LMFDB - Embedded newform 5225.2.a.k.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5225,2,Mod(1,5225)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5225.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5225 = 5^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5225.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-3,8,0,2,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$41.7218350561$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.131947641.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 10x^{4} + 25x^{2} - 3x - 9 $ x^6 - 10x^4 + 25x^2 - 3*x - 9
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1045)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.65636$ of defining polynomial
Character	$\chi$	$=$	5225.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.65636 q^{2} -3.32622 q^{3} +0.743534 q^{4} +5.50942 q^{6} -2.81120 q^{7} +2.08116 q^{8} +8.06374 q^{9} -1.00000 q^{11} -2.47316 q^{12} +5.98258 q^{13} +4.65636 q^{14} -4.93423 q^{16} +6.49551 q^{17} -13.3565 q^{18} -1.00000 q^{19} +9.35067 q^{21} +1.65636 q^{22} -2.77149 q^{23} -6.92241 q^{24} -9.90932 q^{26} -16.8431 q^{27} -2.09022 q^{28} -1.78540 q^{29} -3.15789 q^{31} +4.01054 q^{32} +3.32622 q^{33} -10.7589 q^{34} +5.99567 q^{36} -3.90101 q^{37} +1.65636 q^{38} -19.8994 q^{39} -4.63245 q^{41} -15.4881 q^{42} +2.40738 q^{43} -0.743534 q^{44} +4.59059 q^{46} +4.10602 q^{47} +16.4123 q^{48} +0.902838 q^{49} -21.6055 q^{51} +4.44825 q^{52} -11.5083 q^{53} +27.8983 q^{54} -5.85056 q^{56} +3.32622 q^{57} +2.95726 q^{58} +5.33014 q^{59} +7.18233 q^{61} +5.23060 q^{62} -22.6688 q^{63} +3.22555 q^{64} -5.50942 q^{66} +13.2370 q^{67} +4.82964 q^{68} +9.21858 q^{69} -0.437851 q^{71} +16.7820 q^{72} -11.2457 q^{73} +6.46148 q^{74} -0.743534 q^{76} +2.81120 q^{77} +32.9606 q^{78} +4.44176 q^{79} +31.8327 q^{81} +7.67301 q^{82} -17.7115 q^{83} +6.95254 q^{84} -3.98750 q^{86} +5.93863 q^{87} -2.08116 q^{88} +17.5897 q^{89} -16.8182 q^{91} -2.06070 q^{92} +10.5038 q^{93} -6.80106 q^{94} -13.3399 q^{96} -4.99386 q^{97} -1.49543 q^{98} -8.06374 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{3} + 8 q^{4} + 2 q^{6} - 5 q^{7} + 9 q^{9} - 6 q^{11} - 19 q^{12} + 9 q^{13} + 18 q^{14} + 4 q^{16} + 5 q^{17} - 2 q^{18} - 6 q^{19} + 3 q^{21} - 8 q^{23} - 7 q^{24} - 22 q^{26} - 30 q^{27} - 10 q^{28}+ \cdots - 9 q^{99}+O(q^{100}) $ 6 * q - 3 * q^3 + 8 * q^4 + 2 * q^6 - 5 * q^7 + 9 * q^9 - 6 * q^11 - 19 * q^12 + 9 * q^13 + 18 * q^14 + 4 * q^16 + 5 * q^17 - 2 * q^18 - 6 * q^19 + 3 * q^21 - 8 * q^23 - 7 * q^24 - 22 * q^26 - 30 * q^27 - 10 * q^28 - 5 * q^29 - q^31 - 15 * q^32 + 3 * q^33 - 22 * q^34 + 12 * q^36 - 9 * q^37 - 32 * q^39 + 25 * q^41 - 11 * q^42 - 15 * q^43 - 8 * q^44 - 16 * q^46 - 24 * q^47 + 4 * q^48 + 13 * q^49 + 27 * q^52 - 5 * q^53 - 11 * q^54 - 12 * q^56 + 3 * q^57 - 13 * q^58 + 39 * q^59 - 11 * q^61 + 42 * q^62 - 38 * q^63 - 14 * q^64 - 2 * q^66 - 24 * q^67 - 45 * q^68 + 14 * q^69 - 24 * q^71 + 61 * q^72 + 26 * q^73 + q^74 - 8 * q^76 + 5 * q^77 + 29 * q^78 + 11 * q^79 + 30 * q^81 - 8 * q^82 - 39 * q^83 + 25 * q^84 + 18 * q^86 + 16 * q^87 + 22 * q^89 - 26 * q^91 + 11 * q^92 + 6 * q^93 - 30 * q^94 - 15 * q^96 - 22 * q^97 - 33 * q^98 - 9 * q^99

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 $

$2$	−1.65636	−1.17122	−0.585612	−	0.810591i	$-0.699146\pi$
$2$	−1.65636	−1.17122	−0.585612	+	0.810591i	$0.699146\pi$
$3$	−3.32622	−1.92039	−0.960197	−	0.279323i	$-0.909890\pi$
$3$	−3.32622	−1.92039	−0.960197	+	0.279323i	$0.909890\pi$
$4$	0.743534	0.371767
$5$	0	0
$5$	0	0
$6$	5.50942	2.24921
$7$	−2.81120	−1.06253	−0.531267	−	0.847205i	$-0.678284\pi$
$7$	−2.81120	−1.06253	−0.531267	+	0.847205i	$0.678284\pi$
$8$	2.08116	0.735802
$9$	8.06374	2.68791
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−2.47316	−0.713939
$13$	5.98258	1.65927	0.829635	−	0.558306i	$-0.188549\pi$
$13$	5.98258	1.65927	0.829635	+	0.558306i	$0.188549\pi$
$14$	4.65636	1.24446
$15$	0	0
$16$	−4.93423	−1.23356
$17$	6.49551	1.57539	0.787697	−	0.616063i	$-0.211273\pi$
$17$	6.49551	1.57539	0.787697	+	0.616063i	$0.211273\pi$
$18$	−13.3565	−3.14815
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	9.35067	2.04048
$22$	1.65636	0.353137
$23$	−2.77149	−0.577895	−0.288948	−	0.957345i	$-0.593305\pi$
$23$	−2.77149	−0.577895	−0.288948	+	0.957345i	$0.593305\pi$
$24$	−6.92241	−1.41303
$25$	0	0
$26$	−9.90932	−1.94338
$27$	−16.8431	−3.24146
$28$	−2.09022	−0.395015
$29$	−1.78540	−0.331540	−0.165770	−	0.986164i	$-0.553011\pi$
$29$	−1.78540	−0.331540	−0.165770	+	0.986164i	$0.553011\pi$
$30$	0	0
$31$	−3.15789	−0.567173	−0.283587	−	0.958947i	$-0.591524\pi$
$31$	−3.15789	−0.567173	−0.283587	+	0.958947i	$0.591524\pi$
$32$	4.01054	0.708969
$33$	3.32622	0.579021
$34$	−10.7589	−1.84514
$35$	0	0
$36$	5.99567	0.999278
$37$	−3.90101	−0.641322	−0.320661	−	0.947194i	$-0.603905\pi$
$37$	−3.90101	−0.641322	−0.320661	+	0.947194i	$0.603905\pi$
$38$	1.65636	0.268697
$39$	−19.8994	−3.18645
$40$	0	0
$41$	−4.63245	−0.723467	−0.361734	−	0.932281i	$-0.617815\pi$
$41$	−4.63245	−0.723467	−0.361734	+	0.932281i	$0.617815\pi$
$42$	−15.4881	−2.38986
$43$	2.40738	0.367122	0.183561	−	0.983008i	$-0.441237\pi$
$43$	2.40738	0.367122	0.183561	+	0.983008i	$0.441237\pi$
$44$	−0.743534	−0.112092
$45$	0	0
$46$	4.59059	0.676845
$47$	4.10602	0.598925	0.299462	−	0.954108i	$-0.403193\pi$
$47$	4.10602	0.598925	0.299462	+	0.954108i	$0.403193\pi$
$48$	16.4123	2.36891
$49$	0.902838	0.128977
$50$	0	0
$51$	−21.6055	−3.02538
$52$	4.44825	0.616862
$53$	−11.5083	−1.58078	−0.790391	−	0.612603i	$-0.790123\pi$
$53$	−11.5083	−1.58078	−0.790391	+	0.612603i	$0.790123\pi$
$54$	27.8983	3.79648
$55$	0	0
$56$	−5.85056	−0.781814
$57$	3.32622	0.440569
$58$	2.95726	0.388308
$59$	5.33014	0.693925	0.346963	−	0.937879i	$-0.387213\pi$
$59$	5.33014	0.693925	0.346963	+	0.937879i	$0.387213\pi$
$60$	0	0
$61$	7.18233	0.919603	0.459802	−	0.888022i	$-0.347920\pi$
$61$	7.18233	0.919603	0.459802	+	0.888022i	$0.347920\pi$
$62$	5.23060	0.664287
$63$	−22.6688	−2.85600
$64$	3.22555	0.403194
$65$	0	0
$66$	−5.50942	−0.678163
$67$	13.2370	1.61716	0.808580	−	0.588387i	$-0.200237\pi$
$67$	13.2370	1.61716	0.808580	+	0.588387i	$0.200237\pi$
$68$	4.82964	0.585679
$69$	9.21858	1.10979
$70$	0	0
$71$	−0.437851	−0.0519633	−0.0259817	−	0.999662i	$-0.508271\pi$
$71$	−0.437851	−0.0519633	−0.0259817	+	0.999662i	$0.508271\pi$
$72$	16.7820	1.97777
$73$	−11.2457	−1.31621	−0.658103	−	0.752928i	$-0.728641\pi$
$73$	−11.2457	−1.31621	−0.658103	+	0.752928i	$0.728641\pi$
$74$	6.46148	0.751132
$75$	0	0
$76$	−0.743534	−0.0852892
$77$	2.81120	0.320366
$78$	32.9606	3.73205
$79$	4.44176	0.499737	0.249869	−	0.968280i	$-0.419613\pi$
$79$	4.44176	0.499737	0.249869	+	0.968280i	$0.419613\pi$
$80$	0	0
$81$	31.8327	3.53697
$82$	7.67301	0.847343
$83$	−17.7115	−1.94409	−0.972046	−	0.234790i	$-0.924560\pi$
$83$	−17.7115	−1.94409	−0.972046	+	0.234790i	$0.924560\pi$
$84$	6.95254	0.758584
$85$	0	0
$86$	−3.98750	−0.429983
$87$	5.93863	0.636688
$88$	−2.08116	−0.221853
$89$	17.5897	1.86451	0.932253	−	0.361807i	$-0.117840\pi$
$89$	17.5897	1.86451	0.932253	+	0.361807i	$0.117840\pi$
$90$	0	0
$91$	−16.8182	−1.76303
$92$	−2.06070	−0.214842
$93$	10.5038	1.08920
$94$	−6.80106	−0.701475
$95$	0	0
$96$	−13.3399	−1.36150
$97$	−4.99386	−0.507049	−0.253525	−	0.967329i	$-0.581590\pi$
$97$	−4.99386	−0.507049	−0.253525	+	0.967329i	$0.581590\pi$
$98$	−1.49543	−0.151061
$99$	−8.06374	−0.810437
$100$	0	0
$101$	−16.7267	−1.66437	−0.832186	−	0.554497i	$-0.812911\pi$
$101$	−16.7267	−1.66437	−0.832186	+	0.554497i	$0.812911\pi$
$102$	35.7865	3.54340
$103$	−4.48754	−0.442171	−0.221085	−	0.975254i	$-0.570960\pi$
$103$	−4.48754	−0.442171	−0.221085	+	0.975254i	$0.570960\pi$
$104$	12.4507	1.22089
$105$	0	0
$106$	19.0618	1.85145
$107$	−1.05672	−0.102157	−0.0510783	−	0.998695i	$-0.516266\pi$
$107$	−1.05672	−0.102157	−0.0510783	+	0.998695i	$0.516266\pi$
$108$	−12.5234	−1.20507
$109$	−3.48106	−0.333425	−0.166712	−	0.986006i	$-0.553315\pi$
$109$	−3.48106	−0.333425	−0.166712	+	0.986006i	$0.553315\pi$
$110$	0	0
$111$	12.9756	1.23159
$112$	13.8711	1.31069
$113$	8.94084	0.841084	0.420542	−	0.907273i	$-0.361840\pi$
$113$	8.94084	0.841084	0.420542	+	0.907273i	$0.361840\pi$
$114$	−5.50942	−0.516005
$115$	0	0
$116$	−1.32750	−0.123256
$117$	48.2420	4.45998
$118$	−8.82864	−0.812742
$119$	−18.2602	−1.67391
$120$	0	0
$121$	1.00000	0.0909091
$122$	−11.8965	−1.07706
$123$	15.4086	1.38934
$124$	−2.34800	−0.210856
$125$	0	0
$126$	37.5477	3.34502
$127$	−2.14707	−0.190522	−0.0952609	−	0.995452i	$-0.530369\pi$
$127$	−2.14707	−0.190522	−0.0952609	+	0.995452i	$0.530369\pi$
$128$	−13.3638	−1.18120
$129$	−8.00749	−0.705020
$130$	0	0
$131$	13.1523	1.14912	0.574562	−	0.818461i	$-0.305172\pi$
$131$	13.1523	1.14912	0.574562	+	0.818461i	$0.305172\pi$
$132$	2.47316	0.215261
$133$	2.81120	0.243762
$134$	−21.9253	−1.89406
$135$	0	0
$136$	13.5182	1.15918
$137$	0.316167	0.0270120	0.0135060	−	0.999909i	$-0.495701\pi$
$137$	0.316167	0.0270120	0.0135060	+	0.999909i	$0.495701\pi$
$138$	−15.2693	−1.29981
$139$	−4.07415	−0.345565	−0.172782	−	0.984960i	$-0.555276\pi$
$139$	−4.07415	−0.345565	−0.172782	+	0.984960i	$0.555276\pi$
$140$	0	0
$141$	−13.6575	−1.15017
$142$	0.725239	0.0608607
$143$	−5.98258	−0.500289
$144$	−39.7883	−3.31569
$145$	0	0
$146$	18.6269	1.54157
$147$	−3.00304	−0.247686
$148$	−2.90053	−0.238422
$149$	4.80464	0.393612	0.196806	−	0.980442i	$-0.436943\pi$
$149$	4.80464	0.393612	0.196806	+	0.980442i	$0.436943\pi$
$150$	0	0
$151$	−18.2399	−1.48434	−0.742169	−	0.670213i	$-0.766203\pi$
$151$	−18.2399	−1.48434	−0.742169	+	0.670213i	$0.766203\pi$
$152$	−2.08116	−0.168805
$153$	52.3782	4.23452
$154$	−4.65636	−0.375220
$155$	0	0
$156$	−14.7959	−1.18462
$157$	18.1161	1.44582	0.722910	−	0.690942i	$-0.242804\pi$
$157$	18.1161	1.44582	0.722910	+	0.690942i	$0.242804\pi$
$158$	−7.35716	−0.585304
$159$	38.2790	3.03572
$160$	0	0
$161$	7.79120	0.614033
$162$	−52.7265	−4.14259
$163$	23.9946	1.87940	0.939699	−	0.342002i	$-0.111105\pi$
$163$	23.9946	1.87940	0.939699	+	0.342002i	$0.111105\pi$
$164$	−3.44438	−0.268961
$165$	0	0
$166$	29.3367	2.27697
$167$	−3.10562	−0.240320	−0.120160	−	0.992755i	$-0.538341\pi$
$167$	−3.10562	−0.240320	−0.120160	+	0.992755i	$0.538341\pi$
$168$	19.4603	1.50139
$169$	22.7913	1.75318
$170$	0	0
$171$	−8.06374	−0.616650
$172$	1.78997	0.136484
$173$	−10.9411	−0.831839	−0.415919	−	0.909401i	$-0.636540\pi$
$173$	−10.9411	−0.831839	−0.415919	+	0.909401i	$0.636540\pi$
$174$	−9.83652	−0.745704
$175$	0	0
$176$	4.93423	0.371931
$177$	−17.7292	−1.33261
$178$	−29.1349	−2.18376
$179$	1.88648	0.141002	0.0705011	−	0.997512i	$-0.477540\pi$
$179$	1.88648	0.141002	0.0705011	+	0.997512i	$0.477540\pi$
$180$	0	0
$181$	8.50254	0.631989	0.315995	−	0.948761i	$-0.397662\pi$
$181$	8.50254	0.631989	0.315995	+	0.948761i	$0.397662\pi$
$182$	27.8571	2.06490
$183$	−23.8900	−1.76600
$184$	−5.76792	−0.425216
$185$	0	0
$186$	−17.3981	−1.27569
$187$	−6.49551	−0.474999
$188$	3.05297	0.222660
$189$	47.3494	3.44416
$190$	0	0
$191$	−23.2282	−1.68073	−0.840366	−	0.542019i	$-0.817660\pi$
$191$	−23.2282	−1.68073	−0.840366	+	0.542019i	$0.817660\pi$
$192$	−10.7289	−0.774291
$193$	−15.5701	−1.12076	−0.560380	−	0.828235i	$-0.689345\pi$
$193$	−15.5701	−1.12076	−0.560380	+	0.828235i	$0.689345\pi$
$194$	8.27163	0.593868
$195$	0	0
$196$	0.671291	0.0479493
$197$	22.8694	1.62938	0.814689	−	0.579898i	$-0.196908\pi$
$197$	22.8694	1.62938	0.814689	+	0.579898i	$0.196908\pi$
$198$	13.3565	0.949204
$199$	−23.7525	−1.68377	−0.841885	−	0.539657i	$-0.818554\pi$
$199$	−23.7525	−1.68377	−0.841885	+	0.539657i	$0.818554\pi$
$200$	0	0
$201$	−44.0292	−3.10558
$202$	27.7055	1.94935
$203$	5.01911	0.352272
$204$	−16.0644	−1.12474
$205$	0	0
$206$	7.43300	0.517881
$207$	−22.3486	−1.55333
$208$	−29.5194	−2.04680
$209$	1.00000	0.0691714
$210$	0	0
$211$	7.48728	0.515446	0.257723	−	0.966219i	$-0.417028\pi$
$211$	7.48728	0.515446	0.257723	+	0.966219i	$0.417028\pi$
$212$	−8.55679	−0.587682
$213$	1.45639	0.0997900
$214$	1.75030	0.119648
$215$	0	0
$216$	−35.0533	−2.38507
$217$	8.87745	0.602640
$218$	5.76589	0.390515
$219$	37.4056	2.52763
$220$	0	0
$221$	38.8600	2.61400
$222$	−21.4923	−1.44247
$223$	8.80258	0.589464	0.294732	−	0.955580i	$-0.404770\pi$
$223$	8.80258	0.589464	0.294732	+	0.955580i	$0.404770\pi$
$224$	−11.2744	−0.753304
$225$	0	0
$226$	−14.8093	−0.985098
$227$	12.5157	0.830695	0.415347	−	0.909663i	$-0.363660\pi$
$227$	12.5157	0.830695	0.415347	+	0.909663i	$0.363660\pi$
$228$	2.47316	0.163789
$229$	−2.31366	−0.152891	−0.0764454	−	0.997074i	$-0.524357\pi$
$229$	−2.31366	−0.152891	−0.0764454	+	0.997074i	$0.524357\pi$
$230$	0	0
$231$	−9.35067	−0.615229
$232$	−3.71570	−0.243948
$233$	13.0851	0.857234	0.428617	−	0.903486i	$-0.359001\pi$
$233$	13.0851	0.857234	0.428617	+	0.903486i	$0.359001\pi$
$234$	−79.9062	−5.22363
$235$	0	0
$236$	3.96314	0.257978
$237$	−14.7743	−0.959692
$238$	30.2455	1.96052
$239$	20.9482	1.35503	0.677514	−	0.735510i	$-0.263057\pi$
$239$	20.9482	1.35503	0.677514	+	0.735510i	$0.263057\pi$
$240$	0	0
$241$	−14.2896	−0.920477	−0.460238	−	0.887795i	$-0.652236\pi$
$241$	−14.2896	−0.920477	−0.460238	+	0.887795i	$0.652236\pi$
$242$	−1.65636	−0.106475
$243$	−55.3533	−3.55092
$244$	5.34031	0.341878
$245$	0	0
$246$	−25.5221	−1.62723
$247$	−5.98258	−0.380663
$248$	−6.57207	−0.417327
$249$	58.9124	3.73342
$250$	0	0
$251$	11.1296	0.702495	0.351247	−	0.936283i	$-0.385758\pi$
$251$	11.1296	0.702495	0.351247	+	0.936283i	$0.385758\pi$
$252$	−16.8550	−1.06177
$253$	2.77149	0.174242
$254$	3.55633	0.223144
$255$	0	0
$256$	15.6841	0.980257
$257$	−7.27753	−0.453960	−0.226980	−	0.973899i	$-0.572885\pi$
$257$	−7.27753	−0.453960	−0.226980	+	0.973899i	$0.572885\pi$
$258$	13.2633	0.825737
$259$	10.9665	0.681426
$260$	0	0
$261$	−14.3970	−0.891152
$262$	−21.7850	−1.34588
$263$	16.6526	1.02684	0.513421	−	0.858137i	$-0.328378\pi$
$263$	16.6526	1.02684	0.513421	+	0.858137i	$0.328378\pi$
$264$	6.92241	0.426045
$265$	0	0
$266$	−4.65636	−0.285500
$267$	−58.5073	−3.58059
$268$	9.84217	0.601206
$269$	−7.11358	−0.433723	−0.216861	−	0.976202i	$-0.569582\pi$
$269$	−7.11358	−0.433723	−0.216861	+	0.976202i	$0.569582\pi$
$270$	0	0
$271$	11.2711	0.684669	0.342334	−	0.939578i	$-0.388782\pi$
$271$	11.2711	0.684669	0.342334	+	0.939578i	$0.388782\pi$
$272$	−32.0503	−1.94334
$273$	55.9411	3.38571
$274$	−0.523687	−0.0316371
$275$	0	0
$276$	6.85433	0.412582
$277$	4.24890	0.255292	0.127646	−	0.991820i	$-0.459258\pi$
$277$	4.24890	0.255292	0.127646	+	0.991820i	$0.459258\pi$
$278$	6.74827	0.404734
$279$	−25.4644	−1.52451
$280$	0	0
$281$	0.222832	0.0132931	0.00664653	−	0.999978i	$-0.497884\pi$
$281$	0.222832	0.0132931	0.00664653	+	0.999978i	$0.497884\pi$
$282$	22.6218	1.34711
$283$	−12.1767	−0.723828	−0.361914	−	0.932211i	$-0.617877\pi$
$283$	−12.1767	−0.723828	−0.361914	+	0.932211i	$0.617877\pi$
$284$	−0.325557	−0.0193182
$285$	0	0
$286$	9.90932	0.585950
$287$	13.0227	0.768708
$288$	32.3399	1.90565
$289$	25.1917	1.48187
$290$	0	0
$291$	16.6107	0.973734
$292$	−8.36153	−0.489322
$293$	−24.1175	−1.40896	−0.704480	−	0.709724i	$-0.748820\pi$
$293$	−24.1175	−1.40896	−0.704480	+	0.709724i	$0.748820\pi$
$294$	4.97412	0.290096
$295$	0	0
$296$	−8.11863	−0.471886
$297$	16.8431	0.977338
$298$	−7.95822	−0.461008
$299$	−16.5807	−0.958884
$300$	0	0
$301$	−6.76763	−0.390080
$302$	30.2118	1.73849
$303$	55.6368	3.19625
$304$	4.93423	0.282997
$305$	0	0
$306$	−86.7572	−4.95958
$307$	−19.7231	−1.12566	−0.562829	−	0.826573i	$-0.690287\pi$
$307$	−19.7231	−1.12566	−0.562829	+	0.826573i	$0.690287\pi$
$308$	2.09022	0.119101
$309$	14.9266	0.849142
$310$	0	0
$311$	−26.2571	−1.48890	−0.744451	−	0.667677i	$-0.767289\pi$
$311$	−26.2571	−1.48890	−0.744451	+	0.667677i	$0.767289\pi$
$312$	−41.4139	−2.34460
$313$	−2.77164	−0.156662	−0.0783311	−	0.996927i	$-0.524959\pi$
$313$	−2.77164	−0.156662	−0.0783311	+	0.996927i	$0.524959\pi$
$314$	−30.0068	−1.69338
$315$	0	0
$316$	3.30260	0.185786
$317$	−4.00082	−0.224709	−0.112354	−	0.993668i	$-0.535839\pi$
$317$	−4.00082	−0.224709	−0.112354	+	0.993668i	$0.535839\pi$
$318$	−63.4039	−3.55552
$319$	1.78540	0.0999631
$320$	0	0
$321$	3.51487	0.196181
$322$	−12.9051	−0.719170
$323$	−6.49551	−0.361420
$324$	23.6687	1.31493
$325$	0	0
$326$	−39.7437	−2.20120
$327$	11.5788	0.640307
$328$	−9.64088	−0.532329
$329$	−11.5428	−0.636377
$330$	0	0
$331$	−17.5839	−0.966500	−0.483250	−	0.875483i	$-0.660544\pi$
$331$	−17.5839	−0.966500	−0.483250	+	0.875483i	$0.660544\pi$
$332$	−13.1691	−0.722749
$333$	−31.4567	−1.72382
$334$	5.14403	0.281469
$335$	0	0
$336$	−46.1383	−2.51705
$337$	−32.7681	−1.78499	−0.892497	−	0.451054i	$-0.851048\pi$
$337$	−32.7681	−1.78499	−0.892497	+	0.451054i	$0.851048\pi$
$338$	−37.7506	−2.05336
$339$	−29.7392	−1.61521
$340$	0	0
$341$	3.15789	0.171009
$342$	13.3565	0.722236
$343$	17.1403	0.925491
$344$	5.01015	0.270129
$345$	0	0
$346$	18.1225	0.974270
$347$	19.6495	1.05484	0.527421	−	0.849604i	$-0.323159\pi$
$347$	19.6495	1.05484	0.527421	+	0.849604i	$0.323159\pi$
$348$	4.41557	0.236699
$349$	19.9610	1.06849	0.534243	−	0.845331i	$-0.320597\pi$
$349$	19.9610	1.06849	0.534243	+	0.845331i	$0.320597\pi$
$350$	0	0
$351$	−100.765	−5.37846
$352$	−4.01054	−0.213762
$353$	23.2030	1.23497	0.617486	−	0.786582i	$-0.288151\pi$
$353$	23.2030	1.23497	0.617486	+	0.786582i	$0.288151\pi$
$354$	29.3660	1.56079
$355$	0	0
$356$	13.0786	0.693162
$357$	60.7374	3.21456
$358$	−3.12469	−0.165145
$359$	−9.83399	−0.519018	−0.259509	−	0.965741i	$-0.583561\pi$
$359$	−9.83399	−0.519018	−0.259509	+	0.965741i	$0.583561\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−14.0833	−0.740201
$363$	−3.32622	−0.174581
$364$	−12.5049	−0.655436
$365$	0	0
$366$	39.5705	2.06838
$367$	−1.73835	−0.0907411	−0.0453705	−	0.998970i	$-0.514447\pi$
$367$	−1.73835	−0.0907411	−0.0453705	+	0.998970i	$0.514447\pi$
$368$	13.6751	0.712866
$369$	−37.3549	−1.94462
$370$	0	0
$371$	32.3520	1.67963
$372$	7.80995	0.404927
$373$	6.58428	0.340921	0.170460	−	0.985365i	$-0.445474\pi$
$373$	6.58428	0.340921	0.170460	+	0.985365i	$0.445474\pi$
$374$	10.7589	0.556331
$375$	0	0
$376$	8.54530	0.440690
$377$	−10.6813	−0.550114
$378$	−78.4277	−4.03389
$379$	−3.74050	−0.192137	−0.0960684	−	0.995375i	$-0.530627\pi$
$379$	−3.74050	−0.192137	−0.0960684	+	0.995375i	$0.530627\pi$
$380$	0	0
$381$	7.14163	0.365877
$382$	38.4743	1.96852
$383$	−27.5711	−1.40882	−0.704409	−	0.709795i	$-0.748788\pi$
$383$	−27.5711	−1.40882	−0.704409	+	0.709795i	$0.748788\pi$
$384$	44.4508	2.26837
$385$	0	0
$386$	25.7897	1.31266
$387$	19.4125	0.986794
$388$	−3.71310	−0.188504
$389$	−17.1560	−0.869844	−0.434922	−	0.900468i	$-0.643224\pi$
$389$	−17.1560	−0.869844	−0.434922	+	0.900468i	$0.643224\pi$
$390$	0	0
$391$	−18.0022	−0.910413
$392$	1.87895	0.0949014
$393$	−43.7475	−2.20677
$394$	−37.8800	−1.90837
$395$	0	0
$396$	−5.99567	−0.301294
$397$	14.1506	0.710201	0.355100	−	0.934828i	$-0.384447\pi$
$397$	14.1506	0.710201	0.355100	+	0.934828i	$0.384447\pi$
$398$	39.3427	1.97207
$399$	−9.35067	−0.468119
$400$	0	0
$401$	10.3891	0.518807	0.259403	−	0.965769i	$-0.416474\pi$
$401$	10.3891	0.518807	0.259403	+	0.965769i	$0.416474\pi$
$402$	72.9284	3.63734
$403$	−18.8923	−0.941093
$404$	−12.4369	−0.618758
$405$	0	0
$406$	−8.31346	−0.412590
$407$	3.90101	0.193366
$408$	−44.9646	−2.22608
$409$	4.11163	0.203307	0.101653	−	0.994820i	$-0.467587\pi$
$409$	4.11163	0.203307	0.101653	+	0.994820i	$0.467587\pi$
$410$	0	0
$411$	−1.05164	−0.0518737
$412$	−3.33664	−0.164385
$413$	−14.9841	−0.737319
$414$	37.0173	1.81930
$415$	0	0
$416$	23.9934	1.17637
$417$	13.5515	0.663621
$418$	−1.65636	−0.0810153
$419$	16.4494	0.803604	0.401802	−	0.915726i	$-0.368384\pi$
$419$	16.4494	0.803604	0.401802	+	0.915726i	$0.368384\pi$
$420$	0	0
$421$	−38.6187	−1.88216	−0.941079	−	0.338187i	$-0.890186\pi$
$421$	−38.6187	−1.88216	−0.941079	+	0.338187i	$0.890186\pi$
$422$	−12.4016	−0.603703
$423$	33.1099	1.60986
$424$	−23.9506	−1.16314
$425$	0	0
$426$	−2.41230	−0.116877
$427$	−20.1910	−0.977109
$428$	−0.785704	−0.0379784
$429$	19.8994	0.960752
$430$	0	0
$431$	−24.2965	−1.17032	−0.585161	−	0.810917i	$-0.698969\pi$
$431$	−24.2965	−1.17032	−0.585161	+	0.810917i	$0.698969\pi$
$432$	83.1078	3.99853
$433$	−22.8348	−1.09737	−0.548686	−	0.836028i	$-0.684872\pi$
$433$	−22.8348	−1.09737	−0.548686	+	0.836028i	$0.684872\pi$
$434$	−14.7043	−0.705827
$435$	0	0
$436$	−2.58828	−0.123956
$437$	2.77149	0.132578
$438$	−61.9571	−2.96043
$439$	−23.3798	−1.11586	−0.557929	−	0.829889i	$-0.688404\pi$
$439$	−23.3798	−1.11586	−0.557929	+	0.829889i	$0.688404\pi$
$440$	0	0
$441$	7.28025	0.346679
$442$	−64.3661	−3.06158
$443$	−14.6539	−0.696228	−0.348114	−	0.937452i	$-0.613178\pi$
$443$	−14.6539	−0.696228	−0.348114	+	0.937452i	$0.613178\pi$
$444$	9.64781	0.457865
$445$	0	0
$446$	−14.5803	−0.690395
$447$	−15.9813	−0.755890
$448$	−9.06766	−0.428407
$449$	19.5934	0.924672	0.462336	−	0.886705i	$-0.347011\pi$
$449$	19.5934	0.924672	0.462336	+	0.886705i	$0.347011\pi$
$450$	0	0
$451$	4.63245	0.218134
$452$	6.64782	0.312687
$453$	60.6698	2.85052
$454$	−20.7305	−0.972930
$455$	0	0
$456$	6.92241	0.324171
$457$	32.4549	1.51817	0.759087	−	0.650989i	$-0.225646\pi$
$457$	32.4549	1.51817	0.759087	+	0.650989i	$0.225646\pi$
$458$	3.83225	0.179069
$459$	−109.405	−5.10658
$460$	0	0
$461$	9.88505	0.460393	0.230196	−	0.973144i	$-0.426063\pi$
$461$	9.88505	0.460393	0.230196	+	0.973144i	$0.426063\pi$
$462$	15.4881	0.720571
$463$	−9.99709	−0.464604	−0.232302	−	0.972644i	$-0.574626\pi$
$463$	−9.99709	−0.464604	−0.232302	+	0.972644i	$0.574626\pi$
$464$	8.80956	0.408973
$465$	0	0
$466$	−21.6737	−1.00401
$467$	−26.3637	−1.21997	−0.609984	−	0.792414i	$-0.708824\pi$
$467$	−26.3637	−1.21997	−0.609984	+	0.792414i	$0.708824\pi$
$468$	35.8696	1.65807
$469$	−37.2119	−1.71829
$470$	0	0
$471$	−60.2581	−2.77655
$472$	11.0929	0.510592
$473$	−2.40738	−0.110692
$474$	24.4715	1.12402
$475$	0	0
$476$	−13.5771	−0.622304
$477$	−92.7997	−4.24901
$478$	−34.6978	−1.58704
$479$	19.3573	0.884459	0.442230	−	0.896902i	$-0.354188\pi$
$479$	19.3573	0.884459	0.442230	+	0.896902i	$0.354188\pi$
$480$	0	0
$481$	−23.3381	−1.06413
$482$	23.6688	1.07809
$483$	−25.9153	−1.17919
$484$	0.743534	0.0337970
$485$	0	0
$486$	91.6851	4.15892
$487$	−15.4579	−0.700462	−0.350231	−	0.936663i	$-0.613897\pi$
$487$	−15.4579	−0.700462	−0.350231	+	0.936663i	$0.613897\pi$
$488$	14.9476	0.676646
$489$	−79.8112	−3.60919
$490$	0	0
$491$	33.7098	1.52130	0.760650	−	0.649162i	$-0.224880\pi$
$491$	33.7098	1.52130	0.760650	+	0.649162i	$0.224880\pi$
$492$	11.4568	0.516512
$493$	−11.5971	−0.522306
$494$	9.90932	0.445841
$495$	0	0
$496$	15.5817	0.699640
$497$	1.23089	0.0552127
$498$	−97.5803	−4.37268
$499$	32.9388	1.47454	0.737272	−	0.675596i	$-0.236114\pi$
$499$	32.9388	1.47454	0.737272	+	0.675596i	$0.236114\pi$
$500$	0	0
$501$	10.3300	0.461509
$502$	−18.4347	−0.822779
$503$	−8.51652	−0.379733	−0.189866	−	0.981810i	$-0.560805\pi$
$503$	−8.51652	−0.379733	−0.189866	+	0.981810i	$0.560805\pi$
$504$	−47.1774	−2.10145
$505$	0	0
$506$	−4.59059	−0.204076
$507$	−75.8089	−3.36679
$508$	−1.59642	−0.0708297
$509$	19.8390	0.879350	0.439675	−	0.898157i	$-0.355094\pi$
$509$	19.8390	0.879350	0.439675	+	0.898157i	$0.355094\pi$
$510$	0	0
$511$	31.6138	1.39851
$512$	0.748955	0.0330994
$513$	16.8431	0.743642
$514$	12.0542	0.531689
$515$	0	0
$516$	−5.95384	−0.262103
$517$	−4.10602	−0.180583
$518$	−18.1645	−0.798102
$519$	36.3926	1.59746
$520$	0	0
$521$	42.2657	1.85169	0.925846	−	0.377900i	$-0.123354\pi$
$521$	42.2657	1.85169	0.925846	+	0.377900i	$0.123354\pi$
$522$	23.8466	1.04374
$523$	28.4461	1.24386	0.621931	−	0.783072i	$-0.286348\pi$
$523$	28.4461	1.24386	0.621931	+	0.783072i	$0.286348\pi$
$524$	9.77920	0.427206
$525$	0	0
$526$	−27.5827	−1.20266
$527$	−20.5121	−0.893521
$528$	−16.4123	−0.714255
$529$	−15.3189	−0.666037
$530$	0	0
$531$	42.9809	1.86521
$532$	2.09022	0.0906226
$533$	−27.7140	−1.20043
$534$	96.9092	4.19367
$535$	0	0
$536$	27.5484	1.18991
$537$	−6.27485	−0.270780
$538$	11.7827	0.507987
$539$	−0.902838	−0.0388880
$540$	0	0
$541$	18.7126	0.804515	0.402258	−	0.915526i	$-0.368226\pi$
$541$	18.7126	0.804515	0.402258	+	0.915526i	$0.368226\pi$
$542$	−18.6690	−0.801901
$543$	−28.2813	−1.21367
$544$	26.0505	1.11691
$545$	0	0
$546$	−92.6588	−3.96543
$547$	−22.8994	−0.979109	−0.489555	−	0.871973i	$-0.662841\pi$
$547$	−22.8994	−0.979109	−0.489555	+	0.871973i	$0.662841\pi$
$548$	0.235081	0.0100422
$549$	57.9165	2.47182
$550$	0	0
$551$	1.78540	0.0760605
$552$	19.1854	0.816583
$553$	−12.4867	−0.530987
$554$	−7.03772	−0.299004
$555$	0	0
$556$	−3.02927	−0.128470
$557$	−13.6065	−0.576524	−0.288262	−	0.957552i	$-0.593077\pi$
$557$	−13.6065	−0.576524	−0.288262	+	0.957552i	$0.593077\pi$
$558$	42.1782	1.78555
$559$	14.4024	0.609155
$560$	0	0
$561$	21.6055	0.912186
$562$	−0.369091	−0.0155692
$563$	38.8581	1.63767	0.818836	−	0.574028i	$-0.194620\pi$
$563$	38.8581	1.63767	0.818836	+	0.574028i	$0.194620\pi$
$564$	−10.1548	−0.427596
$565$	0	0
$566$	20.1690	0.847765
$567$	−89.4882	−3.75815
$568$	−0.911238	−0.0382347
$569$	−27.9139	−1.17021	−0.585107	−	0.810956i	$-0.698947\pi$
$569$	−27.9139	−1.17021	−0.585107	+	0.810956i	$0.698947\pi$
$570$	0	0
$571$	−39.5183	−1.65379	−0.826895	−	0.562357i	$-0.809895\pi$
$571$	−39.5183	−1.65379	−0.826895	+	0.562357i	$0.809895\pi$
$572$	−4.44825	−0.185991
$573$	77.2621	3.22767
$574$	−21.5704	−0.900330
$575$	0	0
$576$	26.0100	1.08375
$577$	−12.3632	−0.514685	−0.257343	−	0.966320i	$-0.582847\pi$
$577$	−12.3632	−0.514685	−0.257343	+	0.966320i	$0.582847\pi$
$578$	−41.7266	−1.73560
$579$	51.7896	2.15230
$580$	0	0
$581$	49.7906	2.06566
$582$	−27.5133	−1.14046
$583$	11.5083	0.476624
$584$	−23.4041	−0.968467
$585$	0	0
$586$	39.9473	1.65021
$587$	−6.03747	−0.249193	−0.124597	−	0.992207i	$-0.539764\pi$
$587$	−6.03747	−0.249193	−0.124597	+	0.992207i	$0.539764\pi$
$588$	−2.23286	−0.0920816
$589$	3.15789	0.130118
$590$	0	0
$591$	−76.0687	−3.12905
$592$	19.2485	0.791106
$593$	1.44341	0.0592739	0.0296369	−	0.999561i	$-0.490565\pi$
$593$	1.44341	0.0592739	0.0296369	+	0.999561i	$0.490565\pi$
$594$	−27.8983	−1.14468
$595$	0	0
$596$	3.57241	0.146332
$597$	79.0061	3.23350
$598$	27.4636	1.12307
$599$	20.4093	0.833902	0.416951	−	0.908929i	$-0.363099\pi$
$599$	20.4093	0.833902	0.416951	+	0.908929i	$0.363099\pi$
$600$	0	0
$601$	−33.1258	−1.35123	−0.675616	−	0.737254i	$-0.736122\pi$
$601$	−33.1258	−1.35123	−0.675616	+	0.737254i	$0.736122\pi$
$602$	11.2096	0.456871
$603$	106.740	4.34679
$604$	−13.5620	−0.551828
$605$	0	0
$606$	−92.1546	−3.74353
$607$	−7.79823	−0.316520	−0.158260	−	0.987397i	$-0.550588\pi$
$607$	−7.79823	−0.316520	−0.158260	+	0.987397i	$0.550588\pi$
$608$	−4.01054	−0.162649
$609$	−16.6947	−0.676502
$610$	0	0
$611$	24.5646	0.993778
$612$	38.9449	1.57426
$613$	−28.4672	−1.14978	−0.574890	−	0.818231i	$-0.694955\pi$
$613$	−28.4672	−1.14978	−0.574890	+	0.818231i	$0.694955\pi$
$614$	32.6686	1.31840
$615$	0	0
$616$	5.85056	0.235726
$617$	−1.26620	−0.0509751	−0.0254876	−	0.999675i	$-0.508114\pi$
$617$	−1.26620	−0.0509751	−0.0254876	+	0.999675i	$0.508114\pi$
$618$	−24.7238	−0.994536
$619$	2.57985	0.103693	0.0518464	−	0.998655i	$-0.483489\pi$
$619$	2.57985	0.103693	0.0518464	+	0.998655i	$0.483489\pi$
$620$	0	0
$621$	46.6805	1.87323
$622$	43.4912	1.74384
$623$	−49.4482	−1.98110
$624$	98.1881	3.93067
$625$	0	0
$626$	4.59083	0.183487
$627$	−3.32622	−0.132836
$628$	13.4699	0.537508
$629$	−25.3391	−1.01033
$630$	0	0
$631$	−18.3838	−0.731848	−0.365924	−	0.930645i	$-0.619247\pi$
$631$	−18.3838	−0.731848	−0.365924	+	0.930645i	$0.619247\pi$
$632$	9.24402	0.367707
$633$	−24.9043	−0.989859
$634$	6.62681	0.263184
$635$	0	0
$636$	28.4618	1.12858
$637$	5.40130	0.214007
$638$	−2.95726	−0.117079
$639$	−3.53072	−0.139673
$640$	0	0
$641$	20.9430	0.827199	0.413600	−	0.910459i	$-0.364271\pi$
$641$	20.9430	0.827199	0.413600	+	0.910459i	$0.364271\pi$
$642$	−5.82189	−0.229772
$643$	−9.37012	−0.369522	−0.184761	−	0.982784i	$-0.559151\pi$
$643$	−9.37012	−0.369522	−0.184761	+	0.982784i	$0.559151\pi$
$644$	5.79302	0.228277
$645$	0	0
$646$	10.7589	0.423304
$647$	15.3143	0.602069	0.301034	−	0.953613i	$-0.402668\pi$
$647$	15.3143	0.602069	0.301034	+	0.953613i	$0.402668\pi$
$648$	66.2491	2.60251
$649$	−5.33014	−0.209226
$650$	0	0
$651$	−29.5283	−1.15731
$652$	17.8408	0.698698
$653$	−7.61281	−0.297912	−0.148956	−	0.988844i	$-0.547591\pi$
$653$	−7.61281	−0.297912	−0.148956	+	0.988844i	$0.547591\pi$
$654$	−19.1786	−0.749944
$655$	0	0
$656$	22.8576	0.892438
$657$	−90.6822	−3.53785
$658$	19.1191	0.745341
$659$	27.8742	1.08582	0.542912	−	0.839789i	$-0.317322\pi$
$659$	27.8742	1.08582	0.542912	+	0.839789i	$0.317322\pi$
$660$	0	0
$661$	−40.0185	−1.55654	−0.778269	−	0.627931i	$-0.783902\pi$
$661$	−40.0185	−1.55654	−0.778269	+	0.627931i	$0.783902\pi$
$662$	29.1253	1.13199
$663$	−129.257	−5.01992
$664$	−36.8606	−1.43047
$665$	0	0
$666$	52.1037	2.01898
$667$	4.94821	0.191595
$668$	−2.30913	−0.0893431
$669$	−29.2793	−1.13200
$670$	0	0
$671$	−7.18233	−0.277271
$672$	37.5012	1.44664
$673$	41.4748	1.59874	0.799369	−	0.600841i	$-0.205168\pi$
$673$	41.4748	1.59874	0.799369	+	0.600841i	$0.205168\pi$
$674$	54.2759	2.09063
$675$	0	0
$676$	16.9461	0.651773
$677$	0.266167	0.0102296	0.00511482	−	0.999987i	$-0.498372\pi$
$677$	0.266167	0.0102296	0.00511482	+	0.999987i	$0.498372\pi$
$678$	49.2589	1.89178
$679$	14.0387	0.538757
$680$	0	0
$681$	−41.6299	−1.59526
$682$	−5.23060	−0.200290
$683$	−29.3614	−1.12348	−0.561742	−	0.827312i	$-0.689869\pi$
$683$	−29.3614	−1.12348	−0.561742	+	0.827312i	$0.689869\pi$
$684$	−5.99567	−0.229250
$685$	0	0
$686$	−28.3906	−1.08396
$687$	7.69574	0.293611
$688$	−11.8786	−0.452866
$689$	−68.8491	−2.62294
$690$	0	0
$691$	−22.7603	−0.865844	−0.432922	−	0.901431i	$-0.642517\pi$
$691$	−22.7603	−0.865844	−0.432922	+	0.901431i	$0.642517\pi$
$692$	−8.13510	−0.309250
$693$	22.6688	0.861116
$694$	−32.5468	−1.23546
$695$	0	0
$696$	12.3592	0.468476
$697$	−30.0902	−1.13975
$698$	−33.0626	−1.25144
$699$	−43.5240	−1.64623
$700$	0	0
$701$	−13.6101	−0.514046	−0.257023	−	0.966405i	$-0.582742\pi$
$701$	−13.6101	−0.514046	−0.257023	+	0.966405i	$0.582742\pi$
$702$	166.904	6.29939
$703$	3.90101	0.147129
$704$	−3.22555	−0.121568
$705$	0	0
$706$	−38.4326	−1.44643
$707$	47.0221	1.76845
$708$	−13.1823	−0.495420
$709$	−32.9904	−1.23898	−0.619491	−	0.785004i	$-0.712661\pi$
$709$	−32.9904	−1.23898	−0.619491	+	0.785004i	$0.712661\pi$
$710$	0	0
$711$	35.8172	1.34325
$712$	36.6071	1.37191
$713$	8.75204	0.327767
$714$	−100.603	−3.76498
$715$	0	0
$716$	1.40266	0.0524199
$717$	−69.6784	−2.60219
$718$	16.2886	0.607886
$719$	−3.85276	−0.143684	−0.0718418	−	0.997416i	$-0.522888\pi$
$719$	−3.85276	−0.143684	−0.0718418	+	0.997416i	$0.522888\pi$
$720$	0	0
$721$	12.6154	0.469821
$722$	−1.65636	−0.0616434
$723$	47.5305	1.76768
$724$	6.32193	0.234953
$725$	0	0
$726$	5.50942	0.204474
$727$	35.5951	1.32015	0.660074	−	0.751200i	$-0.270525\pi$
$727$	35.5951	1.32015	0.660074	+	0.751200i	$0.270525\pi$
$728$	−35.0015	−1.29724
$729$	88.6192	3.28219
$730$	0	0
$731$	15.6372	0.578362
$732$	−17.7630	−0.656541
$733$	22.8053	0.842332	0.421166	−	0.906984i	$-0.361621\pi$
$733$	22.8053	0.842332	0.421166	+	0.906984i	$0.361621\pi$
$734$	2.87933	0.106278
$735$	0	0
$736$	−11.1152	−0.409710
$737$	−13.2370	−0.487592
$738$	61.8732	2.27758
$739$	−37.2882	−1.37167	−0.685835	−	0.727757i	$-0.740563\pi$
$739$	−37.2882	−1.37167	−0.685835	+	0.727757i	$0.740563\pi$
$740$	0	0
$741$	19.8994	0.731022
$742$	−53.5866	−1.96723
$743$	−24.9220	−0.914301	−0.457150	−	0.889389i	$-0.651130\pi$
$743$	−24.9220	−0.914301	−0.457150	+	0.889389i	$0.651130\pi$
$744$	21.8602	0.801433
$745$	0	0
$746$	−10.9059	−0.399295
$747$	−142.821	−5.22555
$748$	−4.82964	−0.176589
$749$	2.97064	0.108545
$750$	0	0
$751$	−1.46203	−0.0533501	−0.0266751	−	0.999644i	$-0.508492\pi$
$751$	−1.46203	−0.0533501	−0.0266751	+	0.999644i	$0.508492\pi$
$752$	−20.2600	−0.738807
$753$	−37.0195	−1.34907
$754$	17.6921	0.644308
$755$	0	0
$756$	35.2059	1.28043
$757$	−4.66986	−0.169729	−0.0848645	−	0.996393i	$-0.527046\pi$
$757$	−4.66986	−0.169729	−0.0848645	+	0.996393i	$0.527046\pi$
$758$	6.19562	0.225035
$759$	−9.21858	−0.334613
$760$	0	0
$761$	18.4084	0.667303	0.333652	−	0.942696i	$-0.391719\pi$
$761$	18.4084	0.667303	0.333652	+	0.942696i	$0.391719\pi$
$762$	−11.8291	−0.428524
$763$	9.78595	0.354275
$764$	−17.2709	−0.624841
$765$	0	0
$766$	45.6677	1.65004
$767$	31.8880	1.15141
$768$	−52.1688	−1.88248
$769$	−3.57354	−0.128865	−0.0644326	−	0.997922i	$-0.520524\pi$
$769$	−3.57354	−0.128865	−0.0644326	+	0.997922i	$0.520524\pi$
$770$	0	0
$771$	24.2067	0.871782
$772$	−11.5769	−0.416662
$773$	−34.6561	−1.24649	−0.623246	−	0.782026i	$-0.714186\pi$
$773$	−34.6561	−1.24649	−0.623246	+	0.782026i	$0.714186\pi$
$774$	−32.1542	−1.15576
$775$	0	0
$776$	−10.3930	−0.373088
$777$	−36.4770	−1.30861
$778$	28.4166	1.01878
$779$	4.63245	0.165975
$780$	0	0
$781$	0.437851	0.0156675
$782$	29.8182	1.06630
$783$	30.0717	1.07467
$784$	−4.45481	−0.159100
$785$	0	0
$786$	72.4617	2.58462
$787$	−30.3693	−1.08255	−0.541274	−	0.840846i	$-0.682058\pi$
$787$	−30.3693	−1.08255	−0.541274	+	0.840846i	$0.682058\pi$
$788$	17.0042	0.605749
$789$	−55.3902	−1.97194
$790$	0	0
$791$	−25.1345	−0.893679
$792$	−16.7820	−0.596321
$793$	42.9689	1.52587
$794$	−23.4386	−0.831804
$795$	0	0
$796$	−17.6608	−0.625970
$797$	34.4392	1.21990	0.609950	−	0.792440i	$-0.291190\pi$
$797$	34.4392	1.21990	0.609950	+	0.792440i	$0.291190\pi$
$798$	15.4881	0.548272
$799$	26.6707	0.943542
$800$	0	0
$801$	141.839	5.01163
$802$	−17.2081	−0.607639
$803$	11.2457	0.396851
$804$	−32.7372	−1.15455
$805$	0	0
$806$	31.2925	1.10223
$807$	23.6613	0.832918
$808$	−34.8110	−1.22465
$809$	−8.23292	−0.289454	−0.144727	−	0.989472i	$-0.546230\pi$
$809$	−8.23292	−0.289454	−0.144727	+	0.989472i	$0.546230\pi$
$810$	0	0
$811$	35.9943	1.26393	0.631966	−	0.774996i	$-0.282248\pi$
$811$	35.9943	1.26393	0.631966	+	0.774996i	$0.282248\pi$
$812$	3.73188	0.130963
$813$	−37.4901	−1.31483
$814$	−6.46148	−0.226475
$815$	0	0
$816$	106.606	3.73197
$817$	−2.40738	−0.0842237
$818$	−6.81034	−0.238118
$819$	−135.618	−4.73887
$820$	0	0
$821$	31.2285	1.08988	0.544941	−	0.838474i	$-0.316552\pi$
$821$	31.2285	1.08988	0.544941	+	0.838474i	$0.316552\pi$
$822$	1.74190	0.0607557
$823$	8.20565	0.286031	0.143015	−	0.989720i	$-0.454320\pi$
$823$	8.20565	0.286031	0.143015	+	0.989720i	$0.454320\pi$
$824$	−9.33931	−0.325350
$825$	0	0
$826$	24.8191	0.863566
$827$	15.5275	0.539943	0.269972	−	0.962868i	$-0.412986\pi$
$827$	15.5275	0.539943	0.269972	+	0.962868i	$0.412986\pi$
$828$	−16.6169	−0.577478
$829$	3.31730	0.115215	0.0576074	−	0.998339i	$-0.481653\pi$
$829$	3.31730	0.115215	0.0576074	+	0.998339i	$0.481653\pi$
$830$	0	0
$831$	−14.1328	−0.490261
$832$	19.2971	0.669007
$833$	5.86440	0.203189
$834$	−22.4462	−0.777249
$835$	0	0
$836$	0.743534	0.0257157
$837$	53.1887	1.83847
$838$	−27.2461	−0.941201
$839$	7.02051	0.242375	0.121188	−	0.992630i	$-0.461330\pi$
$839$	7.02051	0.242375	0.121188	+	0.992630i	$0.461330\pi$
$840$	0	0
$841$	−25.8124	−0.890081
$842$	63.9665	2.20443
$843$	−0.741189	−0.0255279
$844$	5.56705	0.191626
$845$	0	0
$846$	−54.8420	−1.88551
$847$	−2.81120	−0.0965939
$848$	56.7844	1.94998
$849$	40.5023	1.39004
$850$	0	0
$851$	10.8116	0.370617
$852$	1.08287	0.0370986
$853$	−3.62821	−0.124227	−0.0621137	−	0.998069i	$-0.519784\pi$
$853$	−3.62821	−0.124227	−0.0621137	+	0.998069i	$0.519784\pi$
$854$	33.4435	1.14441
$855$	0	0
$856$	−2.19920	−0.0751670
$857$	−16.1864	−0.552918	−0.276459	−	0.961026i	$-0.589161\pi$
$857$	−16.1864	−0.552918	−0.276459	+	0.961026i	$0.589161\pi$
$858$	−32.9606	−1.12526
$859$	25.5004	0.870063	0.435032	−	0.900415i	$-0.356737\pi$
$859$	25.5004	0.870063	0.435032	+	0.900415i	$0.356737\pi$
$860$	0	0
$861$	−43.3165	−1.47622
$862$	40.2438	1.37071
$863$	39.9012	1.35825	0.679126	−	0.734022i	$-0.262359\pi$
$863$	39.9012	1.35825	0.679126	+	0.734022i	$0.262359\pi$
$864$	−67.5500	−2.29810
$865$	0	0
$866$	37.8228	1.28527
$867$	−83.7932	−2.84577
$868$	6.60068	0.224042
$869$	−4.44176	−0.150676
$870$	0	0
$871$	79.1916	2.68330
$872$	−7.24465	−0.245335
$873$	−40.2692	−1.36291
$874$	−4.59059	−0.155279
$875$	0	0
$876$	27.8123	0.939691
$877$	−11.1804	−0.377537	−0.188768	−	0.982022i	$-0.560450\pi$
$877$	−11.1804	−0.377537	−0.188768	+	0.982022i	$0.560450\pi$
$878$	38.7254	1.30692
$879$	80.2202	2.70576
$880$	0	0
$881$	2.82033	0.0950195	0.0475097	−	0.998871i	$-0.484871\pi$
$881$	2.82033	0.0950195	0.0475097	+	0.998871i	$0.484871\pi$
$882$	−12.0587	−0.406039
$883$	17.0119	0.572495	0.286247	−	0.958156i	$-0.407592\pi$
$883$	17.0119	0.572495	0.286247	+	0.958156i	$0.407592\pi$
$884$	28.8937	0.971800
$885$	0	0
$886$	24.2722	0.815439
$887$	−10.8354	−0.363817	−0.181908	−	0.983316i	$-0.558227\pi$
$887$	−10.8354	−0.363817	−0.181908	+	0.983316i	$0.558227\pi$
$888$	27.0044	0.906207
$889$	6.03584	0.202436
$890$	0	0
$891$	−31.8327	−1.06644
$892$	6.54501	0.219143
$893$	−4.10602	−0.137403
$894$	26.4708	0.885316
$895$	0	0
$896$	37.5682	1.25506
$897$	55.1509	1.84144
$898$	−32.4538	−1.08300
$899$	5.63808	0.188041
$900$	0	0
$901$	−74.7521	−2.49035
$902$	−7.67301	−0.255483
$903$	22.5106	0.749107
$904$	18.6073	0.618871
$905$	0	0
$906$	−100.491	−3.33859
$907$	−38.8963	−1.29153	−0.645766	−	0.763536i	$-0.723462\pi$
$907$	−38.8963	−1.29153	−0.645766	+	0.763536i	$0.723462\pi$
$908$	9.30583	0.308825
$909$	−134.880	−4.47369
$910$	0	0
$911$	11.2431	0.372502	0.186251	−	0.982502i	$-0.440366\pi$
$911$	11.2431	0.372502	0.186251	+	0.982502i	$0.440366\pi$
$912$	−16.4123	−0.543466
$913$	17.7115	0.586166
$914$	−53.7570	−1.77812
$915$	0	0
$916$	−1.72028	−0.0568398
$917$	−36.9738	−1.22098
$918$	181.214	5.98095
$919$	−9.48069	−0.312739	−0.156370	−	0.987699i	$-0.549979\pi$
$919$	−9.48069	−0.312739	−0.156370	+	0.987699i	$0.549979\pi$
$920$	0	0
$921$	65.6035	2.16171
$922$	−16.3732	−0.539223
$923$	−2.61948	−0.0862211
$924$	−6.95254	−0.228722
$925$	0	0
$926$	16.5588	0.544156
$927$	−36.1864	−1.18852
$928$	−7.16040	−0.235052
$929$	−39.8543	−1.30758	−0.653789	−	0.756677i	$-0.726822\pi$
$929$	−39.8543	−1.30758	−0.653789	+	0.756677i	$0.726822\pi$
$930$	0	0
$931$	−0.902838	−0.0295893
$932$	9.72922	0.318691
$933$	87.3369	2.85928
$934$	43.6679	1.42886
$935$	0	0
$936$	100.399	3.28166
$937$	56.3053	1.83942	0.919708	−	0.392604i	$-0.128426\pi$
$937$	56.3053	1.83942	0.919708	+	0.392604i	$0.128426\pi$
$938$	61.6363	2.01250
$939$	9.21908	0.300853
$940$	0	0
$941$	−21.4276	−0.698519	−0.349259	−	0.937026i	$-0.613567\pi$
$941$	−21.4276	−0.698519	−0.349259	+	0.937026i	$0.613567\pi$
$942$	99.8092	3.25196
$943$	12.8388	0.418088
$944$	−26.3001	−0.855996
$945$	0	0
$946$	3.98750	0.129645
$947$	−16.2072	−0.526662	−0.263331	−	0.964706i	$-0.584821\pi$
$947$	−16.2072	−0.526662	−0.263331	+	0.964706i	$0.584821\pi$
$948$	−10.9852	−0.356782
$949$	−67.2781	−2.18394
$950$	0	0
$951$	13.3076	0.431529
$952$	−38.0024	−1.23166
$953$	−58.8844	−1.90745	−0.953726	−	0.300676i	$-0.902788\pi$
$953$	−58.8844	−1.90745	−0.953726	+	0.300676i	$0.902788\pi$
$954$	153.710	4.97654
$955$	0	0
$956$	15.5757	0.503755
$957$	−5.93863	−0.191969
$958$	−32.0628	−1.03590
$959$	−0.888809	−0.0287011
$960$	0	0
$961$	−21.0278	−0.678315
$962$	38.6563	1.24633
$963$	−8.52108	−0.274588
$964$	−10.6248	−0.342203
$965$	0	0
$966$	42.9251	1.38109
$967$	24.4443	0.786074	0.393037	−	0.919523i	$-0.371424\pi$
$967$	24.4443	0.786074	0.393037	+	0.919523i	$0.371424\pi$
$968$	2.08116	0.0668911
$969$	21.6055	0.694069
$970$	0	0
$971$	59.3586	1.90491	0.952454	−	0.304683i	$-0.0985506\pi$
$971$	59.3586	1.90491	0.952454	+	0.304683i	$0.0985506\pi$
$972$	−41.1571	−1.32011
$973$	11.4532	0.367174
$974$	25.6038	0.820399
$975$	0	0
$976$	−35.4392	−1.13438
$977$	5.33048	0.170537	0.0852686	−	0.996358i	$-0.472825\pi$
$977$	5.33048	0.170537	0.0852686	+	0.996358i	$0.472825\pi$
$978$	132.196	4.22717
$979$	−17.5897	−0.562170
$980$	0	0
$981$	−28.0704	−0.896218
$982$	−55.8356	−1.78178
$983$	31.3817	1.00092	0.500460	−	0.865760i	$-0.333164\pi$
$983$	31.3817	1.00092	0.500460	+	0.865760i	$0.333164\pi$
$984$	32.0677	1.02228
$985$	0	0
$986$	19.2090	0.611738
$987$	38.3940	1.22210
$988$	−4.44825	−0.141518
$989$	−6.67203	−0.212158
$990$	0	0
$991$	−25.0162	−0.794667	−0.397334	−	0.917674i	$-0.630064\pi$
$991$	−25.0162	−0.794667	−0.397334	+	0.917674i	$0.630064\pi$
$992$	−12.6648	−0.402108
$993$	58.4880	1.85606
$994$	−2.03879	−0.0646665
$995$	0	0
$996$	43.8034	1.38796
$997$	16.2641	0.515091	0.257545	−	0.966266i	$-0.417086\pi$
$997$	16.2641	0.515091	0.257545	+	0.966266i	$0.417086\pi$
$998$	−54.5586	−1.72702
$999$	65.7052	2.07882

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5225.2.a.k.1.2		6
5.4	even	2		1045.2.a.g.1.5	✓	6
15.14	odd	2		9405.2.a.w.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.a.g.1.5	✓	6	5.4	even	2
5225.2.a.k.1.2		6	1.1	even	1	trivial
9405.2.a.w.1.2		6	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5225.a (trivial)

\(f(q)\)	\(=\)	\(q-1.65636 q^{2} -3.32622 q^{3} +0.743534 q^{4} +5.50942 q^{6} -2.81120 q^{7} +2.08116 q^{8} +8.06374 q^{9} -1.00000 q^{11} -2.47316 q^{12} +5.98258 q^{13} +4.65636 q^{14} -4.93423 q^{16} +6.49551 q^{17} -13.3565 q^{18} -1.00000 q^{19} +9.35067 q^{21} +1.65636 q^{22} -2.77149 q^{23} -6.92241 q^{24} -9.90932 q^{26} -16.8431 q^{27} -2.09022 q^{28} -1.78540 q^{29} -3.15789 q^{31} +4.01054 q^{32} +3.32622 q^{33} -10.7589 q^{34} +5.99567 q^{36} -3.90101 q^{37} +1.65636 q^{38} -19.8994 q^{39} -4.63245 q^{41} -15.4881 q^{42} +2.40738 q^{43} -0.743534 q^{44} +4.59059 q^{46} +4.10602 q^{47} +16.4123 q^{48} +0.902838 q^{49} -21.6055 q^{51} +4.44825 q^{52} -11.5083 q^{53} +27.8983 q^{54} -5.85056 q^{56} +3.32622 q^{57} +2.95726 q^{58} +5.33014 q^{59} +7.18233 q^{61} +5.23060 q^{62} -22.6688 q^{63} +3.22555 q^{64} -5.50942 q^{66} +13.2370 q^{67} +4.82964 q^{68} +9.21858 q^{69} -0.437851 q^{71} +16.7820 q^{72} -11.2457 q^{73} +6.46148 q^{74} -0.743534 q^{76} +2.81120 q^{77} +32.9606 q^{78} +4.44176 q^{79} +31.8327 q^{81} +7.67301 q^{82} -17.7115 q^{83} +6.95254 q^{84} -3.98750 q^{86} +5.93863 q^{87} -2.08116 q^{88} +17.5897 q^{89} -16.8182 q^{91} -2.06070 q^{92} +10.5038 q^{93} -6.80106 q^{94} -13.3399 q^{96} -4.99386 q^{97} -1.49543 q^{98} -8.06374 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{3} + 8 q^{4} + 2 q^{6} - 5 q^{7} + 9 q^{9} - 6 q^{11} - 19 q^{12} + 9 q^{13} + 18 q^{14} + 4 q^{16} + 5 q^{17} - 2 q^{18} - 6 q^{19} + 3 q^{21} - 8 q^{23} - 7 q^{24} - 22 q^{26} - 30 q^{27} - 10 q^{28}+ \cdots - 9 q^{99}+O(q^{100}) \) 6 * q - 3 * q^3 + 8 * q^4 + 2 * q^6 - 5 * q^7 + 9 * q^9 - 6 * q^11 - 19 * q^12 + 9 * q^13 + 18 * q^14 + 4 * q^16 + 5 * q^17 - 2 * q^18 - 6 * q^19 + 3 * q^21 - 8 * q^23 - 7 * q^24 - 22 * q^26 - 30 * q^27 - 10 * q^28 - 5 * q^29 - q^31 - 15 * q^32 + 3 * q^33 - 22 * q^34 + 12 * q^36 - 9 * q^37 - 32 * q^39 + 25 * q^41 - 11 * q^42 - 15 * q^43 - 8 * q^44 - 16 * q^46 - 24 * q^47 + 4 * q^48 + 13 * q^49 + 27 * q^52 - 5 * q^53 - 11 * q^54 - 12 * q^56 + 3 * q^57 - 13 * q^58 + 39 * q^59 - 11 * q^61 + 42 * q^62 - 38 * q^63 - 14 * q^64 - 2 * q^66 - 24 * q^67 - 45 * q^68 + 14 * q^69 - 24 * q^71 + 61 * q^72 + 26 * q^73 + q^74 - 8 * q^76 + 5 * q^77 + 29 * q^78 + 11 * q^79 + 30 * q^81 - 8 * q^82 - 39 * q^83 + 25 * q^84 + 18 * q^86 + 16 * q^87 + 22 * q^89 - 26 * q^91 + 11 * q^92 + 6 * q^93 - 30 * q^94 - 15 * q^96 - 22 * q^97 - 33 * q^98 - 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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