LMFDB - Embedded newform 1225.4.a.z.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1225,4,Mod(1,1225)] 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("1225.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1225 = 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1225.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,-4,-3,36,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

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

Self dual:	yes
Analytic conductor:	$72.2773397570$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 32x^{2} - 35x + 120 $ x^4 - 32x^2 - 35x + 120
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 175)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$5.87199$ of defining polynomial
Character	$\chi$	$=$	1225.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.87199 q^{2} -4.14916 q^{3} +15.7363 q^{4} -20.2147 q^{6} +37.6910 q^{8} -9.78444 q^{9} +36.9922 q^{11} -65.2924 q^{12} -61.3165 q^{13} +57.7401 q^{16} -44.8345 q^{17} -47.6697 q^{18} +139.701 q^{19} +180.226 q^{22} -217.580 q^{23} -156.386 q^{24} -298.733 q^{26} +152.625 q^{27} -33.8226 q^{29} -124.437 q^{31} -20.2192 q^{32} -153.487 q^{33} -218.433 q^{34} -153.971 q^{36} -237.270 q^{37} +680.620 q^{38} +254.412 q^{39} -195.117 q^{41} +343.725 q^{43} +582.119 q^{44} -1060.05 q^{46} +16.8224 q^{47} -239.573 q^{48} +186.026 q^{51} -964.894 q^{52} -346.965 q^{53} +743.586 q^{54} -579.641 q^{57} -164.783 q^{58} -135.340 q^{59} -490.414 q^{61} -606.255 q^{62} -560.428 q^{64} -747.785 q^{66} -477.969 q^{67} -705.529 q^{68} +902.777 q^{69} +45.2557 q^{71} -368.786 q^{72} +100.781 q^{73} -1155.98 q^{74} +2198.37 q^{76} +1239.49 q^{78} +880.534 q^{79} -369.085 q^{81} -950.609 q^{82} -1155.03 q^{83} +1674.62 q^{86} +140.336 q^{87} +1394.27 q^{88} -619.374 q^{89} -3423.90 q^{92} +516.309 q^{93} +81.9585 q^{94} +83.8929 q^{96} -231.195 q^{97} -361.948 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} - 3 q^{3} + 36 q^{4} - q^{6} - 27 q^{8} + 61 q^{9} + 100 q^{11} - 165 q^{12} + 44 q^{13} + 160 q^{16} - 53 q^{17} - 433 q^{18} + 29 q^{19} + 152 q^{22} - 295 q^{23} + 21 q^{24} - 700 q^{26}+ \cdots + 2383 q^{99}+O(q^{100}) $ 4 * q - 4 * q^2 - 3 * q^3 + 36 * q^4 - q^6 - 27 * q^8 + 61 * q^9 + 100 * q^11 - 165 * q^12 + 44 * q^13 + 160 * q^16 - 53 * q^17 - 433 * q^18 + 29 * q^19 + 152 * q^22 - 295 * q^23 + 21 * q^24 - 700 * q^26 - 441 * q^27 + 129 * q^29 - 114 * q^31 + 310 * q^32 - 865 * q^33 - 203 * q^34 + 1101 * q^36 - 403 * q^37 + 555 * q^38 + 674 * q^39 - 671 * q^41 + 411 * q^43 + 438 * q^44 - 997 * q^46 - 8 * q^47 - 523 * q^48 - 885 * q^51 + 74 * q^52 - 90 * q^53 + 2777 * q^54 - 233 * q^57 - 673 * q^58 - 1018 * q^59 - 50 * q^61 - 1626 * q^62 - 2421 * q^64 + 3841 * q^66 - 424 * q^67 - 617 * q^68 - 1080 * q^69 + 215 * q^71 - 2940 * q^72 - 1207 * q^73 + 623 * q^74 + 3257 * q^76 - 278 * q^78 - 951 * q^79 + 28 * q^81 + 1695 * q^82 - 3035 * q^83 - 99 * q^86 + 2210 * q^87 - 163 * q^88 - 2819 * q^89 - 3073 * q^92 + 852 * q^93 + 3056 * q^94 + 1345 * q^96 - 1100 * q^97 + 2383 * 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 $

$2$	4.87199	1.72251	0.861254	−	0.508175i	$-0.169680\pi$
$2$	4.87199	1.72251	0.861254	+	0.508175i	$0.169680\pi$
$3$	−4.14916	−0.798507	−0.399253	−	0.916841i	$-0.630731\pi$
$3$	−4.14916	−0.798507	−0.399253	+	0.916841i	$0.630731\pi$
$4$	15.7363	1.96703
$5$	0	0
$5$	0	0
$6$	−20.2147	−1.37543
$7$	0	0
$7$	0	0
$8$	37.6910	1.66572
$9$	−9.78444	−0.362387
$10$	0	0
$11$	36.9922	1.01396	0.506980	−	0.861958i	$-0.330762\pi$
$11$	36.9922	1.01396	0.506980	+	0.861958i	$0.330762\pi$
$12$	−65.2924	−1.57069
$13$	−61.3165	−1.30817	−0.654083	−	0.756423i	$-0.726945\pi$
$13$	−61.3165	−1.30817	−0.654083	+	0.756423i	$0.726945\pi$
$14$	0	0
$15$	0	0
$16$	57.7401	0.902189
$17$	−44.8345	−0.639646	−0.319823	−	0.947477i	$-0.603623\pi$
$17$	−44.8345	−0.639646	−0.319823	+	0.947477i	$0.603623\pi$
$18$	−47.6697	−0.624214
$19$	139.701	1.68682	0.843408	−	0.537273i	$-0.180545\pi$
$19$	139.701	1.68682	0.843408	+	0.537273i	$0.180545\pi$
$20$	0	0
$21$	0	0
$22$	180.226	1.74656
$23$	−217.580	−1.97255	−0.986275	−	0.165110i	$-0.947202\pi$
$23$	−217.580	−1.97255	−0.986275	+	0.165110i	$0.947202\pi$
$24$	−156.386	−1.33009
$25$	0	0
$26$	−298.733	−2.25333
$27$	152.625	1.08788
$28$	0	0
$29$	−33.8226	−0.216576	−0.108288	−	0.994120i	$-0.534537\pi$
$29$	−33.8226	−0.216576	−0.108288	+	0.994120i	$0.534537\pi$
$30$	0	0
$31$	−124.437	−0.720952	−0.360476	−	0.932769i	$-0.617386\pi$
$31$	−124.437	−0.720952	−0.360476	+	0.932769i	$0.617386\pi$
$32$	−20.2192	−0.111697
$33$	−153.487	−0.809655
$34$	−218.433	−1.10179
$35$	0	0
$36$	−153.971	−0.712827
$37$	−237.270	−1.05424	−0.527121	−	0.849790i	$-0.676729\pi$
$37$	−237.270	−1.05424	−0.527121	+	0.849790i	$0.676729\pi$
$38$	680.620	2.90556
$39$	254.412	1.04458
$40$	0	0
$41$	−195.117	−0.743224	−0.371612	−	0.928388i	$-0.621195\pi$
$41$	−195.117	−0.743224	−0.371612	+	0.928388i	$0.621195\pi$
$42$	0	0
$43$	343.725	1.21901	0.609506	−	0.792781i	$-0.291368\pi$
$43$	343.725	1.21901	0.609506	+	0.792781i	$0.291368\pi$
$44$	582.119	1.99450
$45$	0	0
$46$	−1060.05	−3.39773
$47$	16.8224	0.0522085	0.0261042	−	0.999659i	$-0.491690\pi$
$47$	16.8224	0.0522085	0.0261042	+	0.999659i	$0.491690\pi$
$48$	−239.573	−0.720404
$49$	0	0
$50$	0	0
$51$	186.026	0.510761
$52$	−964.894	−2.57321
$53$	−346.965	−0.899231	−0.449616	−	0.893222i	$-0.648439\pi$
$53$	−346.965	−0.899231	−0.449616	+	0.893222i	$0.648439\pi$
$54$	743.586	1.87387
$55$	0	0
$56$	0	0
$57$	−579.641	−1.34694
$58$	−164.783	−0.373054
$59$	−135.340	−0.298640	−0.149320	−	0.988789i	$-0.547708\pi$
$59$	−135.340	−0.298640	−0.149320	+	0.988789i	$0.547708\pi$
$60$	0	0
$61$	−490.414	−1.02936	−0.514681	−	0.857382i	$-0.672089\pi$
$61$	−490.414	−1.02936	−0.514681	+	0.857382i	$0.672089\pi$
$62$	−606.255	−1.24185
$63$	0	0
$64$	−560.428	−1.09459
$65$	0	0
$66$	−747.785	−1.39464
$67$	−477.969	−0.871540	−0.435770	−	0.900058i	$-0.643524\pi$
$67$	−477.969	−0.871540	−0.435770	+	0.900058i	$0.643524\pi$
$68$	−705.529	−1.25820
$69$	902.777	1.57510
$70$	0	0
$71$	45.2557	0.0756460	0.0378230	−	0.999284i	$-0.487958\pi$
$71$	45.2557	0.0756460	0.0378230	+	0.999284i	$0.487958\pi$
$72$	−368.786	−0.603636
$73$	100.781	0.161583	0.0807913	−	0.996731i	$-0.474255\pi$
$73$	100.781	0.161583	0.0807913	+	0.996731i	$0.474255\pi$
$74$	−1155.98	−1.81594
$75$	0	0
$76$	2198.37	3.31803
$77$	0	0
$78$	1239.49	1.79930
$79$	880.534	1.25402	0.627012	−	0.779010i	$-0.284278\pi$
$79$	880.534	1.25402	0.627012	+	0.779010i	$0.284278\pi$
$80$	0	0
$81$	−369.085	−0.506289
$82$	−950.609	−1.28021
$83$	−1155.03	−1.52748	−0.763739	−	0.645525i	$-0.776638\pi$
$83$	−1155.03	−1.52748	−0.763739	+	0.645525i	$0.776638\pi$
$84$	0	0
$85$	0	0
$86$	1674.62	2.09976
$87$	140.336	0.172937
$88$	1394.27	1.68898
$89$	−619.374	−0.737680	−0.368840	−	0.929493i	$-0.620245\pi$
$89$	−619.374	−0.737680	−0.368840	+	0.929493i	$0.620245\pi$
$90$	0	0
$91$	0	0
$92$	−3423.90	−3.88007
$93$	516.309	0.575685
$94$	81.9585	0.0899295
$95$	0	0
$96$	83.8929	0.0891904
$97$	−231.195	−0.242003	−0.121001	−	0.992652i	$-0.538611\pi$
$97$	−231.195	−0.242003	−0.121001	+	0.992652i	$0.538611\pi$
$98$	0	0
$99$	−361.948	−0.367446
$100$	0	0
$101$	−1875.99	−1.84820	−0.924101	−	0.382149i	$-0.875184\pi$
$101$	−1875.99	−1.84820	−0.924101	+	0.382149i	$0.875184\pi$
$102$	906.316	0.879791
$103$	1855.94	1.77545	0.887723	−	0.460377i	$-0.152286\pi$
$103$	1855.94	1.77545	0.887723	+	0.460377i	$0.152286\pi$
$104$	−2311.08	−2.17904
$105$	0	0
$106$	−1690.41	−1.54893
$107$	218.016	0.196976	0.0984879	−	0.995138i	$-0.468599\pi$
$107$	218.016	0.196976	0.0984879	+	0.995138i	$0.468599\pi$
$108$	2401.74	2.13989
$109$	847.217	0.744484	0.372242	−	0.928136i	$-0.378589\pi$
$109$	847.217	0.744484	0.372242	+	0.928136i	$0.378589\pi$
$110$	0	0
$111$	984.473	0.841820
$112$	0	0
$113$	−1430.05	−1.19051	−0.595254	−	0.803537i	$-0.702949\pi$
$113$	−1430.05	−1.19051	−0.595254	+	0.803537i	$0.702949\pi$
$114$	−2824.00	−2.32011
$115$	0	0
$116$	−532.242	−0.426012
$117$	599.948	0.474062
$118$	−659.374	−0.514409
$119$	0	0
$120$	0	0
$121$	37.4229	0.0281164
$122$	−2389.29	−1.77308
$123$	809.573	0.593470
$124$	−1958.17	−1.41814
$125$	0	0
$126$	0	0
$127$	1732.84	1.21075	0.605374	−	0.795941i	$-0.293024\pi$
$127$	1732.84	1.21075	0.605374	+	0.795941i	$0.293024\pi$
$128$	−2568.65	−1.77374
$129$	−1426.17	−0.973390
$130$	0	0
$131$	2429.76	1.62052	0.810262	−	0.586068i	$-0.199325\pi$
$131$	2429.76	1.62052	0.810262	+	0.586068i	$0.199325\pi$
$132$	−2415.31	−1.59262
$133$	0	0
$134$	−2328.66	−1.50123
$135$	0	0
$136$	−1689.86	−1.06547
$137$	−1452.84	−0.906018	−0.453009	−	0.891506i	$-0.649649\pi$
$137$	−1452.84	−0.906018	−0.453009	+	0.891506i	$0.649649\pi$
$138$	4398.32	2.71311
$139$	−2927.47	−1.78637	−0.893183	−	0.449693i	$-0.851533\pi$
$139$	−2927.47	−1.78637	−0.893183	+	0.449693i	$0.851533\pi$
$140$	0	0
$141$	−69.7989	−0.0416888
$142$	220.485	0.130301
$143$	−2268.23	−1.32643
$144$	−564.954	−0.326941
$145$	0	0
$146$	491.004	0.278327
$147$	0	0
$148$	−3733.75	−2.07373
$149$	901.724	0.495786	0.247893	−	0.968787i	$-0.420262\pi$
$149$	901.724	0.495786	0.247893	+	0.968787i	$0.420262\pi$
$150$	0	0
$151$	−1357.63	−0.731669	−0.365834	−	0.930680i	$-0.619216\pi$
$151$	−1357.63	−0.731669	−0.365834	+	0.930680i	$0.619216\pi$
$152$	5265.46	2.80977
$153$	438.681	0.231799
$154$	0	0
$155$	0	0
$156$	4003.50	2.05472
$157$	−2318.72	−1.17869	−0.589343	−	0.807883i	$-0.700613\pi$
$157$	−2318.72	−1.17869	−0.589343	+	0.807883i	$0.700613\pi$
$158$	4289.95	2.16006
$159$	1439.61	0.718042
$160$	0	0
$161$	0	0
$162$	−1798.18	−0.872087
$163$	1577.42	0.757996	0.378998	−	0.925397i	$-0.376269\pi$
$163$	1577.42	0.757996	0.378998	+	0.925397i	$0.376269\pi$
$164$	−3070.42	−1.46195
$165$	0	0
$166$	−5627.28	−2.63109
$167$	1038.02	0.480982	0.240491	−	0.970651i	$-0.422691\pi$
$167$	1038.02	0.480982	0.240491	+	0.970651i	$0.422691\pi$
$168$	0	0
$169$	1562.72	0.711296
$170$	0	0
$171$	−1366.89	−0.611280
$172$	5408.95	2.39784
$173$	−482.641	−0.212107	−0.106054	−	0.994360i	$-0.533822\pi$
$173$	−482.641	−0.212107	−0.106054	+	0.994360i	$0.533822\pi$
$174$	683.714	0.297886
$175$	0	0
$176$	2135.93	0.914784
$177$	561.547	0.238466
$178$	−3017.58	−1.27066
$179$	2407.19	1.00515	0.502575	−	0.864534i	$-0.332386\pi$
$179$	2407.19	1.00515	0.502575	+	0.864534i	$0.332386\pi$
$180$	0	0
$181$	−532.600	−0.218717	−0.109359	−	0.994002i	$-0.534880\pi$
$181$	−532.600	−0.218717	−0.109359	+	0.994002i	$0.534880\pi$
$182$	0	0
$183$	2034.81	0.821952
$184$	−8200.83	−3.28572
$185$	0	0
$186$	2515.45	0.991622
$187$	−1658.53	−0.648575
$188$	264.722	0.102696
$189$	0	0
$190$	0	0
$191$	336.675	0.127544	0.0637721	−	0.997964i	$-0.479687\pi$
$191$	336.675	0.127544	0.0637721	+	0.997964i	$0.479687\pi$
$192$	2325.31	0.874035
$193$	−22.7366	−0.00847988	−0.00423994	−	0.999991i	$-0.501350\pi$
$193$	−22.7366	−0.00847988	−0.00423994	+	0.999991i	$0.501350\pi$
$194$	−1126.38	−0.416852
$195$	0	0
$196$	0	0
$197$	1085.33	0.392520	0.196260	−	0.980552i	$-0.437120\pi$
$197$	1085.33	0.392520	0.196260	+	0.980552i	$0.437120\pi$
$198$	−1763.41	−0.632928
$199$	2630.28	0.936963	0.468482	−	0.883473i	$-0.344801\pi$
$199$	2630.28	0.936963	0.468482	+	0.883473i	$0.344801\pi$
$200$	0	0
$201$	1983.17	0.695931
$202$	−9139.82	−3.18354
$203$	0	0
$204$	2927.35	1.00469
$205$	0	0
$206$	9042.11	3.05822
$207$	2128.90	0.714826
$208$	−3540.42	−1.18021
$209$	5167.83	1.71037
$210$	0	0
$211$	614.178	0.200388	0.100194	−	0.994968i	$-0.468054\pi$
$211$	614.178	0.200388	0.100194	+	0.994968i	$0.468054\pi$
$212$	−5459.93	−1.76882
$213$	−187.773	−0.0604039
$214$	1062.17	0.339292
$215$	0	0
$216$	5752.58	1.81210
$217$	0	0
$218$	4127.63	1.28238
$219$	−418.157	−0.129025
$220$	0	0
$221$	2749.10	0.836762
$222$	4796.34	1.45004
$223$	1930.30	0.579651	0.289826	−	0.957079i	$-0.406403\pi$
$223$	1930.30	0.579651	0.289826	+	0.957079i	$0.406403\pi$
$224$	0	0
$225$	0	0
$226$	−6967.17	−2.05066
$227$	765.365	0.223784	0.111892	−	0.993720i	$-0.464309\pi$
$227$	765.365	0.223784	0.111892	+	0.993720i	$0.464309\pi$
$228$	−9121.39	−2.64947
$229$	5712.66	1.64849	0.824243	−	0.566237i	$-0.191601\pi$
$229$	5712.66	1.64849	0.824243	+	0.566237i	$0.191601\pi$
$230$	0	0
$231$	0	0
$232$	−1274.81	−0.360756
$233$	864.418	0.243047	0.121523	−	0.992589i	$-0.461222\pi$
$233$	864.418	0.243047	0.121523	+	0.992589i	$0.461222\pi$
$234$	2922.94	0.816575
$235$	0	0
$236$	−2129.74	−0.587435
$237$	−3653.48	−1.00135
$238$	0	0
$239$	−1816.64	−0.491669	−0.245834	−	0.969312i	$-0.579062\pi$
$239$	−1816.64	−0.491669	−0.245834	+	0.969312i	$0.579062\pi$
$240$	0	0
$241$	2354.53	0.629330	0.314665	−	0.949203i	$-0.398108\pi$
$241$	2354.53	0.629330	0.314665	+	0.949203i	$0.398108\pi$
$242$	182.324	0.0484307
$243$	−2589.47	−0.683600
$244$	−7717.28	−2.02479
$245$	0	0
$246$	3944.23	1.02226
$247$	−8565.96	−2.20664
$248$	−4690.15	−1.20091
$249$	4792.39	1.21970
$250$	0	0
$251$	2983.66	0.750306	0.375153	−	0.926963i	$-0.377590\pi$
$251$	2983.66	0.750306	0.375153	+	0.926963i	$0.377590\pi$
$252$	0	0
$253$	−8048.78	−2.00009
$254$	8442.39	2.08552
$255$	0	0
$256$	−8030.99	−1.96069
$257$	−5121.50	−1.24307	−0.621537	−	0.783385i	$-0.713492\pi$
$257$	−5121.50	−1.24307	−0.621537	+	0.783385i	$0.713492\pi$
$258$	−6948.29	−1.67667
$259$	0	0
$260$	0	0
$261$	330.935	0.0784843
$262$	11837.7	2.79137
$263$	−4663.66	−1.09344	−0.546718	−	0.837317i	$-0.684123\pi$
$263$	−4663.66	−1.09344	−0.546718	+	0.837317i	$0.684123\pi$
$264$	−5785.07	−1.34866
$265$	0	0
$266$	0	0
$267$	2569.88	0.589043
$268$	−7521.45	−1.71435
$269$	−5018.86	−1.13757	−0.568783	−	0.822488i	$-0.692586\pi$
$269$	−5018.86	−1.13757	−0.568783	+	0.822488i	$0.692586\pi$
$270$	0	0
$271$	2512.66	0.563222	0.281611	−	0.959529i	$-0.409131\pi$
$271$	2512.66	0.563222	0.281611	+	0.959529i	$0.409131\pi$
$272$	−2588.75	−0.577081
$273$	0	0
$274$	−7078.21	−1.56062
$275$	0	0
$276$	14206.3	3.09827
$277$	−6286.82	−1.36368	−0.681839	−	0.731503i	$-0.738819\pi$
$277$	−6286.82	−1.36368	−0.681839	+	0.731503i	$0.738819\pi$
$278$	−14262.6	−3.07703
$279$	1217.54	0.261263
$280$	0	0
$281$	8804.33	1.86912	0.934560	−	0.355807i	$-0.115794\pi$
$281$	8804.33	1.86912	0.934560	+	0.355807i	$0.115794\pi$
$282$	−340.059	−0.0718093
$283$	485.298	0.101936	0.0509681	−	0.998700i	$-0.483769\pi$
$283$	485.298	0.101936	0.0509681	+	0.998700i	$0.483769\pi$
$284$	712.157	0.148798
$285$	0	0
$286$	−11050.8	−2.28478
$287$	0	0
$288$	197.834	0.0404773
$289$	−2902.86	−0.590854
$290$	0	0
$291$	959.265	0.193241
$292$	1585.92	0.317838
$293$	4004.48	0.798444	0.399222	−	0.916854i	$-0.369280\pi$
$293$	4004.48	0.798444	0.399222	+	0.916854i	$0.369280\pi$
$294$	0	0
$295$	0	0
$296$	−8942.96	−1.75608
$297$	5645.92	1.10306
$298$	4393.19	0.853995
$299$	13341.3	2.58042
$300$	0	0
$301$	0	0
$302$	−6614.33	−1.26031
$303$	7783.81	1.47580
$304$	8066.32	1.52183
$305$	0	0
$306$	2137.25	0.399276
$307$	3154.13	0.586370	0.293185	−	0.956056i	$-0.405285\pi$
$307$	3154.13	0.586370	0.293185	+	0.956056i	$0.405285\pi$
$308$	0	0
$309$	−7700.59	−1.41771
$310$	0	0
$311$	1738.29	0.316944	0.158472	−	0.987363i	$-0.449343\pi$
$311$	1738.29	0.316944	0.158472	+	0.987363i	$0.449343\pi$
$312$	9589.06	1.73998
$313$	5092.30	0.919597	0.459799	−	0.888023i	$-0.347922\pi$
$313$	5092.30	0.919597	0.459799	+	0.888023i	$0.347922\pi$
$314$	−11296.8	−2.03030
$315$	0	0
$316$	13856.3	2.46671
$317$	4874.67	0.863686	0.431843	−	0.901949i	$-0.357863\pi$
$317$	4874.67	0.863686	0.431843	+	0.901949i	$0.357863\pi$
$318$	7013.78	1.23683
$319$	−1251.17	−0.219600
$320$	0	0
$321$	−904.584	−0.157287
$322$	0	0
$323$	−6263.41	−1.07896
$324$	−5808.02	−0.995888
$325$	0	0
$326$	7685.19	1.30565
$327$	−3515.24	−0.594475
$328$	−7354.17	−1.23801
$329$	0	0
$330$	0	0
$331$	−2448.46	−0.406585	−0.203292	−	0.979118i	$-0.565164\pi$
$331$	−2448.46	−0.406585	−0.203292	+	0.979118i	$0.565164\pi$
$332$	−18175.8	−3.00460
$333$	2321.56	0.382043
$334$	5057.20	0.828496
$335$	0	0
$336$	0	0
$337$	6271.50	1.01374	0.506870	−	0.862023i	$-0.330803\pi$
$337$	6271.50	1.01374	0.506870	+	0.862023i	$0.330803\pi$
$338$	7613.54	1.22521
$339$	5933.50	0.950629
$340$	0	0
$341$	−4603.19	−0.731017
$342$	−6659.48	−1.05293
$343$	0	0
$344$	12955.3	2.03054
$345$	0	0
$346$	−2351.42	−0.365356
$347$	−417.338	−0.0645645	−0.0322822	−	0.999479i	$-0.510278\pi$
$347$	−417.338	−0.0645645	−0.0322822	+	0.999479i	$0.510278\pi$
$348$	2208.36	0.340174
$349$	−5971.08	−0.915829	−0.457915	−	0.888996i	$-0.651403\pi$
$349$	−5971.08	−0.915829	−0.457915	+	0.888996i	$0.651403\pi$
$350$	0	0
$351$	−9358.42	−1.42312
$352$	−747.954	−0.113256
$353$	−12012.6	−1.81124	−0.905620	−	0.424089i	$-0.860594\pi$
$353$	−12012.6	−1.81124	−0.905620	+	0.424089i	$0.860594\pi$
$354$	2735.85	0.410759
$355$	0	0
$356$	−9746.64	−1.45104
$357$	0	0
$358$	11727.8	1.73138
$359$	6312.46	0.928020	0.464010	−	0.885830i	$-0.346410\pi$
$359$	6312.46	0.928020	0.464010	+	0.885830i	$0.346410\pi$
$360$	0	0
$361$	12657.3	1.84535
$362$	−2594.82	−0.376743
$363$	−155.274	−0.0224511
$364$	0	0
$365$	0	0
$366$	9913.55	1.41582
$367$	12228.6	1.73932	0.869659	−	0.493652i	$-0.164338\pi$
$367$	12228.6	1.73932	0.869659	+	0.493652i	$0.164338\pi$
$368$	−12563.1	−1.77961
$369$	1909.11	0.269335
$370$	0	0
$371$	0	0
$372$	8124.77	1.13239
$373$	−7876.24	−1.09334	−0.546671	−	0.837348i	$-0.684105\pi$
$373$	−7876.24	−1.09334	−0.546671	+	0.837348i	$0.684105\pi$
$374$	−8080.33	−1.11718
$375$	0	0
$376$	634.053	0.0869649
$377$	2073.89	0.283317
$378$	0	0
$379$	5992.79	0.812214	0.406107	−	0.913826i	$-0.366886\pi$
$379$	5992.79	0.812214	0.406107	+	0.913826i	$0.366886\pi$
$380$	0	0
$381$	−7189.85	−0.966790
$382$	1640.28	0.219696
$383$	7373.61	0.983743	0.491872	−	0.870668i	$-0.336313\pi$
$383$	7373.61	0.983743	0.491872	+	0.870668i	$0.336313\pi$
$384$	10657.7	1.41634
$385$	0	0
$386$	−110.772	−0.0146067
$387$	−3363.15	−0.441754
$388$	−3638.14	−0.476028
$389$	1896.88	0.247238	0.123619	−	0.992330i	$-0.460550\pi$
$389$	1896.88	0.247238	0.123619	+	0.992330i	$0.460550\pi$
$390$	0	0
$391$	9755.12	1.26173
$392$	0	0
$393$	−10081.5	−1.29400
$394$	5287.71	0.676119
$395$	0	0
$396$	−5695.71	−0.722778
$397$	−791.156	−0.100018	−0.0500088	−	0.998749i	$-0.515925\pi$
$397$	−791.156	−0.100018	−0.0500088	+	0.998749i	$0.515925\pi$
$398$	12814.7	1.61393
$399$	0	0
$400$	0	0
$401$	4410.55	0.549258	0.274629	−	0.961550i	$-0.411445\pi$
$401$	4410.55	0.549258	0.274629	+	0.961550i	$0.411445\pi$
$402$	9661.99	1.19875
$403$	7630.03	0.943124
$404$	−29521.1	−3.63548
$405$	0	0
$406$	0	0
$407$	−8777.14	−1.06896
$408$	7011.51	0.850787
$409$	4403.73	0.532397	0.266199	−	0.963918i	$-0.414232\pi$
$409$	4403.73	0.532397	0.266199	+	0.963918i	$0.414232\pi$
$410$	0	0
$411$	6028.07	0.723461
$412$	29205.5	3.49236
$413$	0	0
$414$	10372.0	1.23129
$415$	0	0
$416$	1239.77	0.146117
$417$	12146.6	1.42643
$418$	25177.6	2.94612
$419$	14272.6	1.66411	0.832057	−	0.554691i	$-0.187163\pi$
$419$	14272.6	1.66411	0.832057	+	0.554691i	$0.187163\pi$
$420$	0	0
$421$	−15530.4	−1.79788	−0.898939	−	0.438074i	$-0.855661\pi$
$421$	−15530.4	−1.79788	−0.898939	+	0.438074i	$0.855661\pi$
$422$	2992.27	0.345169
$423$	−164.598	−0.0189197
$424$	−13077.5	−1.49787
$425$	0	0
$426$	−914.830	−0.104046
$427$	0	0
$428$	3430.76	0.387458
$429$	9411.27	1.05916
$430$	0	0
$431$	−5260.75	−0.587938	−0.293969	−	0.955815i	$-0.594976\pi$
$431$	−5260.75	−0.587938	−0.293969	+	0.955815i	$0.594976\pi$
$432$	8812.56	0.981469
$433$	6610.65	0.733690	0.366845	−	0.930282i	$-0.380438\pi$
$433$	6610.65	0.733690	0.366845	+	0.930282i	$0.380438\pi$
$434$	0	0
$435$	0	0
$436$	13332.0	1.46442
$437$	−30396.1	−3.32733
$438$	−2037.26	−0.222246
$439$	8870.46	0.964383	0.482191	−	0.876066i	$-0.339841\pi$
$439$	8870.46	0.964383	0.482191	+	0.876066i	$0.339841\pi$
$440$	0	0
$441$	0	0
$442$	13393.6	1.44133
$443$	13221.8	1.41802	0.709012	−	0.705196i	$-0.249141\pi$
$443$	13221.8	1.41802	0.709012	+	0.705196i	$0.249141\pi$
$444$	15491.9	1.65589
$445$	0	0
$446$	9404.38	0.998454
$447$	−3741.40	−0.395889
$448$	0	0
$449$	4192.70	0.440681	0.220341	−	0.975423i	$-0.429283\pi$
$449$	4192.70	0.440681	0.220341	+	0.975423i	$0.429283\pi$
$450$	0	0
$451$	−7217.82	−0.753600
$452$	−22503.6	−2.34177
$453$	5633.01	0.584243
$454$	3728.85	0.385470
$455$	0	0
$456$	−21847.3	−2.24362
$457$	4701.78	0.481270	0.240635	−	0.970616i	$-0.422644\pi$
$457$	4701.78	0.481270	0.240635	+	0.970616i	$0.422644\pi$
$458$	27832.0	2.83953
$459$	−6842.86	−0.695855
$460$	0	0
$461$	1949.82	0.196989	0.0984946	−	0.995138i	$-0.468597\pi$
$461$	1949.82	0.196989	0.0984946	+	0.995138i	$0.468597\pi$
$462$	0	0
$463$	−7315.85	−0.734333	−0.367166	−	0.930155i	$-0.619672\pi$
$463$	−7315.85	−0.734333	−0.367166	+	0.930155i	$0.619672\pi$
$464$	−1952.92	−0.195392
$465$	0	0
$466$	4211.43	0.418650
$467$	−1897.54	−0.188025	−0.0940126	−	0.995571i	$-0.529969\pi$
$467$	−1897.54	−0.188025	−0.0940126	+	0.995571i	$0.529969\pi$
$468$	9440.94	0.932495
$469$	0	0
$470$	0	0
$471$	9620.74	0.941189
$472$	−5101.10	−0.497451
$473$	12715.1	1.23603
$474$	−17799.7	−1.72483
$475$	0	0
$476$	0	0
$477$	3394.85	0.325869
$478$	−8850.66	−0.846903
$479$	−10534.4	−1.00486	−0.502429	−	0.864619i	$-0.667560\pi$
$479$	−10534.4	−1.00486	−0.502429	+	0.864619i	$0.667560\pi$
$480$	0	0
$481$	14548.6	1.37912
$482$	11471.2	1.08403
$483$	0	0
$484$	588.897	0.0553058
$485$	0	0
$486$	−12615.9	−1.17751
$487$	−11937.0	−1.11071	−0.555355	−	0.831614i	$-0.687417\pi$
$487$	−11937.0	−1.11071	−0.555355	+	0.831614i	$0.687417\pi$
$488$	−18484.2	−1.71463
$489$	−6544.99	−0.605265
$490$	0	0
$491$	100.060	0.00919683	0.00459841	−	0.999989i	$-0.498536\pi$
$491$	100.060	0.00919683	0.00459841	+	0.999989i	$0.498536\pi$
$492$	12739.7	1.16737
$493$	1516.42	0.138532
$494$	−41733.3	−3.80095
$495$	0	0
$496$	−7184.99	−0.650435
$497$	0	0
$498$	23348.5	2.10095
$499$	−15611.7	−1.40055	−0.700275	−	0.713874i	$-0.746939\pi$
$499$	−15611.7	−1.40055	−0.700275	+	0.713874i	$0.746939\pi$
$500$	0	0
$501$	−4306.90	−0.384068
$502$	14536.3	1.29241
$503$	2270.39	0.201256	0.100628	−	0.994924i	$-0.467915\pi$
$503$	2270.39	0.201256	0.100628	+	0.994924i	$0.467915\pi$
$504$	0	0
$505$	0	0
$506$	−39213.6	−3.44517
$507$	−6483.97	−0.567975
$508$	27268.5	2.38158
$509$	2579.79	0.224651	0.112325	−	0.993671i	$-0.464170\pi$
$509$	2579.79	0.224651	0.112325	+	0.993671i	$0.464170\pi$
$510$	0	0
$511$	0	0
$512$	−18577.7	−1.60357
$513$	21321.8	1.83505
$514$	−24951.9	−2.14121
$515$	0	0
$516$	−22442.6	−1.91469
$517$	622.297	0.0529373
$518$	0	0
$519$	2002.56	0.169369
$520$	0	0
$521$	−12793.9	−1.07584	−0.537918	−	0.842997i	$-0.680789\pi$
$521$	−12793.9	−1.07584	−0.537918	+	0.842997i	$0.680789\pi$
$522$	1612.31	0.135190
$523$	−11273.4	−0.942548	−0.471274	−	0.881987i	$-0.656206\pi$
$523$	−11273.4	−0.942548	−0.471274	+	0.881987i	$0.656206\pi$
$524$	38235.3	3.18763
$525$	0	0
$526$	−22721.3	−1.88345
$527$	5579.07	0.461154
$528$	−8862.33	−0.730461
$529$	35174.2	2.89095
$530$	0	0
$531$	1324.22	0.108223
$532$	0	0
$533$	11963.9	0.972260
$534$	12520.4	1.01463
$535$	0	0
$536$	−18015.1	−1.45175
$537$	−9987.83	−0.802619
$538$	−24451.8	−1.95947
$539$	0	0
$540$	0	0
$541$	10282.9	0.817183	0.408591	−	0.912717i	$-0.366020\pi$
$541$	10282.9	0.817183	0.408591	+	0.912717i	$0.366020\pi$
$542$	12241.6	0.970154
$543$	2209.85	0.174647
$544$	906.520	0.0714462
$545$	0	0
$546$	0	0
$547$	2160.94	0.168912	0.0844562	−	0.996427i	$-0.473085\pi$
$547$	2160.94	0.168912	0.0844562	+	0.996427i	$0.473085\pi$
$548$	−22862.3	−1.78217
$549$	4798.42	0.373027
$550$	0	0
$551$	−4725.04	−0.365324
$552$	34026.6	2.62367
$553$	0	0
$554$	−30629.3	−2.34895
$555$	0	0
$556$	−46067.5	−3.51384
$557$	−15648.4	−1.19038	−0.595190	−	0.803585i	$-0.702923\pi$
$557$	−15648.4	−1.19038	−0.595190	+	0.803585i	$0.702923\pi$
$558$	5931.86	0.450028
$559$	−21076.0	−1.59467
$560$	0	0
$561$	6881.51	0.517892
$562$	42894.6	3.21957
$563$	−21951.6	−1.64325	−0.821623	−	0.570031i	$-0.806931\pi$
$563$	−21951.6	−1.64325	−0.821623	+	0.570031i	$0.806931\pi$
$564$	−1098.37	−0.0820033
$565$	0	0
$566$	2364.36	0.175586
$567$	0	0
$568$	1705.74	0.126005
$569$	14019.2	1.03289	0.516445	−	0.856321i	$-0.327255\pi$
$569$	14019.2	1.03289	0.516445	+	0.856321i	$0.327255\pi$
$570$	0	0
$571$	−21706.3	−1.59086	−0.795430	−	0.606045i	$-0.792755\pi$
$571$	−21706.3	−1.59086	−0.795430	+	0.606045i	$0.792755\pi$
$572$	−35693.5	−2.60913
$573$	−1396.92	−0.101845
$574$	0	0
$575$	0	0
$576$	5483.48	0.396664
$577$	12569.0	0.906853	0.453427	−	0.891294i	$-0.350201\pi$
$577$	12569.0	0.906853	0.453427	+	0.891294i	$0.350201\pi$
$578$	−14142.7	−1.01775
$579$	94.3379	0.00677124
$580$	0	0
$581$	0	0
$582$	4673.53	0.332859
$583$	−12835.0	−0.911785
$584$	3798.54	0.269152
$585$	0	0
$586$	19509.8	1.37533
$587$	−1330.20	−0.0935322	−0.0467661	−	0.998906i	$-0.514892\pi$
$587$	−1330.20	−0.0935322	−0.0467661	+	0.998906i	$0.514892\pi$
$588$	0	0
$589$	−17383.9	−1.21611
$590$	0	0
$591$	−4503.20	−0.313430
$592$	−13700.0	−0.951126
$593$	9866.91	0.683281	0.341640	−	0.939831i	$-0.389018\pi$
$593$	9866.91	0.683281	0.341640	+	0.939831i	$0.389018\pi$
$594$	27506.9	1.90003
$595$	0	0
$596$	14189.8	0.975228
$597$	−10913.5	−0.748172
$598$	64998.6	4.44480
$599$	4185.17	0.285478	0.142739	−	0.989760i	$-0.454409\pi$
$599$	4185.17	0.285478	0.142739	+	0.989760i	$0.454409\pi$
$600$	0	0
$601$	−10599.8	−0.719426	−0.359713	−	0.933063i	$-0.617125\pi$
$601$	−10599.8	−0.719426	−0.359713	+	0.933063i	$0.617125\pi$
$602$	0	0
$603$	4676.66	0.315834
$604$	−21364.0	−1.43922
$605$	0	0
$606$	37922.6	2.54208
$607$	−13290.1	−0.888679	−0.444340	−	0.895858i	$-0.646562\pi$
$607$	−13290.1	−0.888679	−0.444340	+	0.895858i	$0.646562\pi$
$608$	−2824.64	−0.188412
$609$	0	0
$610$	0	0
$611$	−1031.49	−0.0682973
$612$	6903.20	0.455957
$613$	−21327.1	−1.40521	−0.702606	−	0.711579i	$-0.747980\pi$
$613$	−21327.1	−1.40521	−0.702606	+	0.711579i	$0.747980\pi$
$614$	15366.9	1.01003
$615$	0	0
$616$	0	0
$617$	−13741.6	−0.896622	−0.448311	−	0.893878i	$-0.647974\pi$
$617$	−13741.6	−0.896622	−0.448311	+	0.893878i	$0.647974\pi$
$618$	−37517.2	−2.44201
$619$	−19444.3	−1.26257	−0.631285	−	0.775551i	$-0.717472\pi$
$619$	−19444.3	−1.26257	−0.631285	+	0.775551i	$0.717472\pi$
$620$	0	0
$621$	−33208.1	−2.14589
$622$	8468.94	0.545938
$623$	0	0
$624$	14689.8	0.942407
$625$	0	0
$626$	24809.6	1.58401
$627$	−21442.2	−1.36574
$628$	−36488.0	−2.31852
$629$	10637.9	0.674341
$630$	0	0
$631$	−27425.6	−1.73026	−0.865130	−	0.501547i	$-0.832764\pi$
$631$	−27425.6	−1.73026	−0.865130	+	0.501547i	$0.832764\pi$
$632$	33188.2	2.08886
$633$	−2548.33	−0.160011
$634$	23749.3	1.48771
$635$	0	0
$636$	22654.1	1.41241
$637$	0	0
$638$	−6095.70	−0.378262
$639$	−442.802	−0.0274131
$640$	0	0
$641$	−10053.8	−0.619502	−0.309751	−	0.950818i	$-0.600246\pi$
$641$	−10053.8	−0.619502	−0.309751	+	0.950818i	$0.600246\pi$
$642$	−4407.12	−0.270927
$643$	−4044.18	−0.248036	−0.124018	−	0.992280i	$-0.539578\pi$
$643$	−4044.18	−0.248036	−0.124018	+	0.992280i	$0.539578\pi$
$644$	0	0
$645$	0	0
$646$	−30515.3	−1.85853
$647$	1486.90	0.0903491	0.0451746	−	0.998979i	$-0.485616\pi$
$647$	1486.90	0.0903491	0.0451746	+	0.998979i	$0.485616\pi$
$648$	−13911.2	−0.843338
$649$	−5006.52	−0.302809
$650$	0	0
$651$	0	0
$652$	24822.8	1.49100
$653$	−4269.30	−0.255851	−0.127925	−	0.991784i	$-0.540832\pi$
$653$	−4269.30	−0.255851	−0.127925	+	0.991784i	$0.540832\pi$
$654$	−17126.2	−1.02399
$655$	0	0
$656$	−11266.1	−0.670528
$657$	−986.086	−0.0585554
$658$	0	0
$659$	1607.29	0.0950094	0.0475047	−	0.998871i	$-0.484873\pi$
$659$	1607.29	0.0950094	0.0475047	+	0.998871i	$0.484873\pi$
$660$	0	0
$661$	−8001.31	−0.470824	−0.235412	−	0.971896i	$-0.575644\pi$
$661$	−8001.31	−0.470824	−0.235412	+	0.971896i	$0.575644\pi$
$662$	−11928.9	−0.700345
$663$	−11406.5	−0.668160
$664$	−43534.1	−2.54436
$665$	0	0
$666$	11310.6	0.658073
$667$	7359.14	0.427207
$668$	16334.5	0.946109
$669$	−8009.12	−0.462855
$670$	0	0
$671$	−18141.5	−1.04373
$672$	0	0
$673$	−3347.96	−0.191760	−0.0958800	−	0.995393i	$-0.530567\pi$
$673$	−3347.96	−0.191760	−0.0958800	+	0.995393i	$0.530567\pi$
$674$	30554.7	1.74617
$675$	0	0
$676$	24591.4	1.39914
$677$	−7.91810	−0.000449508 0	−0.000224754	−	1.00000i	$-0.500072\pi$
$677$	−7.91810	−0.000449508 0	−0.000224754		1.00000i	$0.500072\pi$
$678$	28907.9	1.63747
$679$	0	0
$680$	0	0
$681$	−3175.62	−0.178693
$682$	−22426.7	−1.25918
$683$	−32136.0	−1.80036	−0.900182	−	0.435513i	$-0.856567\pi$
$683$	−32136.0	−1.80036	−0.900182	+	0.435513i	$0.856567\pi$
$684$	−21509.8	−1.20241
$685$	0	0
$686$	0	0
$687$	−23702.8	−1.31633
$688$	19846.7	1.09978
$689$	21274.7	1.17634
$690$	0	0
$691$	6196.84	0.341156	0.170578	−	0.985344i	$-0.445437\pi$
$691$	6196.84	0.341156	0.170578	+	0.985344i	$0.445437\pi$
$692$	−7594.98	−0.417222
$693$	0	0
$694$	−2033.27	−0.111213
$695$	0	0
$696$	5289.39	0.288066
$697$	8747.99	0.475400
$698$	−29091.0	−1.57752
$699$	−3586.61	−0.194075
$700$	0	0
$701$	1651.38	0.0889755	0.0444878	−	0.999010i	$-0.485834\pi$
$701$	1651.38	0.0889755	0.0444878	+	0.999010i	$0.485834\pi$
$702$	−45594.1	−2.45134
$703$	−33146.8	−1.77831
$704$	−20731.5	−1.10987
$705$	0	0
$706$	−58525.4	−3.11988
$707$	0	0
$708$	8836.66	0.469071
$709$	15399.8	0.815728	0.407864	−	0.913043i	$-0.366274\pi$
$709$	15399.8	0.815728	0.407864	+	0.913043i	$0.366274\pi$
$710$	0	0
$711$	−8615.53	−0.454441
$712$	−23344.8	−1.22877
$713$	27075.0	1.42211
$714$	0	0
$715$	0	0
$716$	37880.2	1.97716
$717$	7537.54	0.392601
$718$	30754.3	1.59852
$719$	−3507.31	−0.181920	−0.0909600	−	0.995855i	$-0.528994\pi$
$719$	−3507.31	−0.181920	−0.0909600	+	0.995855i	$0.528994\pi$
$720$	0	0
$721$	0	0
$722$	61666.0	3.17863
$723$	−9769.32	−0.502524
$724$	−8381.14	−0.430225
$725$	0	0
$726$	−756.492	−0.0386722
$727$	22516.7	1.14869	0.574346	−	0.818613i	$-0.305256\pi$
$727$	22516.7	1.14869	0.574346	+	0.818613i	$0.305256\pi$
$728$	0	0
$729$	20709.4	1.05215
$730$	0	0
$731$	−15410.7	−0.779736
$732$	32020.3	1.61681
$733$	−10964.7	−0.552509	−0.276255	−	0.961084i	$-0.589093\pi$
$733$	−10964.7	−0.552509	−0.276255	+	0.961084i	$0.589093\pi$
$734$	59577.8	2.99599
$735$	0	0
$736$	4399.31	0.220327
$737$	−17681.1	−0.883707
$738$	9301.17	0.463931
$739$	3029.33	0.150793	0.0753963	−	0.997154i	$-0.475978\pi$
$739$	3029.33	0.150793	0.0753963	+	0.997154i	$0.475978\pi$
$740$	0	0
$741$	35541.6	1.76201
$742$	0	0
$743$	8921.04	0.440486	0.220243	−	0.975445i	$-0.429315\pi$
$743$	8921.04	0.440486	0.220243	+	0.975445i	$0.429315\pi$
$744$	19460.2	0.958932
$745$	0	0
$746$	−38373.0	−1.88329
$747$	11301.3	0.553537
$748$	−26099.1	−1.27577
$749$	0	0
$750$	0	0
$751$	−6202.64	−0.301382	−0.150691	−	0.988581i	$-0.548150\pi$
$751$	−6202.64	−0.301382	−0.150691	+	0.988581i	$0.548150\pi$
$752$	971.326	0.0471019
$753$	−12379.7	−0.599124
$754$	10104.0	0.488016
$755$	0	0
$756$	0	0
$757$	12099.3	0.580922	0.290461	−	0.956887i	$-0.406191\pi$
$757$	12099.3	0.580922	0.290461	+	0.956887i	$0.406191\pi$
$758$	29196.8	1.39905
$759$	33395.7	1.59708
$760$	0	0
$761$	−29332.7	−1.39725	−0.698626	−	0.715487i	$-0.746205\pi$
$761$	−29332.7	−1.39725	−0.698626	+	0.715487i	$0.746205\pi$
$762$	−35028.9	−1.66530
$763$	0	0
$764$	5298.01	0.250884
$765$	0	0
$766$	35924.1	1.69451
$767$	8298.57	0.390670
$768$	33321.9	1.56563
$769$	13280.1	0.622748	0.311374	−	0.950287i	$-0.399211\pi$
$769$	13280.1	0.622748	0.311374	+	0.950287i	$0.399211\pi$
$770$	0	0
$771$	21249.9	0.992604
$772$	−357.789	−0.0166802
$773$	11248.1	0.523370	0.261685	−	0.965153i	$-0.415722\pi$
$773$	11248.1	0.523370	0.261685	+	0.965153i	$0.415722\pi$
$774$	−16385.3	−0.760925
$775$	0	0
$776$	−8713.96	−0.403110
$777$	0	0
$778$	9241.57	0.425869
$779$	−27258.0	−1.25368
$780$	0	0
$781$	1674.11	0.0767021
$782$	47526.8	2.17335
$783$	−5162.17	−0.235608
$784$	0	0
$785$	0	0
$786$	−49116.7	−2.22893
$787$	−436.193	−0.0197568	−0.00987839	−	0.999951i	$-0.503144\pi$
$787$	−436.193	−0.0197568	−0.00987839	+	0.999951i	$0.503144\pi$
$788$	17079.0	0.772100
$789$	19350.3	0.873116
$790$	0	0
$791$	0	0
$792$	−13642.2	−0.612063
$793$	30070.5	1.34657
$794$	−3854.50	−0.172281
$795$	0	0
$796$	41390.8	1.84304
$797$	−24034.0	−1.06816	−0.534082	−	0.845433i	$-0.679343\pi$
$797$	−24034.0	−1.06816	−0.534082	+	0.845433i	$0.679343\pi$
$798$	0	0
$799$	−754.224	−0.0333949
$800$	0	0
$801$	6060.23	0.267325
$802$	21488.1	0.946101
$803$	3728.11	0.163838
$804$	31207.7	1.36892
$805$	0	0
$806$	37173.4	1.62454
$807$	20824.1	0.908354
$808$	−70708.1	−3.07859
$809$	16188.0	0.703509	0.351754	−	0.936092i	$-0.385585\pi$
$809$	16188.0	0.703509	0.351754	+	0.936092i	$0.385585\pi$
$810$	0	0
$811$	−3732.42	−0.161607	−0.0808033	−	0.996730i	$-0.525749\pi$
$811$	−3732.42	−0.161607	−0.0808033	+	0.996730i	$0.525749\pi$
$812$	0	0
$813$	−10425.4	−0.449737
$814$	−42762.1	−1.84129
$815$	0	0
$816$	10741.1	0.460803
$817$	48018.6	2.05625
$818$	21454.9	0.917058
$819$	0	0
$820$	0	0
$821$	37226.4	1.58247	0.791237	−	0.611510i	$-0.209438\pi$
$821$	37226.4	1.58247	0.791237	+	0.611510i	$0.209438\pi$
$822$	29368.7	1.24617
$823$	−21355.7	−0.904511	−0.452256	−	0.891888i	$-0.649380\pi$
$823$	−21355.7	−0.904511	−0.452256	+	0.891888i	$0.649380\pi$
$824$	69952.2	2.95740
$825$	0	0
$826$	0	0
$827$	18852.9	0.792718	0.396359	−	0.918096i	$-0.370274\pi$
$827$	18852.9	0.792718	0.396359	+	0.918096i	$0.370274\pi$
$828$	33501.0	1.40609
$829$	43227.9	1.81106	0.905528	−	0.424286i	$-0.139475\pi$
$829$	43227.9	1.81106	0.905528	+	0.424286i	$0.139475\pi$
$830$	0	0
$831$	26085.1	1.08891
$832$	34363.5	1.43190
$833$	0	0
$834$	59177.9	2.45703
$835$	0	0
$836$	81322.4	3.36435
$837$	−18992.1	−0.784306
$838$	69536.1	2.86645
$839$	−30311.3	−1.24727	−0.623637	−	0.781714i	$-0.714346\pi$
$839$	−30311.3	−1.24727	−0.623637	+	0.781714i	$0.714346\pi$
$840$	0	0
$841$	−23245.0	−0.953095
$842$	−75664.0	−3.09686
$843$	−36530.6	−1.49250
$844$	9664.88	0.394169
$845$	0	0
$846$	−801.918	−0.0325893
$847$	0	0
$848$	−20033.8	−0.811276
$849$	−2013.58	−0.0813968
$850$	0	0
$851$	51625.3	2.07955
$852$	−2954.85	−0.118816
$853$	−22797.0	−0.915071	−0.457535	−	0.889191i	$-0.651268\pi$
$853$	−22797.0	−0.915071	−0.457535	+	0.889191i	$0.651268\pi$
$854$	0	0
$855$	0	0
$856$	8217.25	0.328107
$857$	−11280.3	−0.449625	−0.224813	−	0.974402i	$-0.572177\pi$
$857$	−11280.3	−0.449625	−0.224813	+	0.974402i	$0.572177\pi$
$858$	45851.6	1.82442
$859$	−43369.3	−1.72263	−0.861315	−	0.508071i	$-0.830359\pi$
$859$	−43369.3	−1.72263	−0.861315	+	0.508071i	$0.830359\pi$
$860$	0	0
$861$	0	0
$862$	−25630.3	−1.01273
$863$	−9846.22	−0.388377	−0.194189	−	0.980964i	$-0.562207\pi$
$863$	−9846.22	−0.388377	−0.194189	+	0.980964i	$0.562207\pi$
$864$	−3085.95	−0.121512
$865$	0	0
$866$	32207.0	1.26379
$867$	12044.5	0.471801
$868$	0	0
$869$	32572.9	1.27153
$870$	0	0
$871$	29307.4	1.14012
$872$	31932.5	1.24010
$873$	2262.11	0.0876985
$874$	−148090.	−5.73136
$875$	0	0
$876$	−6580.23	−0.253796
$877$	15229.8	0.586400	0.293200	−	0.956051i	$-0.405280\pi$
$877$	15229.8	0.586400	0.293200	+	0.956051i	$0.405280\pi$
$878$	43216.8	1.66116
$879$	−16615.2	−0.637563
$880$	0	0
$881$	−27891.7	−1.06662	−0.533311	−	0.845919i	$-0.679052\pi$
$881$	−27891.7	−1.06662	−0.533311	+	0.845919i	$0.679052\pi$
$882$	0	0
$883$	−18300.3	−0.697456	−0.348728	−	0.937224i	$-0.613386\pi$
$883$	−18300.3	−0.697456	−0.348728	+	0.937224i	$0.613386\pi$
$884$	43260.6	1.64594
$885$	0	0
$886$	64416.3	2.44256
$887$	−19104.8	−0.723197	−0.361598	−	0.932334i	$-0.617769\pi$
$887$	−19104.8	−0.723197	−0.361598	+	0.932334i	$0.617769\pi$
$888$	37105.8	1.40224
$889$	0	0
$890$	0	0
$891$	−13653.3	−0.513357
$892$	30375.7	1.14019
$893$	2350.10	0.0880661
$894$	−18228.1	−0.681921
$895$	0	0
$896$	0	0
$897$	−55355.2	−2.06048
$898$	20426.8	0.759077
$899$	4208.78	0.156141
$900$	0	0
$901$	15556.0	0.575189
$902$	−35165.1	−1.29808
$903$	0	0
$904$	−53899.9	−1.98306
$905$	0	0
$906$	27444.0	1.00636
$907$	29518.8	1.08066	0.540328	−	0.841455i	$-0.318300\pi$
$907$	29518.8	1.08066	0.540328	+	0.841455i	$0.318300\pi$
$908$	12044.0	0.440191
$909$	18355.5	0.669764
$910$	0	0
$911$	−30322.2	−1.10277	−0.551383	−	0.834253i	$-0.685900\pi$
$911$	−30322.2	−1.10277	−0.551383	+	0.834253i	$0.685900\pi$
$912$	−33468.5	−1.21519
$913$	−42727.0	−1.54880
$914$	22907.0	0.828991
$915$	0	0
$916$	89895.9	3.24263
$917$	0	0
$918$	−33338.3	−1.19862
$919$	−37780.9	−1.35612	−0.678062	−	0.735005i	$-0.737180\pi$
$919$	−37780.9	−1.35612	−0.678062	+	0.735005i	$0.737180\pi$
$920$	0	0
$921$	−13087.0	−0.468220
$922$	9499.48	0.339315
$923$	−2774.93	−0.0989575
$924$	0	0
$925$	0	0
$926$	−35642.7	−1.26489
$927$	−18159.3	−0.643398
$928$	683.867	0.0241908
$929$	−48493.3	−1.71261	−0.856305	−	0.516470i	$-0.827246\pi$
$929$	−48493.3	−1.71261	−0.856305	+	0.516470i	$0.827246\pi$
$930$	0	0
$931$	0	0
$932$	13602.7	0.478081
$933$	−7212.46	−0.253082
$934$	−9244.81	−0.323875
$935$	0	0
$936$	22612.7	0.789656
$937$	18661.7	0.650641	0.325320	−	0.945604i	$-0.394528\pi$
$937$	18661.7	0.650641	0.325320	+	0.945604i	$0.394528\pi$
$938$	0	0
$939$	−21128.8	−0.734305
$940$	0	0
$941$	31270.1	1.08329	0.541644	−	0.840608i	$-0.317802\pi$
$941$	31270.1	1.08329	0.541644	+	0.840608i	$0.317802\pi$
$942$	46872.1	1.62121
$943$	42453.7	1.46605
$944$	−7814.53	−0.269429
$945$	0	0
$946$	61948.0	2.12907
$947$	22405.3	0.768823	0.384412	−	0.923162i	$-0.374404\pi$
$947$	22405.3	0.768823	0.384412	+	0.923162i	$0.374404\pi$
$948$	−57492.2	−1.96968
$949$	−6179.55	−0.211377
$950$	0	0
$951$	−20225.8	−0.689660
$952$	0	0
$953$	−21777.9	−0.740247	−0.370123	−	0.928983i	$-0.620685\pi$
$953$	−21777.9	−0.740247	−0.370123	+	0.928983i	$0.620685\pi$
$954$	16539.7	0.561313
$955$	0	0
$956$	−28587.2	−0.967129
$957$	5191.32	0.175352
$958$	−51323.3	−1.73088
$959$	0	0
$960$	0	0
$961$	−14306.5	−0.480229
$962$	70880.5	2.37555
$963$	−2133.16	−0.0713814
$964$	37051.5	1.23791
$965$	0	0
$966$	0	0
$967$	14432.2	0.479945	0.239973	−	0.970780i	$-0.422862\pi$
$967$	14432.2	0.479945	0.239973	+	0.970780i	$0.422862\pi$
$968$	1410.51	0.0468341
$969$	25987.9	0.861561
$970$	0	0
$971$	−6672.57	−0.220528	−0.110264	−	0.993902i	$-0.535170\pi$
$971$	−6672.57	−0.220528	−0.110264	+	0.993902i	$0.535170\pi$
$972$	−40748.6	−1.34466
$973$	0	0
$974$	−58156.7	−1.91321
$975$	0	0
$976$	−28316.5	−0.928678
$977$	33300.2	1.09045	0.545224	−	0.838290i	$-0.316444\pi$
$977$	33300.2	1.09045	0.545224	+	0.838290i	$0.316444\pi$
$978$	−31887.1	−1.04257
$979$	−22912.0	−0.747979
$980$	0	0
$981$	−8289.55	−0.269791
$982$	487.491	0.0158416
$983$	17243.3	0.559486	0.279743	−	0.960075i	$-0.409751\pi$
$983$	17243.3	0.559486	0.279743	+	0.960075i	$0.409751\pi$
$984$	30513.6	0.988557
$985$	0	0
$986$	7387.99	0.238622
$987$	0	0
$988$	−134796.	−4.34053
$989$	−74787.8	−2.40456
$990$	0	0
$991$	−32266.7	−1.03429	−0.517147	−	0.855897i	$-0.673006\pi$
$991$	−32266.7	−1.03429	−0.517147	+	0.855897i	$0.673006\pi$
$992$	2516.02	0.0805278
$993$	10159.1	0.324661
$994$	0	0
$995$	0	0
$996$	75414.4	2.39919
$997$	−14339.8	−0.455514	−0.227757	−	0.973718i	$-0.573139\pi$
$997$	−14339.8	−0.455514	−0.227757	+	0.973718i	$0.573139\pi$
$998$	−76059.8	−2.41246
$999$	−36213.3	−1.14688

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1225.4.a.z.1.4		4
5.4	even	2		1225.4.a.bd.1.1		4
7.6	odd	2		175.4.a.g.1.4	✓	4
21.20	even	2		1575.4.a.bl.1.1		4
35.13	even	4		175.4.b.f.99.1		8
35.27	even	4		175.4.b.f.99.8		8
35.34	odd	2		175.4.a.h.1.1	yes	4
105.104	even	2		1575.4.a.bg.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
175.4.a.g.1.4	✓	4	7.6	odd	2
175.4.a.h.1.1	yes	4	35.34	odd	2
175.4.b.f.99.1		8	35.13	even	4
175.4.b.f.99.8		8	35.27	even	4
1225.4.a.z.1.4		4	1.1	even	1	trivial
1225.4.a.bd.1.1		4	5.4	even	2
1575.4.a.bg.1.4		4	105.104	even	2
1575.4.a.bl.1.1		4	21.20	even	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:	\( N \)	\(=\)	\( 1225 = 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1225.a (trivial)

\(f(q)\)	\(=\)	\(q+4.87199 q^{2} -4.14916 q^{3} +15.7363 q^{4} -20.2147 q^{6} +37.6910 q^{8} -9.78444 q^{9} +36.9922 q^{11} -65.2924 q^{12} -61.3165 q^{13} +57.7401 q^{16} -44.8345 q^{17} -47.6697 q^{18} +139.701 q^{19} +180.226 q^{22} -217.580 q^{23} -156.386 q^{24} -298.733 q^{26} +152.625 q^{27} -33.8226 q^{29} -124.437 q^{31} -20.2192 q^{32} -153.487 q^{33} -218.433 q^{34} -153.971 q^{36} -237.270 q^{37} +680.620 q^{38} +254.412 q^{39} -195.117 q^{41} +343.725 q^{43} +582.119 q^{44} -1060.05 q^{46} +16.8224 q^{47} -239.573 q^{48} +186.026 q^{51} -964.894 q^{52} -346.965 q^{53} +743.586 q^{54} -579.641 q^{57} -164.783 q^{58} -135.340 q^{59} -490.414 q^{61} -606.255 q^{62} -560.428 q^{64} -747.785 q^{66} -477.969 q^{67} -705.529 q^{68} +902.777 q^{69} +45.2557 q^{71} -368.786 q^{72} +100.781 q^{73} -1155.98 q^{74} +2198.37 q^{76} +1239.49 q^{78} +880.534 q^{79} -369.085 q^{81} -950.609 q^{82} -1155.03 q^{83} +1674.62 q^{86} +140.336 q^{87} +1394.27 q^{88} -619.374 q^{89} -3423.90 q^{92} +516.309 q^{93} +81.9585 q^{94} +83.8929 q^{96} -231.195 q^{97} -361.948 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} - 3 q^{3} + 36 q^{4} - q^{6} - 27 q^{8} + 61 q^{9} + 100 q^{11} - 165 q^{12} + 44 q^{13} + 160 q^{16} - 53 q^{17} - 433 q^{18} + 29 q^{19} + 152 q^{22} - 295 q^{23} + 21 q^{24} - 700 q^{26}+ \cdots + 2383 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 - 3 * q^3 + 36 * q^4 - q^6 - 27 * q^8 + 61 * q^9 + 100 * q^11 - 165 * q^12 + 44 * q^13 + 160 * q^16 - 53 * q^17 - 433 * q^18 + 29 * q^19 + 152 * q^22 - 295 * q^23 + 21 * q^24 - 700 * q^26 - 441 * q^27 + 129 * q^29 - 114 * q^31 + 310 * q^32 - 865 * q^33 - 203 * q^34 + 1101 * q^36 - 403 * q^37 + 555 * q^38 + 674 * q^39 - 671 * q^41 + 411 * q^43 + 438 * q^44 - 997 * q^46 - 8 * q^47 - 523 * q^48 - 885 * q^51 + 74 * q^52 - 90 * q^53 + 2777 * q^54 - 233 * q^57 - 673 * q^58 - 1018 * q^59 - 50 * q^61 - 1626 * q^62 - 2421 * q^64 + 3841 * q^66 - 424 * q^67 - 617 * q^68 - 1080 * q^69 + 215 * q^71 - 2940 * q^72 - 1207 * q^73 + 623 * q^74 + 3257 * q^76 - 278 * q^78 - 951 * q^79 + 28 * q^81 + 1695 * q^82 - 3035 * q^83 - 99 * q^86 + 2210 * q^87 - 163 * q^88 - 2819 * q^89 - 3073 * q^92 + 852 * q^93 + 3056 * q^94 + 1345 * q^96 - 1100 * q^97 + 2383 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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