-0.433395\pi$
$641$	10.5183	0.415447	0.207724	+	0.978188i	$0.433395\pi$
$642$	8.11247	0.320173
$643$	−43.1557	−1.70190	−0.850948	−	0.525250i	$-0.823972\pi$
$643$	−43.1557	−1.70190	−0.850948	+	0.525250i	$0.823972\pi$
$644$	−8.27737	−0.326174
$645$	0	0
$646$	31.7593	1.24955
$647$	33.7790	1.32799	0.663995	−	0.747737i	$-0.268860\pi$
$647$	33.7790	1.32799	0.663995	+	0.747737i	$0.268860\pi$
$648$	−22.6227	−0.888703
$649$	24.9640	0.979921
$650$	0	0
$651$	−2.97336	−0.116535
$652$	−15.1683	−0.594035
$653$	3.37444	0.132052	0.0660260	−	0.997818i	$-0.478968\pi$
$653$	3.37444	0.132052	0.0660260	+	0.997818i	$0.478968\pi$
$654$	9.57360	0.374358
$655$	0	0
$656$	−54.6831	−2.13501
$657$	11.6757	0.455512
$658$	−19.2966	−0.752261
$659$	−5.11576	−0.199282	−0.0996410	−	0.995023i	$-0.531769\pi$
$659$	−5.11576	−0.199282	−0.0996410	+	0.995023i	$0.531769\pi$
$660$	0	0
$661$	−8.32139	−0.323664	−0.161832	−	0.986818i	$-0.551740\pi$
$661$	−8.32139	−0.323664	−0.161832	+	0.986818i	$0.551740\pi$
$662$	−52.1079	−2.02523
$663$	−11.2961	−0.438702
$664$	−18.4535	−0.716136
$665$	0	0
$666$	23.5228	0.911489
$667$	6.94408	0.268876
$668$	4.67951	0.181056
$669$	−54.7392	−2.11634
$670$	0	0
$671$	−0.362873	−0.0140086
$672$	19.3526	0.746543
$673$	37.9588	1.46320	0.731602	−	0.681732i	$-0.238773\pi$
$673$	37.9588	1.46320	0.731602	+	0.681732i	$0.238773\pi$
$674$	−8.77816	−0.338122
$675$	0	0
$676$	−9.08824	−0.349548
$677$	26.5637	1.02093	0.510464	−	0.859899i	$-0.329474\pi$
$677$	26.5637	1.02093	0.510464	+	0.859899i	$0.329474\pi$
$678$	−58.8582	−2.26044
$679$	−2.97879	−0.114315
$680$	0	0
$681$	−49.7136	−1.90503
$682$	3.47263	0.132974
$683$	−0.0168420	−0.000644442 0	−0.000322221	−	1.00000i	$-0.500103\pi$
$683$	−0.0168420	−0.000644442 0	−0.000322221		1.00000i	$0.500103\pi$
$684$	5.47680	0.209411
$685$	0	0
$686$	33.7153	1.28726
$687$	−29.7061	−1.13336
$688$	−39.1116	−1.49112
$689$	9.18262	0.349830
$690$	0	0
$691$	−10.7709	−0.409743	−0.204872	−	0.978789i	$-0.565678\pi$
$691$	−10.7709	−0.409743	−0.204872	+	0.978789i	$0.565678\pi$
$692$	−7.56553	−0.287598
$693$	10.2174	0.388126
$694$	44.8732	1.70336
$695$	0	0
$696$	−6.09541	−0.231046
$697$	−46.2298	−1.75108
$698$	4.81520	0.182258
$699$	44.9239	1.69918
$700$	0	0
$701$	1.84635	0.0697358	0.0348679	−	0.999392i	$-0.488899\pi$
$701$	1.84635	0.0697358	0.0348679	+	0.999392i	$0.488899\pi$
$702$	6.68842	0.252438
$703$	−42.0156	−1.58465
$704$	8.76360	0.330291
$705$	0	0
$706$	37.6083	1.41541
$707$	14.6686	0.551668
$708$	13.3821	0.502929
$709$	−13.8085	−0.518588	−0.259294	−	0.965798i	$-0.583490\pi$
$709$	−13.8085	−0.518588	−0.259294	+	0.965798i	$0.583490\pi$
$710$	0	0
$711$	22.5536	0.845827
$712$	23.1291	0.866799
$713$	3.19606	0.119693
$714$	31.7609	1.18862
$715$	0	0
$716$	5.60607	0.209509
$717$	57.1320	2.13363
$718$	40.1066	1.49676
$719$	−2.86391	−0.106806	−0.0534029	−	0.998573i	$-0.517007\pi$
$719$	−2.86391	−0.106806	−0.0534029	+	0.998573i	$0.517007\pi$
$720$	0	0
$721$	2.92238	0.108835
$722$	−2.52768	−0.0940703
$723$	2.12506	0.0790320
$724$	−6.30548	−0.234341
$725$	0	0
$726$	3.55044	0.131769
$727$	−32.4862	−1.20485	−0.602423	−	0.798177i	$-0.705798\pi$
$727$	−32.4862	−1.20485	−0.602423	+	0.798177i	$0.705798\pi$
$728$	5.43412	0.201402
$729$	3.05275	0.113065
$730$	0	0
$731$	−33.0655	−1.22297
$732$	−0.194520	−0.00718968
$733$	−18.6192	−0.687717	−0.343859	−	0.939021i	$-0.611734\pi$
$733$	−18.6192	−0.687717	−0.343859	+	0.939021i	$0.611734\pi$
$734$	5.25243	0.193871
$735$	0	0
$736$	−20.8021	−0.766776
$737$	−7.36290	−0.271216
$738$	−27.9591	−1.02919
$739$	19.5711	0.719936	0.359968	−	0.932965i	$-0.382788\pi$
$739$	19.5711	0.719936	0.359968	+	0.932965i	$0.382788\pi$
$740$	0	0
$741$	12.2025	0.448272
$742$	−25.8186	−0.947830
$743$	−21.0254	−0.771347	−0.385674	−	0.922635i	$-0.626031\pi$
$743$	−21.0254	−0.771347	−0.385674	+	0.922635i	$0.626031\pi$
$744$	−2.80546	−0.102853
$745$	0	0
$746$	5.06555	0.185463
$747$	−13.9106	−0.508964
$748$	−10.5769	−0.386730
$749$	−4.86432	−0.177738
$750$	0	0
$751$	−11.6698	−0.425838	−0.212919	−	0.977070i	$-0.568297\pi$
$751$	−11.6698	−0.425838	−0.212919	+	0.977070i	$0.568297\pi$
$752$	−26.8432	−0.978871
$753$	−0.884064	−0.0322171
$754$	3.02494	0.110162
$755$	0	0
$756$	−5.36222	−0.195022
$757$	−10.9932	−0.399555	−0.199778	−	0.979841i	$-0.564022\pi$
$757$	−10.9932	−0.399555	−0.199778	+	0.979841i	$0.564022\pi$
$758$	−9.98006	−0.362492
$759$	−32.7176	−1.18757
$760$	0	0
$761$	−19.9884	−0.724579	−0.362290	−	0.932066i	$-0.618005\pi$
$761$	−19.9884	−0.724579	−0.362290	+	0.932066i	$0.618005\pi$
$762$	−52.0115	−1.88418
$763$	−5.74043	−0.207818
$764$	−12.4571	−0.450680
$765$	0	0
$766$	0.310232	0.0112091
$767$	10.0086	0.361388
$768$	35.0736	1.26561
$769$	24.6654	0.889457	0.444729	−	0.895665i	$-0.353300\pi$
$769$	24.6654	0.889457	0.444729	+	0.895665i	$0.353300\pi$
$770$	0	0
$771$	−15.0079	−0.540495
$772$	2.95086	0.106204
$773$	−9.65915	−0.347416	−0.173708	−	0.984797i	$-0.555575\pi$
$773$	−9.65915	−0.347416	−0.173708	+	0.984797i	$0.555575\pi$
$774$	−19.9976	−0.718797
$775$	0	0
$776$	−2.81058	−0.100894
$777$	−42.0177	−1.50738
$778$	17.2597	0.618789
$779$	49.9397	1.78928
$780$	0	0
$781$	27.9869	1.00145
$782$	−34.1398	−1.22084
$783$	4.49849	0.160763
$784$	12.1872	0.435258
$785$	0	0
$786$	35.2531	1.25744
$787$	52.7392	1.87995	0.939974	−	0.341245i	$-0.110849\pi$
$787$	52.7392	1.87995	0.939974	+	0.341245i	$0.110849\pi$
$788$	18.2643	0.650640
$789$	38.4905	1.37030
$790$	0	0
$791$	35.2920	1.25484
$792$	9.64040	0.342557
$793$	−0.145483	−0.00516626
$794$	22.9592	0.814793
$795$	0	0
$796$	16.3059	0.577948
$797$	35.0572	1.24179	0.620894	−	0.783894i	$-0.286770\pi$
$797$	35.0572	1.24179	0.620894	+	0.783894i	$0.286770\pi$
$798$	−34.3096	−1.21455
$799$	−22.6936	−0.802843
$800$	0	0
$801$	17.4352	0.616042
$802$	−19.2146	−0.678490
$803$	24.3578	0.859568
$804$	−3.94692	−0.139197
$805$	0	0
$806$	1.39225	0.0490399
$807$	25.1914	0.886780
$808$	13.8402	0.486898
$809$	7.76005	0.272829	0.136414	−	0.990652i	$-0.456442\pi$
$809$	7.76005	0.272829	0.136414	+	0.990652i	$0.456442\pi$
$810$	0	0
$811$	25.5078	0.895699	0.447849	−	0.894109i	$-0.352190\pi$
$811$	25.5078	0.895699	0.447849	+	0.894109i	$0.352190\pi$
$812$	−2.42514	−0.0851059
$813$	43.7250	1.53350
$814$	49.0732	1.72001
$815$	0	0
$816$	44.1820	1.54668
$817$	35.7190	1.24965
$818$	−6.39598	−0.223630
$819$	4.09635	0.143138
$820$	0	0
$821$	17.1740	0.599378	0.299689	−	0.954037i	$-0.403117\pi$
$821$	17.1740	0.599378	0.299689	+	0.954037i	$0.403117\pi$
$822$	46.6296	1.62639
$823$	−9.24256	−0.322175	−0.161088	−	0.986940i	$-0.551500\pi$
$823$	−9.24256	−0.322175	−0.161088	+	0.986940i	$0.551500\pi$
$824$	2.75736	0.0960571
$825$	0	0
$826$	−28.1409	−0.979147
$827$	−37.9168	−1.31850	−0.659248	−	0.751926i	$-0.729125\pi$
$827$	−37.9168	−1.31850	−0.659248	+	0.751926i	$0.729125\pi$
$828$	−5.88731	−0.204598
$829$	42.0959	1.46205	0.731026	−	0.682350i	$-0.239042\pi$
$829$	42.0959	1.46205	0.731026	+	0.682350i	$0.239042\pi$
$830$	0	0
$831$	58.0246	2.01285
$832$	3.51351	0.121809
$833$	10.3032	0.356986
$834$	−23.8358	−0.825367
$835$	0	0
$836$	11.4257	0.395166
$837$	2.07046	0.0715655
$838$	49.2638	1.70179
$839$	−42.6722	−1.47321	−0.736604	−	0.676324i	$-0.763572\pi$
$839$	−42.6722	−1.47321	−0.736604	+	0.676324i	$0.763572\pi$
$840$	0	0
$841$	−26.9655	−0.929844
$842$	−4.19627	−0.144613
$843$	35.4724	1.22173
$844$	2.92767	0.100774
$845$	0	0
$846$	−13.7248	−0.471868
$847$	−2.12888	−0.0731492
$848$	−35.9158	−1.23335
$849$	15.3429	0.526568
$850$	0	0
$851$	45.1648	1.54823
$852$	15.0026	0.513979
$853$	−17.5224	−0.599955	−0.299978	−	0.953946i	$-0.596979\pi$
$853$	−17.5224	−0.599955	−0.299978	+	0.953946i	$0.596979\pi$
$854$	0.409053	0.0139975
$855$	0	0
$856$	−4.58964	−0.156871
$857$	51.1093	1.74586	0.872930	−	0.487845i	$-0.162217\pi$
$857$	51.1093	1.74586	0.872930	+	0.487845i	$0.162217\pi$
$858$	−14.2522	−0.486564
$859$	−17.7914	−0.607033	−0.303517	−	0.952826i	$-0.598161\pi$
$859$	−17.7914	−0.607033	−0.303517	+	0.952826i	$0.598161\pi$
$860$	0	0
$861$	49.9422	1.70203
$862$	13.9493	0.475113
$863$	10.3164	0.351175	0.175587	−	0.984464i	$-0.443818\pi$
$863$	10.3164	0.351175	0.175587	+	0.984464i	$0.443818\pi$
$864$	−13.4759	−0.458461
$865$	0	0
$866$	39.1446	1.33019
$867$	1.22606	0.0416392
$868$	−1.11619	−0.0378859
$869$	47.0513	1.59611
$870$	0	0
$871$	−2.95194	−0.100023
$872$	−5.41628	−0.183418
$873$	−2.11867	−0.0717062
$874$	36.8794	1.24747
$875$	0	0
$876$	13.0571	0.441160
$877$	−10.4095	−0.351505	−0.175753	−	0.984434i	$-0.556236\pi$
$877$	−10.4095	−0.351505	−0.175753	+	0.984434i	$0.556236\pi$
$878$	2.37847	0.0802695
$879$	12.6688	0.427308
$880$	0	0
$881$	51.8731	1.74765	0.873824	−	0.486242i	$-0.161633\pi$
$881$	51.8731	1.74765	0.873824	+	0.486242i	$0.161633\pi$
$882$	6.23125	0.209817
$883$	−23.2765	−0.783315	−0.391658	−	0.920111i	$-0.628098\pi$
$883$	−23.2765	−0.783315	−0.391658	+	0.920111i	$0.628098\pi$
$884$	−4.24050	−0.142624
$885$	0	0
$886$	5.48961	0.184427
$887$	55.3930	1.85991	0.929957	−	0.367669i	$-0.119844\pi$
$887$	55.3930	1.85991	0.929957	+	0.367669i	$0.119844\pi$
$888$	−39.6450	−1.33040
$889$	31.1866	1.04597
$890$	0	0
$891$	−35.5768	−1.19187
$892$	−20.5489	−0.688029
$893$	24.5148	0.820355
$894$	−25.6783	−0.858811
$895$	0	0
$896$	−28.0925	−0.938506
$897$	−13.1172	−0.437969
$898$	−15.0247	−0.501382
$899$	0.936397	0.0312306
$900$	0	0
$901$	−30.3638	−1.01156
$902$	−58.3283	−1.94212
$903$	35.7208	1.18871
$904$	33.2991	1.10751
$905$	0	0
$906$	−17.1684	−0.570381
$907$	17.1818	0.570511	0.285256	−	0.958452i	$-0.407922\pi$
$907$	17.1818	0.570511	0.285256	+	0.958452i	$0.407922\pi$
$908$	−18.6623	−0.619331
$909$	10.4331	0.346043
$910$	0	0
$911$	−20.0139	−0.663090	−0.331545	−	0.943439i	$-0.607570\pi$
$911$	−20.0139	−0.663090	−0.331545	+	0.943439i	$0.607570\pi$
$912$	−47.7276	−1.58042
$913$	−29.0204	−0.960434
$914$	11.0331	0.364943
$915$	0	0
$916$	−11.1516	−0.368458
$917$	−21.1382	−0.698043
$918$	−22.1163	−0.729946
$919$	−49.5879	−1.63575	−0.817877	−	0.575393i	$-0.804849\pi$
$919$	−49.5879	−1.63575	−0.817877	+	0.575393i	$0.804849\pi$
$920$	0	0
$921$	−32.5645	−1.07304
$922$	3.83634	0.126343
$923$	11.2205	0.369329
$924$	11.4263	0.375897
$925$	0	0
$926$	−60.4844	−1.98764
$927$	2.07855	0.0682686
$928$	−6.09470	−0.200068
$929$	−10.0705	−0.330403	−0.165201	−	0.986260i	$-0.552827\pi$
$929$	−10.0705	−0.330403	−0.165201	+	0.986260i	$0.552827\pi$
$930$	0	0
$931$	−11.1301	−0.364773
$932$	16.8643	0.552408
$933$	−8.03222	−0.262963
$934$	61.0318	1.99702
$935$	0	0
$936$	3.86504	0.126333
$937$	7.50748	0.245259	0.122629	−	0.992453i	$-0.460867\pi$
$937$	7.50748	0.245259	0.122629	+	0.992453i	$0.460867\pi$
$938$	8.29990	0.271001
$939$	29.0662	0.948541
$940$	0	0
$941$	16.2989	0.531329	0.265665	−	0.964065i	$-0.414409\pi$
$941$	16.2989	0.531329	0.265665	+	0.964065i	$0.414409\pi$
$942$	48.7630	1.58878
$943$	−53.6828	−1.74815
$944$	−39.1463	−1.27410
$945$	0	0
$946$	−41.7189	−1.35640
$947$	−18.4711	−0.600229	−0.300114	−	0.953903i	$-0.597025\pi$
$947$	−18.4711	−0.600229	−0.300114	+	0.953903i	$0.597025\pi$
$948$	25.2221	0.819177
$949$	9.76554	0.317003
$950$	0	0
$951$	70.1388	2.27441
$952$	−17.9688	−0.582371
$953$	53.3471	1.72808	0.864040	−	0.503423i	$-0.167926\pi$
$953$	53.3471	1.72808	0.864040	+	0.503423i	$0.167926\pi$
$954$	−18.3636	−0.594542
$955$	0	0
$956$	21.4472	0.693651
$957$	−9.58576	−0.309864
$958$	−26.2081	−0.846747
$959$	−27.9596	−0.902863
$960$	0	0
$961$	−30.5690	−0.986097
$962$	19.6744	0.634329
$963$	−3.45976	−0.111489
$964$	0.797742	0.0256936
$965$	0	0
$966$	36.8813	1.18664
$967$	−58.0108	−1.86550	−0.932751	−	0.360522i	$-0.882599\pi$
$967$	−58.0108	−1.86550	−0.932751	+	0.360522i	$0.882599\pi$
$968$	−2.00866	−0.0645609
$969$	−40.3496	−1.29622
$970$	0	0
$971$	7.55522	0.242459	0.121229	−	0.992625i	$-0.461316\pi$
$971$	7.55522	0.242459	0.121229	+	0.992625i	$0.461316\pi$
$972$	−11.5234	−0.369612
$973$	14.2922	0.458187
$974$	54.5667	1.74843
$975$	0	0
$976$	0.569026	0.0182141
$977$	6.25077	0.199980	0.0999900	−	0.994988i	$-0.468119\pi$
$977$	6.25077	0.199980	0.0999900	+	0.994988i	$0.468119\pi$
$978$	67.5848	2.16112
$979$	36.3732	1.16249
$980$	0	0
$981$	−4.08290	−0.130357
$982$	−59.7516	−1.90675
$983$	31.3057	0.998496	0.499248	−	0.866459i	$-0.333610\pi$
$983$	31.3057	0.998496	0.499248	+	0.866459i	$0.333610\pi$
$984$	47.1220	1.50220
$985$	0	0
$986$	−10.0024	−0.318542
$987$	24.5160	0.780352
$988$	4.58080	0.145735
$989$	−38.3962	−1.22093
$990$	0	0
$991$	−11.0433	−0.350802	−0.175401	−	0.984497i	$-0.556122\pi$
$991$	−11.0433	−0.350802	−0.175401	+	0.984497i	$0.556122\pi$
$992$	−2.80513	−0.0890629
$993$	66.2021	2.10086
$994$	−31.5486	−1.00066
$995$	0	0
$996$	−15.5565	−0.492927
$997$	−42.3584	−1.34150	−0.670752	−	0.741682i	$-0.734028\pi$
$997$	−42.3584	−1.34150	−0.670752	+	0.741682i	$0.734028\pi$
$998$	55.4227	1.75438
$999$	29.2585	0.925698

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.k.1.6		25
5.4	even	2		1205.2.a.d.1.20	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1205.2.a.d.1.20	✓	25	5.4	even	2
6025.2.a.k.1.6		25	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:	\( N \)	\(=\)	\( 6025 = 5^{2} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6025.a (trivial)

\(f(q)\)	\(=\)	\(q-1.67265 q^{2} +2.12506 q^{3} +0.797742 q^{4} -3.55448 q^{6} +2.13130 q^{7} +2.01095 q^{8} +1.51590 q^{9} +O(q^{10})\)	\(q-1.67265 q^{2} +2.12506 q^{3} +0.797742 q^{4} -3.55448 q^{6} +2.13130 q^{7} +2.01095 q^{8} +1.51590 q^{9} +3.16246 q^{11} +1.69525 q^{12} +1.26789 q^{13} -3.56491 q^{14} -4.95909 q^{16} -4.19249 q^{17} -2.53555 q^{18} +4.52893 q^{19} +4.52915 q^{21} -5.28967 q^{22} -4.86838 q^{23} +4.27340 q^{24} -2.12074 q^{26} -3.15382 q^{27} +1.70023 q^{28} -1.42636 q^{29} -0.656493 q^{31} +4.27290 q^{32} +6.72042 q^{33} +7.01254 q^{34} +1.20929 q^{36} -9.27717 q^{37} -7.57529 q^{38} +2.69436 q^{39} +11.0268 q^{41} -7.57567 q^{42} +7.88686 q^{43} +2.52283 q^{44} +8.14308 q^{46} +5.41293 q^{47} -10.5384 q^{48} -2.45755 q^{49} -8.90930 q^{51} +1.01145 q^{52} +7.24242 q^{53} +5.27522 q^{54} +4.28594 q^{56} +9.62426 q^{57} +2.38580 q^{58} +7.89385 q^{59} -0.114744 q^{61} +1.09808 q^{62} +3.23083 q^{63} +2.77114 q^{64} -11.2409 q^{66} -2.32822 q^{67} -3.34452 q^{68} -10.3456 q^{69} +8.84974 q^{71} +3.04839 q^{72} +7.70217 q^{73} +15.5174 q^{74} +3.61292 q^{76} +6.74015 q^{77} -4.50670 q^{78} +14.8781 q^{79} -11.2497 q^{81} -18.4440 q^{82} -9.17652 q^{83} +3.61310 q^{84} -13.1919 q^{86} -3.03111 q^{87} +6.35955 q^{88} +11.5016 q^{89} +2.70227 q^{91} -3.88372 q^{92} -1.39509 q^{93} -9.05391 q^{94} +9.08018 q^{96} -1.39764 q^{97} +4.11061 q^{98} +4.79395 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 4 q^{2} - 9 q^{3} + 36 q^{4} + 7 q^{6} - 7 q^{7} + 15 q^{8} + 36 q^{9}+O(q^{10}) \) 25 * q + 4 * q^2 - 9 * q^3 + 36 * q^4 + 7 * q^6 - 7 * q^7 + 15 * q^8 + 36 * q^9	\( 25 q + 4 q^{2} - 9 q^{3} + 36 q^{4} + 7 q^{6} - 7 q^{7} + 15 q^{8} + 36 q^{9} + 10 q^{11} - 22 q^{12} - 10 q^{13} + 13 q^{14} + 54 q^{16} - q^{17} + 13 q^{18} + 50 q^{19} + 9 q^{21} - 11 q^{22} + 31 q^{23} + 22 q^{24} + 8 q^{26} - 42 q^{27} - 14 q^{28} + 4 q^{29} + 34 q^{31} + 44 q^{32} - 28 q^{33} + 33 q^{34} + 83 q^{36} - 14 q^{37} + 10 q^{38} + 23 q^{39} + 11 q^{41} - 23 q^{42} - 49 q^{43} + 20 q^{44} + 27 q^{46} + 28 q^{47} - 30 q^{48} + 66 q^{49} + 49 q^{51} - 39 q^{52} + 16 q^{53} + 5 q^{54} + 51 q^{56} - 10 q^{57} + 8 q^{58} + 30 q^{59} + 35 q^{61} + 18 q^{62} + 73 q^{64} - 13 q^{66} - 37 q^{67} - 11 q^{68} - 4 q^{69} + 12 q^{71} + 90 q^{72} - 36 q^{73} - 12 q^{74} + 57 q^{76} + 31 q^{77} + 9 q^{78} + 16 q^{79} + 65 q^{81} + 11 q^{82} - 43 q^{83} - 62 q^{84} - 9 q^{86} + 22 q^{87} - 20 q^{88} + 38 q^{89} + 86 q^{91} + 119 q^{92} - 10 q^{93} - 18 q^{94} - 34 q^{96} - 17 q^{97} + 32 q^{98} + 26 q^{99}+O(q^{100}) \) 25 * q + 4 * q^2 - 9 * q^3 + 36 * q^4 + 7 * q^6 - 7 * q^7 + 15 * q^8 + 36 * q^9 + 10 * q^11 - 22 * q^12 - 10 * q^13 + 13 * q^14 + 54 * q^16 - q^17 + 13 * q^18 + 50 * q^19 + 9 * q^21 - 11 * q^22 + 31 * q^23 + 22 * q^24 + 8 * q^26 - 42 * q^27 - 14 * q^28 + 4 * q^29 + 34 * q^31 + 44 * q^32 - 28 * q^33 + 33 * q^34 + 83 * q^36 - 14 * q^37 + 10 * q^38 + 23 * q^39 + 11 * q^41 - 23 * q^42 - 49 * q^43 + 20 * q^44 + 27 * q^46 + 28 * q^47 - 30 * q^48 + 66 * q^49 + 49 * q^51 - 39 * q^52 + 16 * q^53 + 5 * q^54 + 51 * q^56 - 10 * q^57 + 8 * q^58 + 30 * q^59 + 35 * q^61 + 18 * q^62 + 73 * q^64 - 13 * q^66 - 37 * q^67 - 11 * q^68 - 4 * q^69 + 12 * q^71 + 90 * q^72 - 36 * q^73 - 12 * q^74 + 57 * q^76 + 31 * q^77 + 9 * q^78 + 16 * q^79 + 65 * q^81 + 11 * q^82 - 43 * q^83 - 62 * q^84 - 9 * q^86 + 22 * q^87 - 20 * q^88 + 38 * q^89 + 86 * q^91 + 119 * q^92 - 10 * q^93 - 18 * q^94 - 34 * q^96 - 17 * q^97 + 32 * q^98 + 26 * q^99

Introduction

L-functions

Modular forms

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Database

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