LMFDB - Embedded newform 5225.2.a.j.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5225,2,Mod(1,5225)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 5225 = 5^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5225.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:	$41.7218350561$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\Q(\zeta_{22})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 $ x^5 - x^4 - 4x^3 + 3x^2 + 3*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
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			1.4
Root			$1.30972$ 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)

$f(q)$	$=$	\(q+1.54620 q^{2} -1.51334 q^{3} +0.390736 q^{4} -2.33992 q^{6} +4.16140 q^{7} -2.48825 q^{8} -0.709811 q^{9} -1.00000 q^{11} -0.591315 q^{12} -1.82862 q^{13} +6.43436 q^{14} -4.62880 q^{16} -4.80511 q^{17} -1.09751 q^{18} +1.00000 q^{19} -6.29760 q^{21} -1.54620 q^{22} +5.53843 q^{23} +3.76555 q^{24} -2.82741 q^{26} +5.61419 q^{27} +1.62601 q^{28} +3.98933 q^{29} -8.51574 q^{31} -2.18056 q^{32} +1.51334 q^{33} -7.42966 q^{34} -0.277348 q^{36} +7.11059 q^{37} +1.54620 q^{38} +2.76732 q^{39} -3.85750 q^{41} -9.73735 q^{42} +12.0581 q^{43} -0.390736 q^{44} +8.56352 q^{46} -5.75936 q^{47} +7.00493 q^{48} +10.3172 q^{49} +7.27175 q^{51} -0.714506 q^{52} +2.82914 q^{53} +8.68067 q^{54} -10.3546 q^{56} -1.51334 q^{57} +6.16831 q^{58} -10.8553 q^{59} +7.35724 q^{61} -13.1670 q^{62} -2.95381 q^{63} +5.88602 q^{64} +2.33992 q^{66} +11.7680 q^{67} -1.87753 q^{68} -8.38151 q^{69} -4.55334 q^{71} +1.76618 q^{72} +6.03445 q^{73} +10.9944 q^{74} +0.390736 q^{76} -4.16140 q^{77} +4.27882 q^{78} +9.68391 q^{79} -6.36674 q^{81} -5.96447 q^{82} -0.508463 q^{83} -2.46070 q^{84} +18.6442 q^{86} -6.03720 q^{87} +2.48825 q^{88} +6.86161 q^{89} -7.60961 q^{91} +2.16406 q^{92} +12.8872 q^{93} -8.90513 q^{94} +3.29992 q^{96} +3.72641 q^{97} +15.9525 q^{98} +0.709811 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 3 q^{2} + 7 q^{3} + 5 q^{4} + 2 q^{6} + 11 q^{7} - 3 q^{8} + 8 q^{9} - 5 q^{11} + 7 q^{12} - q^{13} - 3 q^{16} + 3 q^{17} + 7 q^{18} + 5 q^{19} + 11 q^{21} - 3 q^{22} + 8 q^{23} + 9 q^{24} - 16 q^{26}+ \cdots - 8 q^{99}+O(q^{100}) $ 5 * q + 3 * q^2 + 7 * q^3 + 5 * q^4 + 2 * q^6 + 11 * q^7 - 3 * q^8 + 8 * q^9 - 5 * q^11 + 7 * q^12 - q^13 - 3 * q^16 + 3 * q^17 + 7 * q^18 + 5 * q^19 + 11 * q^21 - 3 * q^22 + 8 * q^23 + 9 * q^24 - 16 * q^26 + 10 * q^27 + 22 * q^28 + 11 * q^29 - 5 * q^31 + 2 * q^32 - 7 * q^33 + 4 * q^34 - 3 * q^36 + 9 * q^37 + 3 * q^38 - 8 * q^39 + 15 * q^41 - 11 * q^42 + 13 * q^43 - 5 * q^44 + 18 * q^46 + 20 * q^47 + 20 * q^48 + 20 * q^49 + 24 * q^51 - q^52 + 5 * q^53 + 17 * q^54 + 7 * q^57 + 33 * q^58 - 17 * q^59 + 3 * q^61 - 14 * q^62 + 22 * q^63 - 17 * q^64 - 2 * q^66 + 28 * q^67 + 25 * q^68 - 2 * q^69 - 6 * q^71 + 26 * q^72 + 16 * q^73 - 21 * q^74 + 5 * q^76 - 11 * q^77 - 29 * q^78 + 3 * q^79 + q^81 - 2 * q^82 + 33 * q^83 + 33 * q^84 + 10 * q^86 + 3 * q^88 - 16 * q^89 - 22 * q^91 + 19 * q^92 + 26 * q^93 - 10 * q^94 + 5 * q^96 + 14 * q^97 - 10 * q^98 - 8 * 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 $

$2$	1.54620	1.09333	0.546664	−	0.837352i	$-0.315897\pi$
$2$	1.54620	1.09333	0.546664	+	0.837352i	$0.315897\pi$
$3$	−1.51334	−0.873726	−0.436863	−	0.899528i	$-0.643910\pi$
$3$	−1.51334	−0.873726	−0.436863	+	0.899528i	$0.643910\pi$
$4$	0.390736	0.195368
$5$	0	0
$5$	0	0
$6$	−2.33992	−0.955269
$7$	4.16140	1.57286	0.786430	−	0.617679i	$-0.211927\pi$
$7$	4.16140	1.57286	0.786430	+	0.617679i	$0.211927\pi$
$8$	−2.48825	−0.879728
$9$	−0.709811	−0.236604
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−0.591315	−0.170698
$13$	−1.82862	−0.507167	−0.253584	−	0.967313i	$-0.581609\pi$
$13$	−1.82862	−0.507167	−0.253584	+	0.967313i	$0.581609\pi$
$14$	6.43436	1.71965
$15$	0	0
$16$	−4.62880	−1.15720
$17$	−4.80511	−1.16541	−0.582705	−	0.812684i	$-0.698006\pi$
$17$	−4.80511	−1.16541	−0.582705	+	0.812684i	$0.698006\pi$
$18$	−1.09751	−0.258686
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−6.29760	−1.37425
$22$	−1.54620	−0.329651
$23$	5.53843	1.15484	0.577421	−	0.816446i	$-0.304059\pi$
$23$	5.53843	1.15484	0.577421	+	0.816446i	$0.304059\pi$
$24$	3.76555	0.768640
$25$	0	0
$26$	−2.82741	−0.554501
$27$	5.61419	1.08045
$28$	1.62601	0.307286
$29$	3.98933	0.740800	0.370400	−	0.928872i	$-0.379221\pi$
$29$	3.98933	0.740800	0.370400	+	0.928872i	$0.379221\pi$
$30$	0	0
$31$	−8.51574	−1.52947	−0.764736	−	0.644343i	$-0.777131\pi$
$31$	−8.51574	−1.52947	−0.764736	+	0.644343i	$0.777131\pi$
$32$	−2.18056	−0.385472
$33$	1.51334	0.263438
$34$	−7.42966	−1.27418
$35$	0	0
$36$	−0.277348	−0.0462247
$37$	7.11059	1.16897	0.584487	−	0.811403i	$-0.301296\pi$
$37$	7.11059	1.16897	0.584487	+	0.811403i	$0.301296\pi$
$38$	1.54620	0.250827
$39$	2.76732	0.443125
$40$	0	0
$41$	−3.85750	−0.602441	−0.301220	−	0.953555i	$-0.597394\pi$
$41$	−3.85750	−0.602441	−0.301220	+	0.953555i	$0.597394\pi$
$42$	−9.73735	−1.50251
$43$	12.0581	1.83884	0.919420	−	0.393277i	$-0.128659\pi$
$43$	12.0581	1.83884	0.919420	+	0.393277i	$0.128659\pi$
$44$	−0.390736	−0.0589056
$45$	0	0
$46$	8.56352	1.26262
$47$	−5.75936	−0.840090	−0.420045	−	0.907503i	$-0.637986\pi$
$47$	−5.75936	−0.840090	−0.420045	+	0.907503i	$0.637986\pi$
$48$	7.00493	1.01107
$49$	10.3172	1.47389
$50$	0	0
$51$	7.27175	1.01825
$52$	−0.714506	−0.0990842
$53$	2.82914	0.388612	0.194306	−	0.980941i	$-0.437754\pi$
$53$	2.82914	0.388612	0.194306	+	0.980941i	$0.437754\pi$
$54$	8.68067	1.18129
$55$	0	0
$56$	−10.3546	−1.38369
$57$	−1.51334	−0.200446
$58$	6.16831	0.809938
$59$	−10.8553	−1.41324	−0.706619	−	0.707594i	$-0.749781\pi$
$59$	−10.8553	−1.41324	−0.706619	+	0.707594i	$0.749781\pi$
$60$	0	0
$61$	7.35724	0.941998	0.470999	−	0.882134i	$-0.343893\pi$
$61$	7.35724	0.941998	0.470999	+	0.882134i	$0.343893\pi$
$62$	−13.1670	−1.67222
$63$	−2.95381	−0.372145
$64$	5.88602	0.735752
$65$	0	0
$66$	2.33992	0.288025
$67$	11.7680	1.43769	0.718845	−	0.695170i	$-0.244671\pi$
$67$	11.7680	1.43769	0.718845	+	0.695170i	$0.244671\pi$
$68$	−1.87753	−0.227684
$69$	−8.38151	−1.00902
$70$	0	0
$71$	−4.55334	−0.540382	−0.270191	−	0.962807i	$-0.587087\pi$
$71$	−4.55334	−0.540382	−0.270191	+	0.962807i	$0.587087\pi$
$72$	1.76618	0.208147
$73$	6.03445	0.706278	0.353139	−	0.935571i	$-0.385114\pi$
$73$	6.03445	0.706278	0.353139	+	0.935571i	$0.385114\pi$
$74$	10.9944	1.27807
$75$	0	0
$76$	0.390736	0.0448204
$77$	−4.16140	−0.474235
$78$	4.27882	0.484481
$79$	9.68391	1.08953	0.544763	−	0.838590i	$-0.316620\pi$
$79$	9.68391	1.08953	0.544763	+	0.838590i	$0.316620\pi$
$80$	0	0
$81$	−6.36674	−0.707415
$82$	−5.96447	−0.658666
$83$	−0.508463	−0.0558110	−0.0279055	−	0.999611i	$-0.508884\pi$
$83$	−0.508463	−0.0558110	−0.0279055	+	0.999611i	$0.508884\pi$
$84$	−2.46070	−0.268484
$85$	0	0
$86$	18.6442	2.01046
$87$	−6.03720	−0.647256
$88$	2.48825	0.265248
$89$	6.86161	0.727329	0.363665	−	0.931530i	$-0.381525\pi$
$89$	6.86161	0.727329	0.363665	+	0.931530i	$0.381525\pi$
$90$	0	0
$91$	−7.60961	−0.797704
$92$	2.16406	0.225619
$93$	12.8872	1.33634
$94$	−8.90513	−0.918494
$95$	0	0
$96$	3.29992	0.336796
$97$	3.72641	0.378360	0.189180	−	0.981942i	$-0.439417\pi$
$97$	3.72641	0.378360	0.189180	+	0.981942i	$0.439417\pi$
$98$	15.9525	1.61145
$99$	0.709811	0.0713387
$100$	0	0
$101$	−14.0934	−1.40234	−0.701171	−	0.712994i	$-0.747339\pi$
$101$	−14.0934	−1.40234	−0.701171	+	0.712994i	$0.747339\pi$
$102$	11.2436	1.11328
$103$	−10.8474	−1.06883	−0.534414	−	0.845223i	$-0.679468\pi$
$103$	−10.8474	−1.06883	−0.534414	+	0.845223i	$0.679468\pi$
$104$	4.55005	0.446169
$105$	0	0
$106$	4.37442	0.424881
$107$	9.49711	0.918120	0.459060	−	0.888405i	$-0.348186\pi$
$107$	9.49711	0.918120	0.459060	+	0.888405i	$0.348186\pi$
$108$	2.19367	0.211086
$109$	10.2467	0.981456	0.490728	−	0.871313i	$-0.336731\pi$
$109$	10.2467	0.981456	0.490728	+	0.871313i	$0.336731\pi$
$110$	0	0
$111$	−10.7607	−1.02136
$112$	−19.2623	−1.82011
$113$	7.19036	0.676412	0.338206	−	0.941072i	$-0.390180\pi$
$113$	7.19036	0.676412	0.338206	+	0.941072i	$0.390180\pi$
$114$	−2.33992	−0.219154
$115$	0	0
$116$	1.55877	0.144728
$117$	1.29797	0.119998
$118$	−16.7845	−1.54513
$119$	−19.9960	−1.83303
$120$	0	0
$121$	1.00000	0.0909091
$122$	11.3758	1.02991
$123$	5.83770	0.526368
$124$	−3.32740	−0.298810
$125$	0	0
$126$	−4.56718	−0.406876
$127$	3.35345	0.297570	0.148785	−	0.988870i	$-0.452464\pi$
$127$	3.35345	0.297570	0.148785	+	0.988870i	$0.452464\pi$
$128$	13.4621	1.18989
$129$	−18.2479	−1.60664
$130$	0	0
$131$	−5.15388	−0.450297	−0.225148	−	0.974324i	$-0.572287\pi$
$131$	−5.15388	−0.450297	−0.225148	+	0.974324i	$0.572287\pi$
$132$	0.591315	0.0514673
$133$	4.16140	0.360839
$134$	18.1957	1.57187
$135$	0	0
$136$	11.9563	1.02524
$137$	11.3104	0.966316	0.483158	−	0.875533i	$-0.339490\pi$
$137$	11.3104	0.966316	0.483158	+	0.875533i	$0.339490\pi$
$138$	−12.9595	−1.10319
$139$	5.14614	0.436490	0.218245	−	0.975894i	$-0.429967\pi$
$139$	5.14614	0.436490	0.218245	+	0.975894i	$0.429967\pi$
$140$	0	0
$141$	8.71586	0.734008
$142$	−7.04038	−0.590816
$143$	1.82862	0.152917
$144$	3.28557	0.273798
$145$	0	0
$146$	9.33046	0.772195
$147$	−15.6135	−1.28778
$148$	2.77836	0.228380
$149$	10.3400	0.847089	0.423545	−	0.905875i	$-0.360786\pi$
$149$	10.3400	0.847089	0.423545	+	0.905875i	$0.360786\pi$
$150$	0	0
$151$	−4.86952	−0.396276	−0.198138	−	0.980174i	$-0.563489\pi$
$151$	−4.86952	−0.396276	−0.198138	+	0.980174i	$0.563489\pi$
$152$	−2.48825	−0.201823
$153$	3.41072	0.275740
$154$	−6.43436	−0.518495
$155$	0	0
$156$	1.08129	0.0865724
$157$	17.3831	1.38732	0.693661	−	0.720302i	$-0.255997\pi$
$157$	17.3831	1.38732	0.693661	+	0.720302i	$0.255997\pi$
$158$	14.9733	1.19121
$159$	−4.28144	−0.339541
$160$	0	0
$161$	23.0476	1.81641
$162$	−9.84425	−0.773437
$163$	−17.0128	−1.33255	−0.666274	−	0.745707i	$-0.732112\pi$
$163$	−17.0128	−1.33255	−0.666274	+	0.745707i	$0.732112\pi$
$164$	−1.50726	−0.117698
$165$	0	0
$166$	−0.786185	−0.0610198
$167$	10.3330	0.799590	0.399795	−	0.916604i	$-0.369081\pi$
$167$	10.3330	0.799590	0.399795	+	0.916604i	$0.369081\pi$
$168$	15.6700	1.20896
$169$	−9.65616	−0.742781
$170$	0	0
$171$	−0.709811	−0.0542806
$172$	4.71152	0.359250
$173$	−15.1982	−1.15550	−0.577748	−	0.816215i	$-0.696068\pi$
$173$	−15.1982	−1.15550	−0.577748	+	0.816215i	$0.696068\pi$
$174$	−9.33473	−0.707664
$175$	0	0
$176$	4.62880	0.348909
$177$	16.4277	1.23478
$178$	10.6094	0.795210
$179$	−10.2858	−0.768794	−0.384397	−	0.923168i	$-0.625591\pi$
$179$	−10.2858	−0.768794	−0.384397	+	0.923168i	$0.625591\pi$
$180$	0	0
$181$	23.0997	1.71699	0.858495	−	0.512823i	$-0.171400\pi$
$181$	23.0997	1.71699	0.858495	+	0.512823i	$0.171400\pi$
$182$	−11.7660	−0.872152
$183$	−11.1340	−0.823048
$184$	−13.7810	−1.01595
$185$	0	0
$186$	19.9262	1.46106
$187$	4.80511	0.351384
$188$	−2.25039	−0.164126
$189$	23.3629	1.69940
$190$	0	0
$191$	13.6788	0.989766	0.494883	−	0.868960i	$-0.335211\pi$
$191$	13.6788	0.989766	0.494883	+	0.868960i	$0.335211\pi$
$192$	−8.90753	−0.642845
$193$	−9.48835	−0.682987	−0.341493	−	0.939884i	$-0.610933\pi$
$193$	−9.48835	−0.682987	−0.341493	+	0.939884i	$0.610933\pi$
$194$	5.76178	0.413672
$195$	0	0
$196$	4.03131	0.287951
$197$	17.2237	1.22713	0.613567	−	0.789643i	$-0.289734\pi$
$197$	17.2237	1.22713	0.613567	+	0.789643i	$0.289734\pi$
$198$	1.09751	0.0779966
$199$	14.4608	1.02510	0.512549	−	0.858658i	$-0.328701\pi$
$199$	14.4608	1.02510	0.512549	+	0.858658i	$0.328701\pi$
$200$	0	0
$201$	−17.8090	−1.25615
$202$	−21.7911	−1.53322
$203$	16.6012	1.16518
$204$	2.84133	0.198933
$205$	0	0
$206$	−16.7723	−1.16858
$207$	−3.93124	−0.273240
$208$	8.46430	0.586894
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	1.14231	0.0786401	0.0393200	−	0.999227i	$-0.487481\pi$
$211$	1.14231	0.0786401	0.0393200	+	0.999227i	$0.487481\pi$
$212$	1.10545	0.0759223
$213$	6.89074	0.472146
$214$	14.6844	1.00381
$215$	0	0
$216$	−13.9695	−0.950504
$217$	−35.4374	−2.40565
$218$	15.8434	1.07305
$219$	−9.13215	−0.617094
$220$	0	0
$221$	8.78671	0.591058
$222$	−16.6382	−1.11668
$223$	−6.86812	−0.459923	−0.229962	−	0.973200i	$-0.573860\pi$
$223$	−6.86812	−0.459923	−0.229962	+	0.973200i	$0.573860\pi$
$224$	−9.07417	−0.606293
$225$	0	0
$226$	11.1177	0.739541
$227$	0.104970	0.00696713	0.00348356	−	0.999994i	$-0.498891\pi$
$227$	0.104970	0.00696713	0.00348356	+	0.999994i	$0.498891\pi$
$228$	−0.591315	−0.0391608
$229$	14.9679	0.989106	0.494553	−	0.869147i	$-0.335332\pi$
$229$	14.9679	0.989106	0.494553	+	0.869147i	$0.335332\pi$
$230$	0	0
$231$	6.29760	0.414352
$232$	−9.92644	−0.651702
$233$	16.8004	1.10063	0.550316	−	0.834956i	$-0.314507\pi$
$233$	16.8004	1.10063	0.550316	+	0.834956i	$0.314507\pi$
$234$	2.00693	0.131197
$235$	0	0
$236$	−4.24155	−0.276101
$237$	−14.6550	−0.951946
$238$	−30.9178	−2.00410
$239$	−3.44343	−0.222737	−0.111369	−	0.993779i	$-0.535523\pi$
$239$	−3.44343	−0.222737	−0.111369	+	0.993779i	$0.535523\pi$
$240$	0	0
$241$	−14.8135	−0.954222	−0.477111	−	0.878843i	$-0.658316\pi$
$241$	−14.8135	−0.954222	−0.477111	+	0.878843i	$0.658316\pi$
$242$	1.54620	0.0993935
$243$	−7.20757	−0.462366
$244$	2.87474	0.184036
$245$	0	0
$246$	9.02626	0.575493
$247$	−1.82862	−0.116352
$248$	21.1893	1.34552
$249$	0.769475	0.0487635
$250$	0	0
$251$	8.76253	0.553086	0.276543	−	0.961002i	$-0.410811\pi$
$251$	8.76253	0.553086	0.276543	+	0.961002i	$0.410811\pi$
$252$	−1.15416	−0.0727050
$253$	−5.53843	−0.348198
$254$	5.18510	0.325342
$255$	0	0
$256$	9.04303	0.565189
$257$	−2.52960	−0.157792	−0.0788960	−	0.996883i	$-0.525140\pi$
$257$	−2.52960	−0.157792	−0.0788960	+	0.996883i	$0.525140\pi$
$258$	−28.2150	−1.75659
$259$	29.5900	1.83863
$260$	0	0
$261$	−2.83167	−0.175276
$262$	−7.96893	−0.492322
$263$	15.3262	0.945053	0.472527	−	0.881316i	$-0.343342\pi$
$263$	15.3262	0.945053	0.472527	+	0.881316i	$0.343342\pi$
$264$	−3.76555	−0.231754
$265$	0	0
$266$	6.43436	0.394516
$267$	−10.3839	−0.635486
$268$	4.59818	0.280878
$269$	19.1624	1.16835	0.584176	−	0.811627i	$-0.301418\pi$
$269$	19.1624	1.16835	0.584176	+	0.811627i	$0.301418\pi$
$270$	0	0
$271$	30.2567	1.83796	0.918982	−	0.394299i	$-0.129012\pi$
$271$	30.2567	1.83796	0.918982	+	0.394299i	$0.129012\pi$
$272$	22.2419	1.34861
$273$	11.5159	0.696974
$274$	17.4882	1.05650
$275$	0	0
$276$	−3.27495	−0.197129
$277$	−4.07877	−0.245069	−0.122535	−	0.992464i	$-0.539102\pi$
$277$	−4.07877	−0.245069	−0.122535	+	0.992464i	$0.539102\pi$
$278$	7.95697	0.477227
$279$	6.04457	0.361879
$280$	0	0
$281$	3.04066	0.181390	0.0906952	−	0.995879i	$-0.471091\pi$
$281$	3.04066	0.181390	0.0906952	+	0.995879i	$0.471091\pi$
$282$	13.4765	0.802512
$283$	−2.24468	−0.133432	−0.0667162	−	0.997772i	$-0.521252\pi$
$283$	−2.24468	−0.133432	−0.0667162	+	0.997772i	$0.521252\pi$
$284$	−1.77915	−0.105573
$285$	0	0
$286$	2.82741	0.167188
$287$	−16.0526	−0.947556
$288$	1.54778	0.0912040
$289$	6.08907	0.358181
$290$	0	0
$291$	−5.63932	−0.330583
$292$	2.35787	0.137984
$293$	−18.3882	−1.07425	−0.537124	−	0.843503i	$-0.680489\pi$
$293$	−18.3882	−1.07425	−0.537124	+	0.843503i	$0.680489\pi$
$294$	−24.1415	−1.40796
$295$	0	0
$296$	−17.6929	−1.02838
$297$	−5.61419	−0.325769
$298$	15.9878	0.926147
$299$	−10.1277	−0.585698
$300$	0	0
$301$	50.1785	2.89224
$302$	−7.52925	−0.433260
$303$	21.3280	1.22526
$304$	−4.62880	−0.265480
$305$	0	0
$306$	5.27365	0.301475
$307$	−13.5196	−0.771607	−0.385804	−	0.922581i	$-0.626076\pi$
$307$	−13.5196	−0.771607	−0.385804	+	0.922581i	$0.626076\pi$
$308$	−1.62601	−0.0926503
$309$	16.4158	0.933862
$310$	0	0
$311$	27.3678	1.55188	0.775941	−	0.630805i	$-0.217275\pi$
$311$	27.3678	1.55188	0.775941	+	0.630805i	$0.217275\pi$
$312$	−6.88576	−0.389829
$313$	21.1505	1.19550	0.597748	−	0.801684i	$-0.296063\pi$
$313$	21.1505	1.19550	0.597748	+	0.801684i	$0.296063\pi$
$314$	26.8777	1.51680
$315$	0	0
$316$	3.78385	0.212858
$317$	4.68075	0.262897	0.131448	−	0.991323i	$-0.458037\pi$
$317$	4.68075	0.262897	0.131448	+	0.991323i	$0.458037\pi$
$318$	−6.61997	−0.371230
$319$	−3.98933	−0.223360
$320$	0	0
$321$	−14.3723	−0.802185
$322$	35.6362	1.98593
$323$	−4.80511	−0.267363
$324$	−2.48771	−0.138206
$325$	0	0
$326$	−26.3053	−1.45691
$327$	−15.5067	−0.857523
$328$	9.59842	0.529984
$329$	−23.9670	−1.32134
$330$	0	0
$331$	24.3025	1.33578	0.667892	−	0.744258i	$-0.267197\pi$
$331$	24.3025	1.33578	0.667892	+	0.744258i	$0.267197\pi$
$332$	−0.198674	−0.0109037
$333$	−5.04717	−0.276583
$334$	15.9769	0.874215
$335$	0	0
$336$	29.1503	1.59028
$337$	−11.6901	−0.636803	−0.318401	−	0.947956i	$-0.603146\pi$
$337$	−11.6901	−0.636803	−0.318401	+	0.947956i	$0.603146\pi$
$338$	−14.9304	−0.812104
$339$	−10.8814	−0.590999
$340$	0	0
$341$	8.51574	0.461153
$342$	−1.09751	−0.0593465
$343$	13.8043	0.745365
$344$	−30.0035	−1.61768
$345$	0	0
$346$	−23.4994	−1.26334
$347$	−9.85130	−0.528845	−0.264423	−	0.964407i	$-0.585181\pi$
$347$	−9.85130	−0.528845	−0.264423	+	0.964407i	$0.585181\pi$
$348$	−2.35895	−0.126453
$349$	−3.99813	−0.214015	−0.107007	−	0.994258i	$-0.534127\pi$
$349$	−3.99813	−0.214015	−0.107007	+	0.994258i	$0.534127\pi$
$350$	0	0
$351$	−10.2662	−0.547970
$352$	2.18056	0.116224
$353$	−7.73116	−0.411488	−0.205744	−	0.978606i	$-0.565961\pi$
$353$	−7.73116	−0.411488	−0.205744	+	0.978606i	$0.565961\pi$
$354$	25.4005	1.35002
$355$	0	0
$356$	2.68107	0.142097
$357$	30.2606	1.60156
$358$	−15.9038	−0.840545
$359$	4.83542	0.255204	0.127602	−	0.991825i	$-0.459272\pi$
$359$	4.83542	0.255204	0.127602	+	0.991825i	$0.459272\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	35.7168	1.87723
$363$	−1.51334	−0.0794296
$364$	−2.97334	−0.155846
$365$	0	0
$366$	−17.2154	−0.899862
$367$	29.8171	1.55644	0.778220	−	0.627992i	$-0.216123\pi$
$367$	29.8171	1.55644	0.778220	+	0.627992i	$0.216123\pi$
$368$	−25.6363	−1.33638
$369$	2.73810	0.142540
$370$	0	0
$371$	11.7732	0.611233
$372$	5.03548	0.261078
$373$	28.2081	1.46056	0.730280	−	0.683148i	$-0.239390\pi$
$373$	28.2081	1.46056	0.730280	+	0.683148i	$0.239390\pi$
$374$	7.42966	0.384179
$375$	0	0
$376$	14.3307	0.739050
$377$	−7.29496	−0.375710
$378$	36.1237	1.85800
$379$	−33.7427	−1.73325	−0.866624	−	0.498962i	$-0.833715\pi$
$379$	−33.7427	−1.73325	−0.866624	+	0.498962i	$0.833715\pi$
$380$	0	0
$381$	−5.07490	−0.259995
$382$	21.1502	1.08214
$383$	29.4579	1.50523	0.752615	−	0.658461i	$-0.228792\pi$
$383$	29.4579	1.50523	0.752615	+	0.658461i	$0.228792\pi$
$384$	−20.3727	−1.03964
$385$	0	0
$386$	−14.6709	−0.746729
$387$	−8.55896	−0.435076
$388$	1.45604	0.0739193
$389$	−24.5828	−1.24640	−0.623199	−	0.782063i	$-0.714167\pi$
$389$	−24.5828	−1.24640	−0.623199	+	0.782063i	$0.714167\pi$
$390$	0	0
$391$	−26.6128	−1.34586
$392$	−25.6718	−1.29662
$393$	7.79956	0.393436
$394$	26.6312	1.34166
$395$	0	0
$396$	0.277348	0.0139373
$397$	−22.5952	−1.13402	−0.567009	−	0.823711i	$-0.691900\pi$
$397$	−22.5952	−1.13402	−0.567009	+	0.823711i	$0.691900\pi$
$398$	22.3593	1.12077
$399$	−6.29760	−0.315274
$400$	0	0
$401$	−10.9449	−0.546562	−0.273281	−	0.961934i	$-0.588109\pi$
$401$	−10.9449	−0.546562	−0.273281	+	0.961934i	$0.588109\pi$
$402$	−27.5362	−1.37338
$403$	15.5720	0.775699
$404$	−5.50677	−0.273972
$405$	0	0
$406$	25.6688	1.27392
$407$	−7.11059	−0.352459
$408$	−18.0939	−0.895781
$409$	30.7338	1.51969	0.759844	−	0.650106i	$-0.225275\pi$
$409$	30.7338	1.51969	0.759844	+	0.650106i	$0.225275\pi$
$410$	0	0
$411$	−17.1165	−0.844295
$412$	−4.23847	−0.208814
$413$	−45.1732	−2.22283
$414$	−6.07848	−0.298741
$415$	0	0
$416$	3.98741	0.195499
$417$	−7.78785	−0.381373
$418$	−1.54620	−0.0756271
$419$	−35.3598	−1.72744	−0.863720	−	0.503972i	$-0.831871\pi$
$419$	−35.3598	−1.72744	−0.863720	+	0.503972i	$0.831871\pi$
$420$	0	0
$421$	−38.7689	−1.88948	−0.944740	−	0.327820i	$-0.893686\pi$
$421$	−38.7689	−1.88948	−0.944740	+	0.327820i	$0.893686\pi$
$422$	1.76624	0.0859795
$423$	4.08806	0.198768
$424$	−7.03960	−0.341873
$425$	0	0
$426$	10.6545	0.516211
$427$	30.6164	1.48163
$428$	3.71086	0.179371
$429$	−2.76732	−0.133607
$430$	0	0
$431$	29.7370	1.43238	0.716191	−	0.697904i	$-0.245884\pi$
$431$	29.7370	1.43238	0.716191	+	0.697904i	$0.245884\pi$
$432$	−25.9870	−1.25030
$433$	8.54319	0.410559	0.205280	−	0.978703i	$-0.434190\pi$
$433$	8.54319	0.410559	0.205280	+	0.978703i	$0.434190\pi$
$434$	−54.7933	−2.63016
$435$	0	0
$436$	4.00375	0.191745
$437$	5.53843	0.264939
$438$	−14.1201	−0.674686
$439$	−2.07249	−0.0989146	−0.0494573	−	0.998776i	$-0.515749\pi$
$439$	−2.07249	−0.0989146	−0.0494573	+	0.998776i	$0.515749\pi$
$440$	0	0
$441$	−7.32329	−0.348728
$442$	13.5860	0.646221
$443$	−23.8457	−1.13295	−0.566473	−	0.824081i	$-0.691692\pi$
$443$	−23.8457	−1.13295	−0.566473	+	0.824081i	$0.691692\pi$
$444$	−4.20459	−0.199541
$445$	0	0
$446$	−10.6195	−0.502848
$447$	−15.6480	−0.740123
$448$	24.4941	1.15724
$449$	5.44413	0.256925	0.128462	−	0.991714i	$-0.458996\pi$
$449$	5.44413	0.256925	0.128462	+	0.991714i	$0.458996\pi$
$450$	0	0
$451$	3.85750	0.181643
$452$	2.80953	0.132149
$453$	7.36923	0.346236
$454$	0.162305	0.00761736
$455$	0	0
$456$	3.76555	0.176338
$457$	23.4484	1.09687	0.548435	−	0.836193i	$-0.315224\pi$
$457$	23.4484	1.09687	0.548435	+	0.836193i	$0.315224\pi$
$458$	23.1434	1.08142
$459$	−26.9768	−1.25917
$460$	0	0
$461$	−37.5982	−1.75112	−0.875560	−	0.483109i	$-0.839508\pi$
$461$	−37.5982	−1.75112	−0.875560	+	0.483109i	$0.839508\pi$
$462$	9.73735	0.453022
$463$	−27.2719	−1.26743	−0.633716	−	0.773566i	$-0.718471\pi$
$463$	−27.2719	−1.26743	−0.633716	+	0.773566i	$0.718471\pi$
$464$	−18.4658	−0.857253
$465$	0	0
$466$	25.9768	1.20335
$467$	−14.2213	−0.658082	−0.329041	−	0.944316i	$-0.606725\pi$
$467$	−14.2213	−0.658082	−0.329041	+	0.944316i	$0.606725\pi$
$468$	0.507164	0.0234437
$469$	48.9713	2.26129
$470$	0	0
$471$	−26.3065	−1.21214
$472$	27.0106	1.24327
$473$	−12.0581	−0.554431
$474$	−22.6596	−1.04079
$475$	0	0
$476$	−7.81314	−0.358115
$477$	−2.00816	−0.0919471
$478$	−5.32424	−0.243525
$479$	1.56056	0.0713037	0.0356518	−	0.999364i	$-0.488649\pi$
$479$	1.56056	0.0713037	0.0356518	+	0.999364i	$0.488649\pi$
$480$	0	0
$481$	−13.0025	−0.592865
$482$	−22.9047	−1.04328
$483$	−34.8788	−1.58704
$484$	0.390736	0.0177607
$485$	0	0
$486$	−11.1443	−0.505518
$487$	17.8046	0.806803	0.403402	−	0.915023i	$-0.367828\pi$
$487$	17.8046	0.806803	0.403402	+	0.915023i	$0.367828\pi$
$488$	−18.3066	−0.828702
$489$	25.7462	1.16428
$490$	0	0
$491$	4.97956	0.224724	0.112362	−	0.993667i	$-0.464158\pi$
$491$	4.97956	0.224724	0.112362	+	0.993667i	$0.464158\pi$
$492$	2.28100	0.102835
$493$	−19.1692	−0.863336
$494$	−2.82741	−0.127211
$495$	0	0
$496$	39.4176	1.76990
$497$	−18.9483	−0.849946
$498$	1.18976	0.0533146
$499$	7.14052	0.319654	0.159827	−	0.987145i	$-0.448906\pi$
$499$	7.14052	0.319654	0.159827	+	0.987145i	$0.448906\pi$
$500$	0	0
$501$	−15.6373	−0.698623
$502$	13.5486	0.604705
$503$	−39.9024	−1.77916	−0.889579	−	0.456781i	$-0.849002\pi$
$503$	−39.9024	−1.77916	−0.889579	+	0.456781i	$0.849002\pi$
$504$	7.34979	0.327386
$505$	0	0
$506$	−8.56352	−0.380695
$507$	14.6130	0.648987
$508$	1.31031	0.0581357
$509$	2.18899	0.0970252	0.0485126	−	0.998823i	$-0.484552\pi$
$509$	2.18899	0.0970252	0.0485126	+	0.998823i	$0.484552\pi$
$510$	0	0
$511$	25.1117	1.11088
$512$	−12.9418	−0.571953
$513$	5.61419	0.247873
$514$	−3.91127	−0.172519
$515$	0	0
$516$	−7.13012	−0.313886
$517$	5.75936	0.253297
$518$	45.7520	2.01023
$519$	23.0000	1.00959
$520$	0	0
$521$	21.6375	0.947957	0.473979	−	0.880536i	$-0.342817\pi$
$521$	21.6375	0.947957	0.473979	+	0.880536i	$0.342817\pi$
$522$	−4.37833	−0.191634
$523$	−26.8411	−1.17368	−0.586840	−	0.809703i	$-0.699628\pi$
$523$	−26.8411	−1.17368	−0.586840	+	0.809703i	$0.699628\pi$
$524$	−2.01380	−0.0879735
$525$	0	0
$526$	23.6974	1.03325
$527$	40.9191	1.78246
$528$	−7.00493	−0.304850
$529$	7.67419	0.333661
$530$	0	0
$531$	7.70520	0.334377
$532$	1.62601	0.0704963
$533$	7.05390	0.305538
$534$	−16.0556	−0.694795
$535$	0	0
$536$	−29.2817	−1.26478
$537$	15.5658	0.671715
$538$	29.6289	1.27739
$539$	−10.3172	−0.444395
$540$	0	0
$541$	−44.0968	−1.89587	−0.947935	−	0.318465i	$-0.896833\pi$
$541$	−44.0968	−1.89587	−0.947935	+	0.318465i	$0.896833\pi$
$542$	46.7829	2.00950
$543$	−34.9577	−1.50018
$544$	10.4778	0.449232
$545$	0	0
$546$	17.8059	0.762022
$547$	21.5021	0.919362	0.459681	−	0.888084i	$-0.347964\pi$
$547$	21.5021	0.919362	0.459681	+	0.888084i	$0.347964\pi$
$548$	4.41939	0.188787
$549$	−5.22225	−0.222880
$550$	0	0
$551$	3.98933	0.169951
$552$	20.8553	0.887658
$553$	40.2986	1.71367
$554$	−6.30659	−0.267941
$555$	0	0
$556$	2.01078	0.0852761
$557$	33.2260	1.40783	0.703915	−	0.710284i	$-0.251434\pi$
$557$	33.2260	1.40783	0.703915	+	0.710284i	$0.251434\pi$
$558$	9.34611	0.395652
$559$	−22.0496	−0.932600
$560$	0	0
$561$	−7.27175	−0.307013
$562$	4.70146	0.198319
$563$	7.31538	0.308306	0.154153	−	0.988047i	$-0.450735\pi$
$563$	7.31538	0.308306	0.154153	+	0.988047i	$0.450735\pi$
$564$	3.40560	0.143401
$565$	0	0
$566$	−3.47073	−0.145885
$567$	−26.4945	−1.11267
$568$	11.3298	0.475389
$569$	7.20515	0.302056	0.151028	−	0.988530i	$-0.451742\pi$
$569$	7.20515	0.302056	0.151028	+	0.988530i	$0.451742\pi$
$570$	0	0
$571$	15.0655	0.630472	0.315236	−	0.949013i	$-0.397916\pi$
$571$	15.0655	0.630472	0.315236	+	0.949013i	$0.397916\pi$
$572$	0.714506	0.0298750
$573$	−20.7007	−0.864783
$574$	−24.8206	−1.03599
$575$	0	0
$576$	−4.17796	−0.174082
$577$	−5.82824	−0.242633	−0.121316	−	0.992614i	$-0.538712\pi$
$577$	−5.82824	−0.242633	−0.121316	+	0.992614i	$0.538712\pi$
$578$	9.41492	0.391609
$579$	14.3591	0.596743
$580$	0	0
$581$	−2.11592	−0.0877830
$582$	−8.71952	−0.361436
$583$	−2.82914	−0.117171
$584$	−15.0152	−0.621333
$585$	0	0
$586$	−28.4318	−1.17451
$587$	40.6364	1.67724	0.838622	−	0.544714i	$-0.183362\pi$
$587$	40.6364	1.67724	0.838622	+	0.544714i	$0.183362\pi$
$588$	−6.10073	−0.251590
$589$	−8.51574	−0.350885
$590$	0	0
$591$	−26.0652	−1.07218
$592$	−32.9135	−1.35273
$593$	32.0839	1.31753	0.658764	−	0.752350i	$-0.271080\pi$
$593$	32.0839	1.31753	0.658764	+	0.752350i	$0.271080\pi$
$594$	−8.68067	−0.356172
$595$	0	0
$596$	4.04022	0.165494
$597$	−21.8840	−0.895654
$598$	−15.6594	−0.640361
$599$	−47.1879	−1.92805	−0.964023	−	0.265820i	$-0.914357\pi$
$599$	−47.1879	−1.92805	−0.964023	+	0.265820i	$0.914357\pi$
$600$	0	0
$601$	−6.75179	−0.275411	−0.137706	−	0.990473i	$-0.543973\pi$
$601$	−6.75179	−0.275411	−0.137706	+	0.990473i	$0.543973\pi$
$602$	77.5860	3.16217
$603$	−8.35305	−0.340163
$604$	−1.90269	−0.0774195
$605$	0	0
$606$	32.9774	1.33961
$607$	−37.4849	−1.52147	−0.760733	−	0.649065i	$-0.775160\pi$
$607$	−37.4849	−1.52147	−0.760733	+	0.649065i	$0.775160\pi$
$608$	−2.18056	−0.0884332
$609$	−25.1232	−1.01804
$610$	0	0
$611$	10.5317	0.426066
$612$	1.33269	0.0538708
$613$	−35.4094	−1.43017	−0.715085	−	0.699037i	$-0.753612\pi$
$613$	−35.4094	−1.43017	−0.715085	+	0.699037i	$0.753612\pi$
$614$	−20.9041	−0.843620
$615$	0	0
$616$	10.3546	0.417198
$617$	27.4510	1.10513	0.552567	−	0.833469i	$-0.313648\pi$
$617$	27.4510	1.10513	0.552567	+	0.833469i	$0.313648\pi$
$618$	25.3821	1.02102
$619$	−7.15830	−0.287716	−0.143858	−	0.989598i	$-0.545951\pi$
$619$	−7.15830	−0.287716	−0.143858	+	0.989598i	$0.545951\pi$
$620$	0	0
$621$	31.0938	1.24775
$622$	42.3160	1.69672
$623$	28.5539	1.14399
$624$	−12.8093	−0.512784
$625$	0	0
$626$	32.7029	1.30707
$627$	1.51334	0.0604369
$628$	6.79219	0.271038
$629$	−34.1671	−1.36233
$630$	0	0
$631$	−28.7158	−1.14316	−0.571578	−	0.820547i	$-0.693669\pi$
$631$	−28.7158	−1.14316	−0.571578	+	0.820547i	$0.693669\pi$
$632$	−24.0959	−0.958485
$633$	−1.72870	−0.0687098
$634$	7.23737	0.287433
$635$	0	0
$636$	−1.67291	−0.0663353
$637$	−18.8663	−0.747509
$638$	−6.16831	−0.244206
$639$	3.23201	0.127856
$640$	0	0
$641$	−42.9018	−1.69452	−0.847259	−	0.531180i	$-0.821749\pi$
$641$	−42.9018	−1.69452	−0.847259	+	0.531180i	$0.821749\pi$
$642$	−22.2225	−0.877052
$643$	26.1604	1.03167	0.515834	−	0.856689i	$-0.327482\pi$
$643$	26.1604	1.03167	0.515834	+	0.856689i	$0.327482\pi$
$644$	9.00552	0.354867
$645$	0	0
$646$	−7.42966	−0.292316
$647$	−32.5031	−1.27783	−0.638915	−	0.769277i	$-0.720616\pi$
$647$	−32.5031	−1.27783	−0.638915	+	0.769277i	$0.720616\pi$
$648$	15.8420	0.622333
$649$	10.8553	0.426107
$650$	0	0
$651$	53.6287	2.10188
$652$	−6.64752	−0.260337
$653$	−18.9389	−0.741137	−0.370569	−	0.928805i	$-0.620837\pi$
$653$	−18.9389	−0.741137	−0.370569	+	0.928805i	$0.620837\pi$
$654$	−23.9765	−0.937554
$655$	0	0
$656$	17.8556	0.697144
$657$	−4.28331	−0.167108
$658$	−37.0578	−1.44466
$659$	43.5740	1.69740	0.848701	−	0.528873i	$-0.177385\pi$
$659$	43.5740	1.69740	0.848701	+	0.528873i	$0.177385\pi$
$660$	0	0
$661$	−34.9291	−1.35858	−0.679292	−	0.733868i	$-0.737713\pi$
$661$	−34.9291	−1.35858	−0.679292	+	0.733868i	$0.737713\pi$
$662$	37.5765	1.46045
$663$	−13.2973	−0.516422
$664$	1.26518	0.0490985
$665$	0	0
$666$	−7.80394	−0.302396
$667$	22.0946	0.855507
$668$	4.03746	0.156214
$669$	10.3938	0.401847
$670$	0	0
$671$	−7.35724	−0.284023
$672$	13.7323	0.529734
$673$	−39.0030	−1.50346	−0.751728	−	0.659473i	$-0.770780\pi$
$673$	−39.0030	−1.50346	−0.751728	+	0.659473i	$0.770780\pi$
$674$	−18.0753	−0.696235
$675$	0	0
$676$	−3.77300	−0.145116
$677$	−27.3864	−1.05255	−0.526273	−	0.850316i	$-0.676411\pi$
$677$	−27.3864	−1.05255	−0.526273	+	0.850316i	$0.676411\pi$
$678$	−16.8249	−0.646156
$679$	15.5071	0.595107
$680$	0	0
$681$	−0.158856	−0.00608736
$682$	13.1670	0.504192
$683$	−44.9382	−1.71951	−0.859755	−	0.510706i	$-0.829384\pi$
$683$	−44.9382	−1.71951	−0.859755	+	0.510706i	$0.829384\pi$
$684$	−0.277348	−0.0106047
$685$	0	0
$686$	21.3443	0.814929
$687$	−22.6515	−0.864207
$688$	−55.8144	−2.12790
$689$	−5.17342	−0.197092
$690$	0	0
$691$	25.4509	0.968199	0.484100	−	0.875013i	$-0.339147\pi$
$691$	25.4509	0.968199	0.484100	+	0.875013i	$0.339147\pi$
$692$	−5.93847	−0.225747
$693$	2.95381	0.112206
$694$	−15.2321	−0.578202
$695$	0	0
$696$	15.0220	0.569409
$697$	18.5357	0.702091
$698$	−6.18191	−0.233989
$699$	−25.4247	−0.961651
$700$	0	0
$701$	50.6146	1.91169	0.955844	−	0.293873i	$-0.0949444\pi$
$701$	50.6146	1.91169	0.955844	+	0.293873i	$0.0949444\pi$
$702$	−15.8736	−0.599111
$703$	7.11059	0.268181
$704$	−5.88602	−0.221838
$705$	0	0
$706$	−11.9539	−0.449892
$707$	−58.6481	−2.20569
$708$	6.41889	0.241237
$709$	25.8266	0.969938	0.484969	−	0.874531i	$-0.338831\pi$
$709$	25.8266	0.969938	0.484969	+	0.874531i	$0.338831\pi$
$710$	0	0
$711$	−6.87374	−0.257786
$712$	−17.0734	−0.639852
$713$	−47.1638	−1.76630
$714$	46.7890	1.75104
$715$	0	0
$716$	−4.01901	−0.150198
$717$	5.21107	0.194611
$718$	7.47653	0.279021
$719$	−26.1220	−0.974187	−0.487093	−	0.873350i	$-0.661943\pi$
$719$	−26.1220	−0.974187	−0.487093	+	0.873350i	$0.661943\pi$
$720$	0	0
$721$	−45.1404	−1.68112
$722$	1.54620	0.0575436
$723$	22.4178	0.833728
$724$	9.02588	0.335444
$725$	0	0
$726$	−2.33992	−0.0868427
$727$	32.1853	1.19369	0.596844	−	0.802357i	$-0.296421\pi$
$727$	32.1853	1.19369	0.596844	+	0.802357i	$0.296421\pi$
$728$	18.9346	0.701762
$729$	30.0077	1.11140
$730$	0	0
$731$	−57.9404	−2.14300
$732$	−4.35044	−0.160797
$733$	24.9746	0.922457	0.461229	−	0.887281i	$-0.347409\pi$
$733$	24.9746	0.922457	0.461229	+	0.887281i	$0.347409\pi$
$734$	46.1032	1.70170
$735$	0	0
$736$	−12.0769	−0.445159
$737$	−11.7680	−0.433480
$738$	4.23365	0.155843
$739$	−21.0642	−0.774859	−0.387429	−	0.921899i	$-0.626637\pi$
$739$	−21.0642	−0.774859	−0.387429	+	0.921899i	$0.626637\pi$
$740$	0	0
$741$	2.76732	0.101660
$742$	18.2037	0.668279
$743$	19.3902	0.711357	0.355679	−	0.934608i	$-0.384250\pi$
$743$	19.3902	0.711357	0.355679	+	0.934608i	$0.384250\pi$
$744$	−32.0665	−1.17561
$745$	0	0
$746$	43.6154	1.59687
$747$	0.360912	0.0132051
$748$	1.87753	0.0686492
$749$	39.5213	1.44408
$750$	0	0
$751$	−32.5618	−1.18820	−0.594098	−	0.804393i	$-0.702491\pi$
$751$	−32.5618	−1.18820	−0.594098	+	0.804393i	$0.702491\pi$
$752$	26.6589	0.972151
$753$	−13.2607	−0.483245
$754$	−11.2795	−0.410774
$755$	0	0
$756$	9.12871	0.332008
$757$	−40.4497	−1.47017	−0.735084	−	0.677976i	$-0.762857\pi$
$757$	−40.4497	−1.47017	−0.735084	+	0.677976i	$0.762857\pi$
$758$	−52.1730	−1.89501
$759$	8.38151	0.304230
$760$	0	0
$761$	20.6529	0.748666	0.374333	−	0.927294i	$-0.377872\pi$
$761$	20.6529	0.748666	0.374333	+	0.927294i	$0.377872\pi$
$762$	−7.84681	−0.284260
$763$	42.6406	1.54369
$764$	5.34481	0.193368
$765$	0	0
$766$	45.5479	1.64571
$767$	19.8502	0.716749
$768$	−13.6852	−0.493820
$769$	−2.53650	−0.0914686	−0.0457343	−	0.998954i	$-0.514563\pi$
$769$	−2.53650	−0.0914686	−0.0457343	+	0.998954i	$0.514563\pi$
$770$	0	0
$771$	3.82813	0.137867
$772$	−3.70744	−0.133434
$773$	−27.4490	−0.987273	−0.493637	−	0.869668i	$-0.664333\pi$
$773$	−27.4490	−0.987273	−0.493637	+	0.869668i	$0.664333\pi$
$774$	−13.2339	−0.475681
$775$	0	0
$776$	−9.27223	−0.332854
$777$	−44.7796	−1.60646
$778$	−38.0100	−1.36272
$779$	−3.85750	−0.138209
$780$	0	0
$781$	4.55334	0.162931
$782$	−41.1486	−1.47147
$783$	22.3969	0.800399
$784$	−47.7564	−1.70559
$785$	0	0
$786$	12.0597	0.430155
$787$	−39.4380	−1.40581	−0.702907	−	0.711281i	$-0.748115\pi$
$787$	−39.4380	−1.40581	−0.702907	+	0.711281i	$0.748115\pi$
$788$	6.72989	0.239742
$789$	−23.1937	−0.825717
$790$	0	0
$791$	29.9220	1.06390
$792$	−1.76618	−0.0627586
$793$	−13.4536	−0.477751
$794$	−34.9367	−1.23986
$795$	0	0
$796$	5.65034	0.200271
$797$	−38.1840	−1.35255	−0.676273	−	0.736651i	$-0.736406\pi$
$797$	−38.1840	−1.35255	−0.676273	+	0.736651i	$0.736406\pi$
$798$	−9.73735	−0.344698
$799$	27.6744	0.979049
$800$	0	0
$801$	−4.87044	−0.172089
$802$	−16.9230	−0.597572
$803$	−6.03445	−0.212951
$804$	−6.95859	−0.245411
$805$	0	0
$806$	24.0775	0.848094
$807$	−28.9992	−1.02082
$808$	35.0677	1.23368
$809$	−15.2742	−0.537011	−0.268506	−	0.963278i	$-0.586530\pi$
$809$	−15.2742	−0.537011	−0.268506	+	0.963278i	$0.586530\pi$
$810$	0	0
$811$	9.16787	0.321928	0.160964	−	0.986960i	$-0.448540\pi$
$811$	9.16787	0.321928	0.160964	+	0.986960i	$0.448540\pi$
$812$	6.48668	0.227638
$813$	−45.7886	−1.60588
$814$	−10.9944	−0.385353
$815$	0	0
$816$	−33.6594	−1.17832
$817$	12.0581	0.421859
$818$	47.5206	1.66152
$819$	5.40138	0.188740
$820$	0	0
$821$	−8.97266	−0.313148	−0.156574	−	0.987666i	$-0.550045\pi$
$821$	−8.97266	−0.313148	−0.156574	+	0.987666i	$0.550045\pi$
$822$	−26.4656	−0.923092
$823$	11.8425	0.412805	0.206402	−	0.978467i	$-0.433824\pi$
$823$	11.8425	0.412805	0.206402	+	0.978467i	$0.433824\pi$
$824$	26.9910	0.940277
$825$	0	0
$826$	−69.8468	−2.43028
$827$	35.1524	1.22237	0.611184	−	0.791488i	$-0.290693\pi$
$827$	35.1524	1.22237	0.611184	+	0.791488i	$0.290693\pi$
$828$	−1.53607	−0.0533823
$829$	−37.8166	−1.31343	−0.656713	−	0.754141i	$-0.728054\pi$
$829$	−37.8166	−1.31343	−0.656713	+	0.754141i	$0.728054\pi$
$830$	0	0
$831$	6.17255	0.214123
$832$	−10.7633	−0.373149
$833$	−49.5754	−1.71769
$834$	−12.0416	−0.416966
$835$	0	0
$836$	−0.390736	−0.0135139
$837$	−47.8090	−1.65252
$838$	−54.6733	−1.88866
$839$	25.0571	0.865067	0.432533	−	0.901618i	$-0.357620\pi$
$839$	25.0571	0.865067	0.432533	+	0.901618i	$0.357620\pi$
$840$	0	0
$841$	−13.0852	−0.451215
$842$	−59.9445	−2.06582
$843$	−4.60154	−0.158485
$844$	0.446342	0.0153637
$845$	0	0
$846$	6.32096	0.217319
$847$	4.16140	0.142987
$848$	−13.0955	−0.449702
$849$	3.39696	0.116583
$850$	0	0
$851$	39.3815	1.34998
$852$	2.69246	0.0922421
$853$	−23.8418	−0.816329	−0.408164	−	0.912908i	$-0.633831\pi$
$853$	−23.8418	−0.816329	−0.408164	+	0.912908i	$0.633831\pi$
$854$	47.3391	1.61991
$855$	0	0
$856$	−23.6311	−0.807696
$857$	34.6121	1.18233	0.591164	−	0.806551i	$-0.298669\pi$
$857$	34.6121	1.18233	0.591164	+	0.806551i	$0.298669\pi$
$858$	−4.27882	−0.146077
$859$	32.6520	1.11407	0.557035	−	0.830489i	$-0.311939\pi$
$859$	32.6520	1.11407	0.557035	+	0.830489i	$0.311939\pi$
$860$	0	0
$861$	24.2930	0.827904
$862$	45.9794	1.56606
$863$	10.1698	0.346182	0.173091	−	0.984906i	$-0.444624\pi$
$863$	10.1698	0.346182	0.173091	+	0.984906i	$0.444624\pi$
$864$	−12.2421	−0.416484
$865$	0	0
$866$	13.2095	0.448876
$867$	−9.21481	−0.312951
$868$	−13.8467	−0.469986
$869$	−9.68391	−0.328504
$870$	0	0
$871$	−21.5192	−0.729150
$872$	−25.4963	−0.863414
$873$	−2.64505	−0.0895213
$874$	8.56352	0.289665
$875$	0	0
$876$	−3.56826	−0.120560
$877$	−8.91797	−0.301138	−0.150569	−	0.988599i	$-0.548111\pi$
$877$	−8.91797	−0.301138	−0.150569	+	0.988599i	$0.548111\pi$
$878$	−3.20449	−0.108146
$879$	27.8275	0.938598
$880$	0	0
$881$	−9.08968	−0.306239	−0.153120	−	0.988208i	$-0.548932\pi$
$881$	−9.08968	−0.306239	−0.153120	+	0.988208i	$0.548932\pi$
$882$	−11.3233	−0.381274
$883$	45.0015	1.51442	0.757210	−	0.653171i	$-0.226562\pi$
$883$	45.0015	1.51442	0.757210	+	0.653171i	$0.226562\pi$
$884$	3.43328	0.115474
$885$	0	0
$886$	−36.8703	−1.23868
$887$	21.4687	0.720850	0.360425	−	0.932788i	$-0.382632\pi$
$887$	21.4687	0.720850	0.360425	+	0.932788i	$0.382632\pi$
$888$	26.7753	0.898520
$889$	13.9550	0.468037
$890$	0	0
$891$	6.36674	0.213294
$892$	−2.68362	−0.0898542
$893$	−5.75936	−0.192730
$894$	−24.1949	−0.809198
$895$	0	0
$896$	56.0211	1.87153
$897$	15.3266	0.511740
$898$	8.41772	0.280903
$899$	−33.9721	−1.13303
$900$	0	0
$901$	−13.5943	−0.452893
$902$	5.96447	0.198595
$903$	−75.9370	−2.52702
$904$	−17.8914	−0.595059
$905$	0	0
$906$	11.3943	0.378550
$907$	−12.5080	−0.415322	−0.207661	−	0.978201i	$-0.566585\pi$
$907$	−12.5080	−0.415322	−0.207661	+	0.978201i	$0.566585\pi$
$908$	0.0410156	0.00136115
$909$	10.0036	0.331799
$910$	0	0
$911$	−4.62780	−0.153326	−0.0766629	−	0.997057i	$-0.524427\pi$
$911$	−4.62780	−0.153326	−0.0766629	+	0.997057i	$0.524427\pi$
$912$	7.00493	0.231956
$913$	0.508463	0.0168277
$914$	36.2559	1.19924
$915$	0	0
$916$	5.84849	0.193239
$917$	−21.4474	−0.708254
$918$	−41.7116	−1.37669
$919$	−11.1376	−0.367395	−0.183698	−	0.982983i	$-0.558807\pi$
$919$	−11.1376	−0.367395	−0.183698	+	0.982983i	$0.558807\pi$
$920$	0	0
$921$	20.4598	0.674173
$922$	−58.1343	−1.91455
$923$	8.32632	0.274064
$924$	2.46070	0.0809509
$925$	0	0
$926$	−42.1678	−1.38572
$927$	7.69961	0.252888
$928$	−8.69896	−0.285557
$929$	−27.3911	−0.898672	−0.449336	−	0.893363i	$-0.648339\pi$
$929$	−27.3911	−0.898672	−0.449336	+	0.893363i	$0.648339\pi$
$930$	0	0
$931$	10.3172	0.338134
$932$	6.56452	0.215028
$933$	−41.4166	−1.35592
$934$	−21.9889	−0.719500
$935$	0	0
$936$	−3.22968	−0.105565
$937$	−41.4439	−1.35391	−0.676957	−	0.736022i	$-0.736702\pi$
$937$	−41.4439	−1.35391	−0.676957	+	0.736022i	$0.736702\pi$
$938$	75.7195	2.47233
$939$	−32.0078	−1.04453
$940$	0	0
$941$	−43.5568	−1.41991	−0.709956	−	0.704246i	$-0.751285\pi$
$941$	−43.5568	−1.41991	−0.709956	+	0.704246i	$0.751285\pi$
$942$	−40.6751	−1.32527
$943$	−21.3645	−0.695724
$944$	50.2469	1.63540
$945$	0	0
$946$	−18.6442	−0.606176
$947$	33.0491	1.07395	0.536976	−	0.843598i	$-0.319567\pi$
$947$	33.0491	1.07395	0.536976	+	0.843598i	$0.319567\pi$
$948$	−5.72624	−0.185980
$949$	−11.0347	−0.358201
$950$	0	0
$951$	−7.08355	−0.229700
$952$	49.7549	1.61257
$953$	−0.505736	−0.0163824	−0.00819120	−	0.999966i	$-0.502607\pi$
$953$	−0.505736	−0.0163824	−0.00819120	+	0.999966i	$0.502607\pi$
$954$	−3.10501	−0.100528
$955$	0	0
$956$	−1.34547	−0.0435157
$957$	6.03720	0.195155
$958$	2.41293	0.0779584
$959$	47.0672	1.51988
$960$	0	0
$961$	41.5179	1.33929
$962$	−20.1045	−0.648196
$963$	−6.74115	−0.217231
$964$	−5.78817	−0.186424
$965$	0	0
$966$	−53.9296	−1.73516
$967$	46.1493	1.48406	0.742031	−	0.670366i	$-0.233863\pi$
$967$	46.1493	1.48406	0.742031	+	0.670366i	$0.233863\pi$
$968$	−2.48825	−0.0799752
$969$	7.27175	0.233602
$970$	0	0
$971$	−27.0101	−0.866795	−0.433398	−	0.901203i	$-0.642685\pi$
$971$	−27.0101	−0.866795	−0.433398	+	0.901203i	$0.642685\pi$
$972$	−2.81625	−0.0903313
$973$	21.4152	0.686538
$974$	27.5295	0.882101
$975$	0	0
$976$	−34.0552	−1.09008
$977$	30.5323	0.976814	0.488407	−	0.872616i	$-0.337578\pi$
$977$	30.5323	0.976814	0.488407	+	0.872616i	$0.337578\pi$
$978$	39.8087	1.27294
$979$	−6.86161	−0.219298
$980$	0	0
$981$	−7.27322	−0.232216
$982$	7.69940	0.245698
$983$	−27.5882	−0.879926	−0.439963	−	0.898016i	$-0.645008\pi$
$983$	−27.5882	−0.879926	−0.439963	+	0.898016i	$0.645008\pi$
$984$	−14.5256	−0.463060
$985$	0	0
$986$	−29.6394	−0.943910
$987$	36.2702	1.15449
$988$	−0.714506	−0.0227315
$989$	66.7828	2.12357
$990$	0	0
$991$	−57.2710	−1.81927	−0.909636	−	0.415405i	$-0.863640\pi$
$991$	−57.2710	−1.81927	−0.909636	+	0.415405i	$0.863640\pi$
$992$	18.5691	0.589568
$993$	−36.7778	−1.16711
$994$	−29.2978	−0.929271
$995$	0	0
$996$	0.300661	0.00952682
$997$	50.4928	1.59912	0.799561	−	0.600584i	$-0.205065\pi$
$997$	50.4928	1.59912	0.799561	+	0.600584i	$0.205065\pi$
$998$	11.0407	0.349486
$999$	39.9202	1.26302

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5225.2.a.j.1.4		5
5.4	even	2		1045.2.a.d.1.2	✓	5
15.14	odd	2		9405.2.a.v.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.a.d.1.2	✓	5	5.4	even	2
5225.2.a.j.1.4		5	1.1	even	1	trivial
9405.2.a.v.1.4		5	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.54620 q^{2} -1.51334 q^{3} +0.390736 q^{4} -2.33992 q^{6} +4.16140 q^{7} -2.48825 q^{8} -0.709811 q^{9} -1.00000 q^{11} -0.591315 q^{12} -1.82862 q^{13} +6.43436 q^{14} -4.62880 q^{16} -4.80511 q^{17} -1.09751 q^{18} +1.00000 q^{19} -6.29760 q^{21} -1.54620 q^{22} +5.53843 q^{23} +3.76555 q^{24} -2.82741 q^{26} +5.61419 q^{27} +1.62601 q^{28} +3.98933 q^{29} -8.51574 q^{31} -2.18056 q^{32} +1.51334 q^{33} -7.42966 q^{34} -0.277348 q^{36} +7.11059 q^{37} +1.54620 q^{38} +2.76732 q^{39} -3.85750 q^{41} -9.73735 q^{42} +12.0581 q^{43} -0.390736 q^{44} +8.56352 q^{46} -5.75936 q^{47} +7.00493 q^{48} +10.3172 q^{49} +7.27175 q^{51} -0.714506 q^{52} +2.82914 q^{53} +8.68067 q^{54} -10.3546 q^{56} -1.51334 q^{57} +6.16831 q^{58} -10.8553 q^{59} +7.35724 q^{61} -13.1670 q^{62} -2.95381 q^{63} +5.88602 q^{64} +2.33992 q^{66} +11.7680 q^{67} -1.87753 q^{68} -8.38151 q^{69} -4.55334 q^{71} +1.76618 q^{72} +6.03445 q^{73} +10.9944 q^{74} +0.390736 q^{76} -4.16140 q^{77} +4.27882 q^{78} +9.68391 q^{79} -6.36674 q^{81} -5.96447 q^{82} -0.508463 q^{83} -2.46070 q^{84} +18.6442 q^{86} -6.03720 q^{87} +2.48825 q^{88} +6.86161 q^{89} -7.60961 q^{91} +2.16406 q^{92} +12.8872 q^{93} -8.90513 q^{94} +3.29992 q^{96} +3.72641 q^{97} +15.9525 q^{98} +0.709811 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 3 q^{2} + 7 q^{3} + 5 q^{4} + 2 q^{6} + 11 q^{7} - 3 q^{8} + 8 q^{9} - 5 q^{11} + 7 q^{12} - q^{13} - 3 q^{16} + 3 q^{17} + 7 q^{18} + 5 q^{19} + 11 q^{21} - 3 q^{22} + 8 q^{23} + 9 q^{24} - 16 q^{26}+ \cdots - 8 q^{99}+O(q^{100}) \) 5 * q + 3 * q^2 + 7 * q^3 + 5 * q^4 + 2 * q^6 + 11 * q^7 - 3 * q^8 + 8 * q^9 - 5 * q^11 + 7 * q^12 - q^13 - 3 * q^16 + 3 * q^17 + 7 * q^18 + 5 * q^19 + 11 * q^21 - 3 * q^22 + 8 * q^23 + 9 * q^24 - 16 * q^26 + 10 * q^27 + 22 * q^28 + 11 * q^29 - 5 * q^31 + 2 * q^32 - 7 * q^33 + 4 * q^34 - 3 * q^36 + 9 * q^37 + 3 * q^38 - 8 * q^39 + 15 * q^41 - 11 * q^42 + 13 * q^43 - 5 * q^44 + 18 * q^46 + 20 * q^47 + 20 * q^48 + 20 * q^49 + 24 * q^51 - q^52 + 5 * q^53 + 17 * q^54 + 7 * q^57 + 33 * q^58 - 17 * q^59 + 3 * q^61 - 14 * q^62 + 22 * q^63 - 17 * q^64 - 2 * q^66 + 28 * q^67 + 25 * q^68 - 2 * q^69 - 6 * q^71 + 26 * q^72 + 16 * q^73 - 21 * q^74 + 5 * q^76 - 11 * q^77 - 29 * q^78 + 3 * q^79 + q^81 - 2 * q^82 + 33 * q^83 + 33 * q^84 + 10 * q^86 + 3 * q^88 - 16 * q^89 - 22 * q^91 + 19 * q^92 + 26 * q^93 - 10 * q^94 + 5 * q^96 + 14 * q^97 - 10 * q^98 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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