LMFDB - Embedded newform 425.6.a.k.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [425,6,Mod(1,425)] 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("425.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(425, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 425 = 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	425.a (trivial)

Newform invariants

comment:select newform

sage:traces = [15,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$68.1631234205$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - x^{14} - 378 x^{13} + 106 x^{12} + 55677 x^{11} + 23739 x^{10} - 4018640 x^{9} + \cdots - 45034730496 $ x^15 - x^14 - 378x^13 + 106x^12 + 55677x^11 + 23739x^10 - 4018640x^9 - 4229556x^8 + 146332768x^7 + 216845216x^6 - 2522980352x^5 - 3824088960x^4 + 19716500736x^3 + 24929041152x^2 - 56205914112*x - 45034730496
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{9}\cdot 5^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-8.07389$ of defining polynomial
Character	$\chi$	$=$	425.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-8.07389 q^{2} +18.0213 q^{3} +33.1877 q^{4} -145.502 q^{6} -142.473 q^{7} -9.58900 q^{8} +81.7682 q^{9} +687.202 q^{11} +598.086 q^{12} +738.458 q^{13} +1150.31 q^{14} -984.584 q^{16} +289.000 q^{17} -660.187 q^{18} -67.1524 q^{19} -2567.56 q^{21} -5548.39 q^{22} -942.982 q^{23} -172.807 q^{24} -5962.23 q^{26} -2905.61 q^{27} -4728.36 q^{28} +7736.96 q^{29} -6185.50 q^{31} +8256.27 q^{32} +12384.3 q^{33} -2333.35 q^{34} +2713.69 q^{36} -8429.23 q^{37} +542.181 q^{38} +13308.0 q^{39} +7390.63 q^{41} +20730.2 q^{42} +11369.6 q^{43} +22806.6 q^{44} +7613.53 q^{46} +6055.20 q^{47} -17743.5 q^{48} +3491.70 q^{49} +5208.16 q^{51} +24507.7 q^{52} -35098.3 q^{53} +23459.6 q^{54} +1366.18 q^{56} -1210.17 q^{57} -62467.4 q^{58} +23999.4 q^{59} +32862.9 q^{61} +49941.1 q^{62} -11649.8 q^{63} -35153.5 q^{64} -99989.4 q^{66} -49249.7 q^{67} +9591.23 q^{68} -16993.8 q^{69} +56753.1 q^{71} -784.075 q^{72} -28007.0 q^{73} +68056.6 q^{74} -2228.63 q^{76} -97908.1 q^{77} -107447. q^{78} +40327.5 q^{79} -72232.6 q^{81} -59671.1 q^{82} -29417.7 q^{83} -85211.3 q^{84} -91797.0 q^{86} +139430. q^{87} -6589.58 q^{88} +64117.3 q^{89} -105211. q^{91} -31295.4 q^{92} -111471. q^{93} -48889.0 q^{94} +148789. q^{96} +98670.9 q^{97} -28191.6 q^{98} +56191.3 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + q^{2} + 9 q^{3} + 277 q^{4} + 169 q^{6} + 181 q^{7} + 753 q^{8} + 1826 q^{9} + 172 q^{11} - 2109 q^{12} - 389 q^{13} + 3635 q^{14} + 6837 q^{16} + 4335 q^{17} + 6742 q^{18} + 5150 q^{19} - 6891 q^{21}+ \cdots - 183214 q^{99}+O(q^{100}) $ 15 * q + q^2 + 9 * q^3 + 277 * q^4 + 169 * q^6 + 181 * q^7 + 753 * q^8 + 1826 * q^9 + 172 * q^11 - 2109 * q^12 - 389 * q^13 + 3635 * q^14 + 6837 * q^16 + 4335 * q^17 + 6742 * q^18 + 5150 * q^19 - 6891 * q^21 - 10522 * q^22 + 3140 * q^23 + 9263 * q^24 + 3181 * q^26 + 1611 * q^27 + 16371 * q^28 + 15214 * q^29 + 18873 * q^31 + 16857 * q^32 - 14700 * q^33 + 289 * q^34 + 83010 * q^36 + 12122 * q^37 - 5302 * q^38 + 17995 * q^39 - 2336 * q^41 - 65201 * q^42 - 660 * q^43 + 40740 * q^44 - 18610 * q^46 + 11494 * q^47 - 48879 * q^48 + 89846 * q^49 + 2601 * q^51 - 71027 * q^52 - 76605 * q^53 - 33675 * q^54 + 71385 * q^56 + 112044 * q^57 - 11270 * q^58 + 7268 * q^59 + 137458 * q^61 + 212145 * q^62 - 25488 * q^63 + 116341 * q^64 + 87900 * q^66 - 5036 * q^67 + 80053 * q^68 - 57176 * q^69 + 180961 * q^71 + 588818 * q^72 - 176062 * q^73 + 11010 * q^74 + 390158 * q^76 + 319456 * q^77 - 339079 * q^78 + 127813 * q^79 + 419847 * q^81 + 200996 * q^82 - 318176 * q^83 - 373745 * q^84 + 214564 * q^86 + 138204 * q^87 - 722070 * q^88 + 97682 * q^89 + 229269 * q^91 + 202080 * q^92 - 514385 * q^93 + 104594 * q^94 + 706465 * q^96 + 381812 * q^97 - 399302 * q^98 - 183214 * 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 $

$2$	−8.07389	−1.42728	−0.713638	−	0.700515i	$-0.752954\pi$
$2$	−8.07389	−1.42728	−0.713638	+	0.700515i	$0.752954\pi$
$3$	18.0213	1.15607	0.578034	−	0.816012i	$-0.303820\pi$
$3$	18.0213	1.15607	0.578034	+	0.816012i	$0.303820\pi$
$4$	33.1877	1.03711
$5$	0	0
$5$	0	0
$6$	−145.502	−1.65003
$7$	−142.473	−1.09898	−0.549489	−	0.835501i	$-0.685178\pi$
$7$	−142.473	−1.09898	−0.549489	+	0.835501i	$0.685178\pi$
$8$	−9.58900	−0.0529723
$9$	81.7682	0.336494
$10$	0	0
$11$	687.202	1.71239	0.856195	−	0.516652i	$-0.172822\pi$
$11$	687.202	1.71239	0.856195	+	0.516652i	$0.172822\pi$
$12$	598.086	1.19898
$13$	738.458	1.21190	0.605951	−	0.795502i	$-0.292793\pi$
$13$	738.458	1.21190	0.605951	+	0.795502i	$0.292793\pi$
$14$	1150.31	1.56854
$15$	0	0
$16$	−984.584	−0.961508
$17$	289.000	0.242536
$17$	289.000	0.242536
$18$	−660.187	−0.480270
$19$	−67.1524	−0.0426754	−0.0213377	−	0.999772i	$-0.506793\pi$
$19$	−67.1524	−0.0426754	−0.0213377	+	0.999772i	$0.506793\pi$
$20$	0	0
$21$	−2567.56	−1.27049
$22$	−5548.39	−2.44405
$23$	−942.982	−0.371692	−0.185846	−	0.982579i	$-0.559503\pi$
$23$	−942.982	−0.371692	−0.185846	+	0.982579i	$0.559503\pi$
$24$	−172.807	−0.0612396
$25$	0	0
$26$	−5962.23	−1.72972
$27$	−2905.61	−0.767058
$28$	−4728.36	−1.13977
$29$	7736.96	1.70834	0.854172	−	0.519990i	$-0.174064\pi$
$29$	7736.96	1.70834	0.854172	+	0.519990i	$0.174064\pi$
$30$	0	0
$31$	−6185.50	−1.15603	−0.578017	−	0.816025i	$-0.696173\pi$
$31$	−6185.50	−1.15603	−0.578017	+	0.816025i	$0.696173\pi$
$32$	8256.27	1.42531
$33$	12384.3	1.97964
$34$	−2333.35	−0.346165
$35$	0	0
$36$	2713.69	0.348983
$37$	−8429.23	−1.01224	−0.506120	−	0.862463i	$-0.668921\pi$
$37$	−8429.23	−1.01224	−0.506120	+	0.862463i	$0.668921\pi$
$38$	542.181	0.0609095
$39$	13308.0	1.40104
$40$	0	0
$41$	7390.63	0.686629	0.343314	−	0.939221i	$-0.388450\pi$
$41$	7390.63	0.686629	0.343314	+	0.939221i	$0.388450\pi$
$42$	20730.2	1.81334
$43$	11369.6	0.937724	0.468862	−	0.883272i	$-0.344664\pi$
$43$	11369.6	0.937724	0.468862	+	0.883272i	$0.344664\pi$
$44$	22806.6	1.77594
$45$	0	0
$46$	7613.53	0.530507
$47$	6055.20	0.399838	0.199919	−	0.979812i	$-0.435932\pi$
$47$	6055.20	0.399838	0.199919	+	0.979812i	$0.435932\pi$
$48$	−17743.5	−1.11157
$49$	3491.70	0.207752
$50$	0	0
$51$	5208.16	0.280388
$52$	24507.7	1.25688
$53$	−35098.3	−1.71631	−0.858156	−	0.513390i	$-0.828390\pi$
$53$	−35098.3	−1.71631	−0.858156	+	0.513390i	$0.828390\pi$
$54$	23459.6	1.09480
$55$	0	0
$56$	1366.18	0.0582153
$57$	−1210.17	−0.0493357
$58$	−62467.4	−2.43828
$59$	23999.4	0.897576	0.448788	−	0.893638i	$-0.351856\pi$
$59$	23999.4	0.897576	0.448788	+	0.893638i	$0.351856\pi$
$60$	0	0
$61$	32862.9	1.13079	0.565394	−	0.824821i	$-0.308724\pi$
$61$	32862.9	1.13079	0.565394	+	0.824821i	$0.308724\pi$
$62$	49941.1	1.64998
$63$	−11649.8	−0.369800
$64$	−35153.5	−1.07280
$65$	0	0
$66$	−99989.4	−2.82549
$67$	−49249.7	−1.34035	−0.670173	−	0.742205i	$-0.733780\pi$
$67$	−49249.7	−1.34035	−0.670173	+	0.742205i	$0.733780\pi$
$68$	9591.23	0.251537
$69$	−16993.8	−0.429702
$70$	0	0
$71$	56753.1	1.33611	0.668057	−	0.744110i	$-0.267126\pi$
$71$	56753.1	1.33611	0.668057	+	0.744110i	$0.267126\pi$
$72$	−784.075	−0.0178249
$73$	−28007.0	−0.615119	−0.307559	−	0.951529i	$-0.599512\pi$
$73$	−28007.0	−0.615119	−0.307559	+	0.951529i	$0.599512\pi$
$74$	68056.6	1.44475
$75$	0	0
$76$	−2228.63	−0.0442592
$77$	−97908.1	−1.88188
$78$	−107447.	−1.99967
$79$	40327.5	0.726999	0.363499	−	0.931594i	$-0.381582\pi$
$79$	40327.5	0.726999	0.363499	+	0.931594i	$0.381582\pi$
$80$	0	0
$81$	−72232.6	−1.22327
$82$	−59671.1	−0.980008
$83$	−29417.7	−0.468720	−0.234360	−	0.972150i	$-0.575299\pi$
$83$	−29417.7	−0.468720	−0.234360	+	0.972150i	$0.575299\pi$
$84$	−85211.3	−1.31765
$85$	0	0
$86$	−91797.0	−1.33839
$87$	139430.	1.97496
$88$	−6589.58	−0.0907092
$89$	64117.3	0.858026	0.429013	−	0.903298i	$-0.358862\pi$
$89$	64117.3	0.858026	0.429013	+	0.903298i	$0.358862\pi$
$90$	0	0
$91$	−105211.	−1.33185
$92$	−31295.4	−0.385488
$93$	−111471.	−1.33646
$94$	−48889.0	−0.570678
$95$	0	0
$96$	148789.	1.64775
$97$	98670.9	1.06478	0.532390	−	0.846499i	$-0.321294\pi$
$97$	98670.9	1.06478	0.532390	+	0.846499i	$0.321294\pi$
$98$	−28191.6	−0.296520
$99$	56191.3	0.576210
$100$	0	0
$101$	157091.	1.53231	0.766157	−	0.642653i	$-0.222166\pi$
$101$	157091.	1.53231	0.766157	+	0.642653i	$0.222166\pi$
$102$	−42050.1	−0.400191
$103$	196343.	1.82357	0.911783	−	0.410672i	$-0.134706\pi$
$103$	196343.	1.82357	0.911783	+	0.410672i	$0.134706\pi$
$104$	−7081.07	−0.0641972
$105$	0	0
$106$	283379.	2.44965
$107$	−135667.	−1.14555	−0.572776	−	0.819712i	$-0.694134\pi$
$107$	−135667.	−1.14555	−0.572776	+	0.819712i	$0.694134\pi$
$108$	−96430.4	−0.795527
$109$	−189284.	−1.52598	−0.762988	−	0.646413i	$-0.776268\pi$
$109$	−189284.	−1.52598	−0.762988	+	0.646413i	$0.776268\pi$
$110$	0	0
$111$	−151906.	−1.17022
$112$	140277.	1.05668
$113$	−102836.	−0.757618	−0.378809	−	0.925475i	$-0.623666\pi$
$113$	−102836.	−0.757618	−0.378809	+	0.925475i	$0.623666\pi$
$114$	9770.81	0.0704156
$115$	0	0
$116$	256772.	1.77175
$117$	60382.3	0.407798
$118$	−193769.	−1.28109
$119$	−41174.8	−0.266541
$120$	0	0
$121$	311196.	1.93228
$122$	−265331.	−1.61395
$123$	133189.	0.793790
$124$	−205282.	−1.19894
$125$	0	0
$126$	94059.1	0.527806
$127$	89049.4	0.489916	0.244958	−	0.969534i	$-0.421226\pi$
$127$	89049.4	0.489916	0.244958	+	0.969534i	$0.421226\pi$
$128$	19624.7	0.105872
$129$	204896.	1.08407
$130$	0	0
$131$	146144.	0.744051	0.372025	−	0.928223i	$-0.378663\pi$
$131$	146144.	0.744051	0.372025	+	0.928223i	$0.378663\pi$
$132$	411006.	2.05311
$133$	9567.43	0.0468993
$134$	397637.	1.91304
$135$	0	0
$136$	−2771.22	−0.0128477
$137$	160824.	0.732063	0.366031	−	0.930603i	$-0.380716\pi$
$137$	160824.	0.732063	0.366031	+	0.930603i	$0.380716\pi$
$138$	137206.	0.613303
$139$	19640.0	0.0862194	0.0431097	−	0.999070i	$-0.486274\pi$
$139$	19640.0	0.0862194	0.0431097	+	0.999070i	$0.486274\pi$
$140$	0	0
$141$	109123.	0.462240
$142$	−458218.	−1.90700
$143$	507470.	2.07525
$144$	−80507.7	−0.323542
$145$	0	0
$146$	226125.	0.877944
$147$	62925.0	0.240176
$148$	−279746.	−1.04981
$149$	157622.	0.581636	0.290818	−	0.956778i	$-0.406073\pi$
$149$	157622.	0.581636	0.290818	+	0.956778i	$0.406073\pi$
$150$	0	0
$151$	71650.8	0.255728	0.127864	−	0.991792i	$-0.459188\pi$
$151$	71650.8	0.255728	0.127864	+	0.991792i	$0.459188\pi$
$152$	643.924	0.00226061
$153$	23631.0	0.0816119
$154$	790499.	2.68596
$155$	0	0
$156$	441661.	1.45304
$157$	133758.	0.433081	0.216541	−	0.976274i	$-0.430523\pi$
$157$	133758.	0.433081	0.216541	+	0.976274i	$0.430523\pi$
$158$	−325600.	−1.03763
$159$	−632517.	−1.98417
$160$	0	0
$161$	134350.	0.408482
$162$	583198.	1.74594
$163$	356947.	1.05229	0.526145	−	0.850395i	$-0.323637\pi$
$163$	356947.	1.05229	0.526145	+	0.850395i	$0.323637\pi$
$164$	245278.	0.712112
$165$	0	0
$166$	237515.	0.668993
$167$	239921.	0.665699	0.332850	−	0.942980i	$-0.391990\pi$
$167$	239921.	0.665699	0.332850	+	0.942980i	$0.391990\pi$
$168$	24620.3	0.0673009
$169$	174027.	0.468706
$170$	0	0
$171$	−5490.93	−0.0143600
$172$	377331.	0.972527
$173$	554668.	1.40902	0.704512	−	0.709693i	$-0.251166\pi$
$173$	554668.	1.40902	0.704512	+	0.709693i	$0.251166\pi$
$174$	−1.12574e6	−2.81882
$175$	0	0
$176$	−676609.	−1.64648
$177$	432502.	1.03766
$178$	−517676.	−1.22464
$179$	−92023.1	−0.214667	−0.107333	−	0.994223i	$-0.534231\pi$
$179$	−92023.1	−0.214667	−0.107333	+	0.994223i	$0.534231\pi$
$180$	0	0
$181$	−3508.83	−0.00796096	−0.00398048	−	0.999992i	$-0.501267\pi$
$181$	−3508.83	−0.00796096	−0.00398048	+	0.999992i	$0.501267\pi$
$182$	849459.	1.90092
$183$	592233.	1.30727
$184$	9042.25	0.0196894
$185$	0	0
$186$	900004.	1.90749
$187$	198601.	0.415316
$188$	200958.	0.414677
$189$	413973.	0.842980
$190$	0	0
$191$	35556.6	0.0705239	0.0352619	−	0.999378i	$-0.488773\pi$
$191$	35556.6	0.0705239	0.0352619	+	0.999378i	$0.488773\pi$
$192$	−633513.	−1.24023
$193$	−87356.2	−0.168811	−0.0844055	−	0.996431i	$-0.526899\pi$
$193$	−87356.2	−0.168811	−0.0844055	+	0.996431i	$0.526899\pi$
$194$	−796658.	−1.51973
$195$	0	0
$196$	115881.	0.215463
$197$	240229.	0.441021	0.220510	−	0.975385i	$-0.429228\pi$
$197$	240229.	0.441021	0.220510	+	0.975385i	$0.429228\pi$
$198$	−453682.	−0.822410
$199$	436482.	0.781329	0.390664	−	0.920533i	$-0.372245\pi$
$199$	436482.	0.781329	0.390664	+	0.920533i	$0.372245\pi$
$200$	0	0
$201$	−887545.	−1.54953
$202$	−1.26833e6	−2.18703
$203$	−1.10231e6	−1.87743
$204$	172847.	0.290794
$205$	0	0
$206$	−1.58525e6	−2.60273
$207$	−77105.9	−0.125072
$208$	−727074.	−1.16525
$209$	−46147.3	−0.0730769
$210$	0	0
$211$	−130210.	−0.201344	−0.100672	−	0.994920i	$-0.532099\pi$
$211$	−130210.	−0.201344	−0.100672	+	0.994920i	$0.532099\pi$
$212$	−1.16483e6	−1.78001
$213$	1.02277e6	1.54464
$214$	1.09536e6	1.63502
$215$	0	0
$216$	27861.9	0.0406328
$217$	881270.	1.27046
$218$	1.52826e6	2.17799
$219$	−504723.	−0.711120
$220$	0	0
$221$	213414.	0.293929
$222$	1.22647e6	1.67022
$223$	1.27729e6	1.71999	0.859995	−	0.510302i	$-0.170466\pi$
$223$	1.27729e6	1.71999	0.859995	+	0.510302i	$0.170466\pi$
$224$	−1.17630e6	−1.56638
$225$	0	0
$226$	830289.	1.08133
$227$	1.10351e6	1.42139	0.710695	−	0.703500i	$-0.248381\pi$
$227$	1.10351e6	1.42139	0.710695	+	0.703500i	$0.248381\pi$
$228$	−40162.9	−0.0511667
$229$	−426559.	−0.537515	−0.268758	−	0.963208i	$-0.586613\pi$
$229$	−426559.	−0.537515	−0.268758	+	0.963208i	$0.586613\pi$
$230$	0	0
$231$	−1.76443e6	−2.17558
$232$	−74189.7	−0.0904949
$233$	−998389.	−1.20479	−0.602393	−	0.798199i	$-0.705786\pi$
$233$	−998389.	−1.20479	−0.602393	+	0.798199i	$0.705786\pi$
$234$	−487520.	−0.582040
$235$	0	0
$236$	796485.	0.930889
$237$	726755.	0.840460
$238$	332441.	0.380428
$239$	621754.	0.704083	0.352042	−	0.935984i	$-0.385488\pi$
$239$	621754.	0.704083	0.352042	+	0.935984i	$0.385488\pi$
$240$	0	0
$241$	1.15433e6	1.28022	0.640112	−	0.768282i	$-0.278888\pi$
$241$	1.15433e6	1.28022	0.640112	+	0.768282i	$0.278888\pi$
$242$	−2.51256e6	−2.75790
$243$	−595664.	−0.647121
$244$	1.09064e6	1.17276
$245$	0	0
$246$	−1.07535e6	−1.13296
$247$	−49589.2	−0.0517184
$248$	59312.8	0.0612378
$249$	−530146.	−0.541873
$250$	0	0
$251$	−894344.	−0.896025	−0.448013	−	0.894027i	$-0.647868\pi$
$251$	−894344.	−0.896025	−0.448013	+	0.894027i	$0.647868\pi$
$252$	−386629.	−0.383525
$253$	−648019.	−0.636483
$254$	−718974.	−0.699245
$255$	0	0
$256$	966464.	0.921692
$257$	−281866.	−0.266201	−0.133101	−	0.991103i	$-0.542493\pi$
$257$	−281866.	−0.266201	−0.133101	+	0.991103i	$0.542493\pi$
$258$	−1.65430e6	−1.54727
$259$	1.20094e6	1.11243
$260$	0	0
$261$	632637.	0.574848
$262$	−1.17995e6	−1.06196
$263$	−100525.	−0.0896160	−0.0448080	−	0.998996i	$-0.514268\pi$
$263$	−100525.	−0.0896160	−0.0448080	+	0.998996i	$0.514268\pi$
$264$	−118753.	−0.104866
$265$	0	0
$266$	−77246.4	−0.0669382
$267$	1.15548e6	0.991936
$268$	−1.63448e6	−1.39009
$269$	1.27364e6	1.07317	0.536584	−	0.843847i	$-0.319715\pi$
$269$	1.27364e6	1.07317	0.536584	+	0.843847i	$0.319715\pi$
$270$	0	0
$271$	1.86952e6	1.54635	0.773175	−	0.634193i	$-0.218668\pi$
$271$	1.86952e6	1.54635	0.773175	+	0.634193i	$0.218668\pi$
$272$	−284545.	−0.233200
$273$	−1.89604e6	−1.53971
$274$	−1.29847e6	−1.04485
$275$	0	0
$276$	−563984.	−0.445650
$277$	−1.61032e6	−1.26099	−0.630496	−	0.776192i	$-0.717149\pi$
$277$	−1.61032e6	−1.26099	−0.630496	+	0.776192i	$0.717149\pi$
$278$	−158571.	−0.123059
$279$	−505777.	−0.388999
$280$	0	0
$281$	−1.96805e6	−1.48686	−0.743432	−	0.668812i	$-0.766803\pi$
$281$	−1.96805e6	−1.48686	−0.743432	+	0.668812i	$0.766803\pi$
$282$	−881044.	−0.659743
$283$	−591303.	−0.438878	−0.219439	−	0.975626i	$-0.570423\pi$
$283$	−591303.	−0.438878	−0.219439	+	0.975626i	$0.570423\pi$
$284$	1.88350e6	1.38570
$285$	0	0
$286$	−4.09726e6	−2.96195
$287$	−1.05297e6	−0.754590
$288$	675100.	0.479609
$289$	83521.0	0.0588235
$290$	0	0
$291$	1.77818e6	1.23096
$292$	−929486.	−0.637949
$293$	956964.	0.651219	0.325609	−	0.945504i	$-0.394431\pi$
$293$	956964.	0.651219	0.325609	+	0.945504i	$0.394431\pi$
$294$	−508049.	−0.342797
$295$	0	0
$296$	80827.9	0.0536206
$297$	−1.99674e6	−1.31350
$298$	−1.27262e6	−0.830155
$299$	−696352.	−0.450455
$300$	0	0
$301$	−1.61987e6	−1.03054
$302$	−578500.	−0.364994
$303$	2.83099e6	1.77146
$304$	66117.2	0.0410327
$305$	0	0
$306$	−190794.	−0.116483
$307$	350909.	0.212495	0.106247	−	0.994340i	$-0.466116\pi$
$307$	350909.	0.212495	0.106247	+	0.994340i	$0.466116\pi$
$308$	−3.24934e6	−1.95172
$309$	3.53835e6	2.10817
$310$	0	0
$311$	2.00404e6	1.17491	0.587456	−	0.809256i	$-0.300129\pi$
$311$	2.00404e6	1.17491	0.587456	+	0.809256i	$0.300129\pi$
$312$	−127610.	−0.0742163
$313$	−971363.	−0.560429	−0.280215	−	0.959937i	$-0.590406\pi$
$313$	−971363.	−0.560429	−0.280215	+	0.959937i	$0.590406\pi$
$314$	−1.07994e6	−0.618126
$315$	0	0
$316$	1.33838e6	0.753981
$317$	−2.40304e6	−1.34311	−0.671557	−	0.740953i	$-0.734374\pi$
$317$	−2.40304e6	−1.34311	−0.671557	+	0.740953i	$0.734374\pi$
$318$	5.10687e6	2.83196
$319$	5.31686e6	2.92535
$320$	0	0
$321$	−2.44490e6	−1.32434
$322$	−1.08473e6	−0.583016
$323$	−19407.0	−0.0103503
$324$	−2.39723e6	−1.26867
$325$	0	0
$326$	−2.88195e6	−1.50191
$327$	−3.41115e6	−1.76413
$328$	−70868.8	−0.0363723
$329$	−862705.	−0.439413
$330$	0	0
$331$	−2.35993e6	−1.18394	−0.591970	−	0.805960i	$-0.701649\pi$
$331$	−2.35993e6	−1.18394	−0.591970	+	0.805960i	$0.701649\pi$
$332$	−976305.	−0.486116
$333$	−689242.	−0.340613
$334$	−1.93710e6	−0.950136
$335$	0	0
$336$	2.52798e6	1.22159
$337$	3.46539e6	1.66218	0.831088	−	0.556141i	$-0.187719\pi$
$337$	3.46539e6	1.66218	0.831088	+	0.556141i	$0.187719\pi$
$338$	−1.40508e6	−0.668972
$339$	−1.85325e6	−0.875858
$340$	0	0
$341$	−4.25069e6	−1.97958
$342$	44333.1	0.0204957
$343$	1.89708e6	0.870663
$344$	−109023.	−0.0496733
$345$	0	0
$346$	−4.47833e6	−2.01106
$347$	−3.98574e6	−1.77699	−0.888495	−	0.458885i	$-0.848249\pi$
$347$	−3.98574e6	−1.77699	−0.888495	+	0.458885i	$0.848249\pi$
$348$	4.62737e6	2.04826
$349$	1.94543e6	0.854972	0.427486	−	0.904022i	$-0.359399\pi$
$349$	1.94543e6	0.854972	0.427486	+	0.904022i	$0.359399\pi$
$350$	0	0
$351$	−2.14567e6	−0.929599
$352$	5.67373e6	2.44069
$353$	1.26368e6	0.539761	0.269881	−	0.962894i	$-0.413016\pi$
$353$	1.26368e6	0.539761	0.269881	+	0.962894i	$0.413016\pi$
$354$	−3.49197e6	−1.48103
$355$	0	0
$356$	2.12790e6	0.889871
$357$	−742025.	−0.308140
$358$	742985.	0.306388
$359$	−2.86737e6	−1.17422	−0.587108	−	0.809509i	$-0.699734\pi$
$359$	−2.86737e6	−1.17422	−0.587108	+	0.809509i	$0.699734\pi$
$360$	0	0
$361$	−2.47159e6	−0.998179
$362$	28329.9	0.0113625
$363$	5.60816e6	2.23385
$364$	−3.49170e6	−1.38128
$365$	0	0
$366$	−4.78162e6	−1.86583
$367$	3.47234e6	1.34573	0.672863	−	0.739767i	$-0.265064\pi$
$367$	3.47234e6	1.34573	0.672863	+	0.739767i	$0.265064\pi$
$368$	928445.	0.357385
$369$	604318.	0.231047
$370$	0	0
$371$	5.00057e6	1.88619
$372$	−3.69946e6	−1.38606
$373$	4.52989e6	1.68584	0.842919	−	0.538041i	$-0.180835\pi$
$373$	4.52989e6	1.68584	0.842919	+	0.538041i	$0.180835\pi$
$374$	−1.60349e6	−0.592770
$375$	0	0
$376$	−58063.3	−0.0211803
$377$	5.71342e6	2.07035
$378$	−3.34237e6	−1.20316
$379$	249026.	0.0890526	0.0445263	−	0.999008i	$-0.485822\pi$
$379$	249026.	0.0890526	0.0445263	+	0.999008i	$0.485822\pi$
$380$	0	0
$381$	1.60479e6	0.566376
$382$	−287080.	−0.100657
$383$	−2.42097e6	−0.843320	−0.421660	−	0.906754i	$-0.638552\pi$
$383$	−2.42097e6	−0.843320	−0.421660	+	0.906754i	$0.638552\pi$
$384$	353664.	0.122395
$385$	0	0
$386$	705304.	0.240940
$387$	929673.	0.315539
$388$	3.27466e6	1.10430
$389$	−4.83276e6	−1.61928	−0.809639	−	0.586928i	$-0.800337\pi$
$389$	−4.83276e6	−1.61928	−0.809639	+	0.586928i	$0.800337\pi$
$390$	0	0
$391$	−272522.	−0.0901487
$392$	−33481.9	−0.0110051
$393$	2.63371e6	0.860173
$394$	−1.93958e6	−0.629458
$395$	0	0
$396$	1.86486e6	0.597596
$397$	4.34939e6	1.38501	0.692503	−	0.721415i	$-0.256508\pi$
$397$	4.34939e6	1.38501	0.692503	+	0.721415i	$0.256508\pi$
$398$	−3.52411e6	−1.11517
$399$	172418.	0.0542188
$400$	0	0
$401$	5.20421e6	1.61620	0.808098	−	0.589048i	$-0.200497\pi$
$401$	5.20421e6	1.61620	0.808098	+	0.589048i	$0.200497\pi$
$402$	7.16594e6	2.21161
$403$	−4.56773e6	−1.40100
$404$	5.21348e6	1.58918
$405$	0	0
$406$	8.89994e6	2.67961
$407$	−5.79258e6	−1.73335
$408$	−49941.1	−0.0148528
$409$	3.68597e6	1.08954	0.544771	−	0.838585i	$-0.316617\pi$
$409$	3.68597e6	1.08954	0.544771	+	0.838585i	$0.316617\pi$
$410$	0	0
$411$	2.89825e6	0.846314
$412$	6.51615e6	1.89125
$413$	−3.41928e6	−0.986416
$414$	622544.	0.178513
$415$	0	0
$416$	6.09691e6	1.72733
$417$	353939.	0.0996755
$418$	372588.	0.104301
$419$	−184836.	−0.0514341	−0.0257171	−	0.999669i	$-0.508187\pi$
$419$	−184836.	−0.0514341	−0.0257171	+	0.999669i	$0.508187\pi$
$420$	0	0
$421$	−3.47811e6	−0.956397	−0.478198	−	0.878252i	$-0.658710\pi$
$421$	−3.47811e6	−0.956397	−0.478198	+	0.878252i	$0.658710\pi$
$422$	1.05130e6	0.287373
$423$	495122.	0.134543
$424$	336557.	0.0909169
$425$	0	0
$426$	−8.25770e6	−2.20463
$427$	−4.68209e6	−1.24271
$428$	−4.50247e6	−1.18807
$429$	9.14528e6	2.39913
$430$	0	0
$431$	−2.01973e6	−0.523721	−0.261860	−	0.965106i	$-0.584336\pi$
$431$	−2.01973e6	−0.523721	−0.261860	+	0.965106i	$0.584336\pi$
$432$	2.86082e6	0.737533
$433$	−153136.	−0.0392516	−0.0196258	−	0.999807i	$-0.506247\pi$
$433$	−153136.	−0.0392516	−0.0196258	+	0.999807i	$0.506247\pi$
$434$	−7.11528e6	−1.81329
$435$	0	0
$436$	−6.28189e6	−1.58261
$437$	63323.5	0.0158621
$438$	4.07507e6	1.01496
$439$	4.10685e6	1.01706	0.508531	−	0.861044i	$-0.330189\pi$
$439$	4.10685e6	1.01706	0.508531	+	0.861044i	$0.330189\pi$
$440$	0	0
$441$	285510.	0.0699076
$442$	−1.72308e6	−0.419518
$443$	1.67255e6	0.404921	0.202460	−	0.979290i	$-0.435106\pi$
$443$	1.67255e6	0.404921	0.202460	+	0.979290i	$0.435106\pi$
$444$	−5.04140e6	−1.21365
$445$	0	0
$446$	−1.03127e7	−2.45490
$447$	2.84056e6	0.672412
$448$	5.00844e6	1.17898
$449$	5.87992e6	1.37643	0.688217	−	0.725505i	$-0.258394\pi$
$449$	5.87992e6	1.37643	0.688217	+	0.725505i	$0.258394\pi$
$450$	0	0
$451$	5.07886e6	1.17578
$452$	−3.41289e6	−0.785736
$453$	1.29124e6	0.295639
$454$	−8.90965e6	−2.02871
$455$	0	0
$456$	11604.4	0.00261342
$457$	−3.94831e6	−0.884342	−0.442171	−	0.896931i	$-0.645792\pi$
$457$	−3.94831e6	−0.884342	−0.442171	+	0.896931i	$0.645792\pi$
$458$	3.44399e6	0.767182
$459$	−839722.	−0.186039
$460$	0	0
$461$	5.70622e6	1.25054	0.625268	−	0.780410i	$-0.284990\pi$
$461$	5.70622e6	1.25054	0.625268	+	0.780410i	$0.284990\pi$
$462$	1.42458e7	3.10515
$463$	−2.81944e6	−0.611237	−0.305619	−	0.952154i	$-0.598863\pi$
$463$	−2.81944e6	−0.611237	−0.305619	+	0.952154i	$0.598863\pi$
$464$	−7.61769e6	−1.64259
$465$	0	0
$466$	8.06088e6	1.71956
$467$	2.78639e6	0.591220	0.295610	−	0.955309i	$-0.404477\pi$
$467$	2.78639e6	0.591220	0.295610	+	0.955309i	$0.404477\pi$
$468$	2.00395e6	0.422933
$469$	7.01678e6	1.47301
$470$	0	0
$471$	2.41049e6	0.500672
$472$	−230131.	−0.0475466
$473$	7.81323e6	1.60575
$474$	−5.86774e6	−1.19957
$475$	0	0
$476$	−1.36650e6	−0.276434
$477$	−2.86992e6	−0.577529
$478$	−5.01997e6	−1.00492
$479$	−6.30161e6	−1.25491	−0.627456	−	0.778652i	$-0.715904\pi$
$479$	−6.30161e6	−1.25491	−0.627456	+	0.778652i	$0.715904\pi$
$480$	0	0
$481$	−6.22463e6	−1.22674
$482$	−9.31990e6	−1.82723
$483$	2.42116e6	0.472233
$484$	1.03279e7	2.00400
$485$	0	0
$486$	4.80932e6	0.923620
$487$	−7.39430e6	−1.41278	−0.706390	−	0.707823i	$-0.749677\pi$
$487$	−7.39430e6	−1.41278	−0.706390	+	0.707823i	$0.749677\pi$
$488$	−315122.	−0.0599004
$489$	6.43266e6	1.21652
$490$	0	0
$491$	4.20165e6	0.786531	0.393266	−	0.919425i	$-0.371345\pi$
$491$	4.20165e6	0.786531	0.393266	+	0.919425i	$0.371345\pi$
$492$	4.42023e6	0.823251
$493$	2.23598e6	0.414334
$494$	400378.	0.0738163
$495$	0	0
$496$	6.09015e6	1.11154
$497$	−8.08581e6	−1.46836
$498$	4.28034e6	0.773401
$499$	−6.73674e6	−1.21115	−0.605576	−	0.795787i	$-0.707057\pi$
$499$	−6.73674e6	−1.21115	−0.605576	+	0.795787i	$0.707057\pi$
$500$	0	0
$501$	4.32370e6	0.769594
$502$	7.22083e6	1.27887
$503$	−1.02213e7	−1.80130	−0.900652	−	0.434540i	$-0.856911\pi$
$503$	−1.02213e7	−1.80130	−0.900652	+	0.434540i	$0.856911\pi$
$504$	111710.	0.0195891
$505$	0	0
$506$	5.23203e6	0.908436
$507$	3.13620e6	0.541856
$508$	2.95534e6	0.508099
$509$	−3.08184e6	−0.527248	−0.263624	−	0.964625i	$-0.584918\pi$
$509$	−3.08184e6	−0.527248	−0.263624	+	0.964625i	$0.584918\pi$
$510$	0	0
$511$	3.99025e6	0.676002
$512$	−8.43112e6	−1.42138
$513$	195119.	0.0327345
$514$	2.27575e6	0.379942
$515$	0	0
$516$	6.80001e6	1.12431
$517$	4.16115e6	0.684678
$518$	−9.69627e6	−1.58774
$519$	9.99586e6	1.62893
$520$	0	0
$521$	−1.11138e7	−1.79377	−0.896887	−	0.442261i	$-0.854177\pi$
$521$	−1.11138e7	−1.79377	−0.896887	+	0.442261i	$0.854177\pi$
$522$	−5.10784e6	−0.820467
$523$	2.30091e6	0.367828	0.183914	−	0.982942i	$-0.441123\pi$
$523$	2.30091e6	0.367828	0.183914	+	0.982942i	$0.441123\pi$
$524$	4.85017e6	0.771665
$525$	0	0
$526$	811629.	0.127907
$527$	−1.78761e6	−0.280380
$528$	−1.21934e7	−1.90344
$529$	−5.54713e6	−0.861845
$530$	0	0
$531$	1.96239e6	0.302029
$532$	317521.	0.0486399
$533$	5.45767e6	0.832126
$534$	−9.32921e6	−1.41577
$535$	0	0
$536$	472256.	0.0710011
$537$	−1.65838e6	−0.248169
$538$	−1.02833e7	−1.53170
$539$	2.39950e6	0.355753
$540$	0	0
$541$	−7.55092e6	−1.10919	−0.554596	−	0.832120i	$-0.687127\pi$
$541$	−7.55092e6	−1.10919	−0.554596	+	0.832120i	$0.687127\pi$
$542$	−1.50943e7	−2.20707
$543$	−63233.7	−0.00920341
$544$	2.38606e6	0.345688
$545$	0	0
$546$	1.53084e7	2.19759
$547$	−3.95362e6	−0.564972	−0.282486	−	0.959271i	$-0.591159\pi$
$547$	−3.95362e6	−0.564972	−0.282486	+	0.959271i	$0.591159\pi$
$548$	5.33736e6	0.759232
$549$	2.68714e6	0.380504
$550$	0	0
$551$	−519555.	−0.0729043
$552$	162953.	0.0227623
$553$	−5.74560e6	−0.798956
$554$	1.30015e7	1.79978
$555$	0	0
$556$	651806.	0.0894194
$557$	−8.92128e6	−1.21840	−0.609199	−	0.793017i	$-0.708509\pi$
$557$	−8.92128e6	−1.21840	−0.609199	+	0.793017i	$0.708509\pi$
$558$	4.08359e6	0.555209
$559$	8.39599e6	1.13643
$560$	0	0
$561$	3.57906e6	0.480133
$562$	1.58898e7	2.12216
$563$	−7.62831e6	−1.01428	−0.507139	−	0.861864i	$-0.669297\pi$
$563$	−7.62831e6	−1.01428	−0.507139	+	0.861864i	$0.669297\pi$
$564$	3.62153e6	0.479395
$565$	0	0
$566$	4.77411e6	0.626400
$567$	1.02912e7	1.34434
$568$	−544206.	−0.0707770
$569$	−1.44032e7	−1.86500	−0.932501	−	0.361168i	$-0.882378\pi$
$569$	−1.44032e7	−1.86500	−0.932501	+	0.361168i	$0.882378\pi$
$570$	0	0
$571$	−8.41065e6	−1.07954	−0.539771	−	0.841812i	$-0.681489\pi$
$571$	−8.41065e6	−1.07954	−0.539771	+	0.841812i	$0.681489\pi$
$572$	1.68417e7	2.15227
$573$	640776.	0.0815305
$574$	8.50155e6	1.07701
$575$	0	0
$576$	−2.87444e6	−0.360991
$577$	−3.17986e6	−0.397620	−0.198810	−	0.980038i	$-0.563708\pi$
$577$	−3.17986e6	−0.397620	−0.198810	+	0.980038i	$0.563708\pi$
$578$	−674339.	−0.0839574
$579$	−1.57428e6	−0.195157
$580$	0	0
$581$	4.19124e6	0.515113
$582$	−1.43568e7	−1.75692
$583$	−2.41196e7	−2.93900
$584$	268559.	0.0325842
$585$	0	0
$586$	−7.72642e6	−0.929468
$587$	7.17871e6	0.859906	0.429953	−	0.902851i	$-0.358530\pi$
$587$	7.17871e6	0.859906	0.429953	+	0.902851i	$0.358530\pi$
$588$	2.08833e6	0.249090
$589$	415371.	0.0493342
$590$	0	0
$591$	4.32924e6	0.509850
$592$	8.29929e6	0.973277
$593$	−9.25867e6	−1.08121	−0.540607	−	0.841275i	$-0.681806\pi$
$593$	−9.25867e6	−1.08121	−0.540607	+	0.841275i	$0.681806\pi$
$594$	1.61215e7	1.87473
$595$	0	0
$596$	5.23111e6	0.603223
$597$	7.86599e6	0.903270
$598$	5.62227e6	0.642923
$599$	−399594.	−0.0455042	−0.0227521	−	0.999741i	$-0.507243\pi$
$599$	−399594.	−0.0455042	−0.0227521	+	0.999741i	$0.507243\pi$
$600$	0	0
$601$	1.33812e6	0.151115	0.0755577	−	0.997141i	$-0.475926\pi$
$601$	1.33812e6	0.151115	0.0755577	+	0.997141i	$0.475926\pi$
$602$	1.30786e7	1.47086
$603$	−4.02706e6	−0.451019
$604$	2.37792e6	0.265219
$605$	0	0
$606$	−2.28571e7	−2.52836
$607$	−1.48763e7	−1.63878	−0.819392	−	0.573233i	$-0.805689\pi$
$607$	−1.48763e7	−1.63878	−0.819392	+	0.573233i	$0.805689\pi$
$608$	−554428.	−0.0608256
$609$	−1.98651e7	−2.17044
$610$	0	0
$611$	4.47151e6	0.484564
$612$	784257.	0.0846409
$613$	−2.23381e6	−0.240102	−0.120051	−	0.992768i	$-0.538306\pi$
$613$	−2.23381e6	−0.240102	−0.120051	+	0.992768i	$0.538306\pi$
$614$	−2.83320e6	−0.303288
$615$	0	0
$616$	938841.	0.0996874
$617$	8.43176e6	0.891672	0.445836	−	0.895115i	$-0.352906\pi$
$617$	8.43176e6	0.891672	0.445836	+	0.895115i	$0.352906\pi$
$618$	−2.85683e7	−3.00893
$619$	−6.74772e6	−0.707832	−0.353916	−	0.935277i	$-0.615150\pi$
$619$	−6.74772e6	−0.707832	−0.353916	+	0.935277i	$0.615150\pi$
$620$	0	0
$621$	2.73994e6	0.285110
$622$	−1.61804e7	−1.67692
$623$	−9.13502e6	−0.942951
$624$	−1.31028e7	−1.34711
$625$	0	0
$626$	7.84267e6	0.799886
$627$	−831635.	−0.0844819
$628$	4.43910e6	0.449155
$629$	−2.43605e6	−0.245504
$630$	0	0
$631$	−1.02310e6	−0.102293	−0.0511466	−	0.998691i	$-0.516288\pi$
$631$	−1.02310e6	−0.102293	−0.0511466	+	0.998691i	$0.516288\pi$
$632$	−386701.	−0.0385108
$633$	−2.34656e6	−0.232767
$634$	1.94019e7	1.91699
$635$	0	0
$636$	−2.09918e7	−2.05781
$637$	2.57847e6	0.251776
$638$	−4.29277e7	−4.17528
$639$	4.64060e6	0.449595
$640$	0	0
$641$	3.27709e6	0.315024	0.157512	−	0.987517i	$-0.449653\pi$
$641$	3.27709e6	0.315024	0.157512	+	0.987517i	$0.449653\pi$
$642$	1.97398e7	1.89019
$643$	1.81703e7	1.73314	0.866571	−	0.499054i	$-0.166319\pi$
$643$	1.81703e7	1.73314	0.866571	+	0.499054i	$0.166319\pi$
$644$	4.45876e6	0.423642
$645$	0	0
$646$	156690.	0.0147727
$647$	−1.11777e7	−1.04976	−0.524882	−	0.851175i	$-0.675891\pi$
$647$	−1.11777e7	−1.04976	−0.524882	+	0.851175i	$0.675891\pi$
$648$	692639.	0.0647992
$649$	1.64925e7	1.53700
$650$	0	0
$651$	1.58817e7	1.46873
$652$	1.18462e7	1.09134
$653$	1.46182e7	1.34156	0.670782	−	0.741655i	$-0.265959\pi$
$653$	1.46182e7	1.34156	0.670782	+	0.741655i	$0.265959\pi$
$654$	2.75412e7	2.51790
$655$	0	0
$656$	−7.27670e6	−0.660199
$657$	−2.29008e6	−0.206984
$658$	6.96538e6	0.627163
$659$	3.39727e6	0.304731	0.152365	−	0.988324i	$-0.451311\pi$
$659$	3.39727e6	0.304731	0.152365	+	0.988324i	$0.451311\pi$
$660$	0	0
$661$	−1.77397e7	−1.57922	−0.789611	−	0.613608i	$-0.789718\pi$
$661$	−1.77397e7	−1.57922	−0.789611	+	0.613608i	$0.789718\pi$
$662$	1.90538e7	1.68981
$663$	3.84601e6	0.339802
$664$	282087.	0.0248292
$665$	0	0
$666$	5.56487e6	0.486149
$667$	−7.29582e6	−0.634979
$668$	7.96243e6	0.690406
$669$	2.30184e7	1.98843
$670$	0	0
$671$	2.25835e7	1.93635
$672$	−2.11985e7	−1.81085
$673$	6.45698e6	0.549530	0.274765	−	0.961511i	$-0.411400\pi$
$673$	6.45698e6	0.549530	0.274765	+	0.961511i	$0.411400\pi$
$674$	−2.79791e7	−2.37238
$675$	0	0
$676$	5.77555e6	0.486101
$677$	−1.08873e7	−0.912953	−0.456477	−	0.889735i	$-0.650889\pi$
$677$	−1.08873e7	−0.912953	−0.456477	+	0.889735i	$0.650889\pi$
$678$	1.49629e7	1.25009
$679$	−1.40580e7	−1.17017
$680$	0	0
$681$	1.98868e7	1.64322
$682$	3.43196e7	2.82541
$683$	6.23075e6	0.511080	0.255540	−	0.966798i	$-0.417747\pi$
$683$	6.23075e6	0.511080	0.255540	+	0.966798i	$0.417747\pi$
$684$	−182231.	−0.0148930
$685$	0	0
$686$	−1.53168e7	−1.24267
$687$	−7.68716e6	−0.621404
$688$	−1.11944e7	−0.901629
$689$	−2.59186e7	−2.08000
$690$	0	0
$691$	−1.16135e7	−0.925270	−0.462635	−	0.886549i	$-0.653096\pi$
$691$	−1.16135e7	−0.925270	−0.462635	+	0.886549i	$0.653096\pi$
$692$	1.84081e7	1.46132
$693$	−8.00576e6	−0.633242
$694$	3.21804e7	2.53626
$695$	0	0
$696$	−1.33700e6	−0.104618
$697$	2.13589e6	0.166532
$698$	−1.57072e7	−1.22028
$699$	−1.79923e7	−1.39282
$700$	0	0
$701$	2.42677e7	1.86523	0.932617	−	0.360869i	$-0.117520\pi$
$701$	2.42677e7	1.86523	0.932617	+	0.360869i	$0.117520\pi$
$702$	1.73239e7	1.32679
$703$	566043.	0.0431977
$704$	−2.41576e7	−1.83705
$705$	0	0
$706$	−1.02028e7	−0.770388
$707$	−2.23813e7	−1.68398
$708$	1.43537e7	1.07617
$709$	−4.96806e6	−0.371168	−0.185584	−	0.982628i	$-0.559418\pi$
$709$	−4.96806e6	−0.371168	−0.185584	+	0.982628i	$0.559418\pi$
$710$	0	0
$711$	3.29751e6	0.244631
$712$	−614821.	−0.0454516
$713$	5.83282e6	0.429689
$714$	5.99103e6	0.439801
$715$	0	0
$716$	−3.05403e6	−0.222634
$717$	1.12048e7	0.813969
$718$	2.31508e7	1.67593
$719$	2.62046e7	1.89041	0.945204	−	0.326481i	$-0.105863\pi$
$719$	2.62046e7	1.89041	0.945204	+	0.326481i	$0.105863\pi$
$720$	0	0
$721$	−2.79736e7	−2.00406
$722$	1.99553e7	1.42468
$723$	2.08025e7	1.48003
$724$	−116450.	−0.00825642
$725$	0	0
$726$	−4.52797e7	−3.18832
$727$	−1.77981e7	−1.24893	−0.624465	−	0.781053i	$-0.714683\pi$
$727$	−1.77981e7	−1.24893	−0.624465	+	0.781053i	$0.714683\pi$
$728$	1.00887e6	0.0705513
$729$	6.81787e6	0.475149
$730$	0	0
$731$	3.28582e6	0.227431
$732$	1.96548e7	1.35579
$733$	2.30093e7	1.58177	0.790884	−	0.611966i	$-0.209621\pi$
$733$	2.30093e7	1.58177	0.790884	+	0.611966i	$0.209621\pi$
$734$	−2.80353e7	−1.92072
$735$	0	0
$736$	−7.78551e6	−0.529777
$737$	−3.38445e7	−2.29519
$738$	−4.87920e6	−0.329767
$739$	1.89314e7	1.27518	0.637591	−	0.770375i	$-0.279931\pi$
$739$	1.89314e7	1.27518	0.637591	+	0.770375i	$0.279931\pi$
$740$	0	0
$741$	−893663.	−0.0597900
$742$	−4.03741e7	−2.69211
$743$	8.23748e6	0.547422	0.273711	−	0.961812i	$-0.411749\pi$
$743$	8.23748e6	0.547422	0.273711	+	0.961812i	$0.411749\pi$
$744$	1.06890e6	0.0707950
$745$	0	0
$746$	−3.65738e7	−2.40615
$747$	−2.40543e6	−0.157722
$748$	6.59112e6	0.430730
$749$	1.93290e7	1.25894
$750$	0	0
$751$	1.40581e7	0.909552	0.454776	−	0.890606i	$-0.349719\pi$
$751$	1.40581e7	0.909552	0.454776	+	0.890606i	$0.349719\pi$
$752$	−5.96185e6	−0.384447
$753$	−1.61173e7	−1.03587
$754$	−4.61295e7	−2.95495
$755$	0	0
$756$	1.37388e7	0.874266
$757$	9.05387e6	0.574241	0.287121	−	0.957894i	$-0.407302\pi$
$757$	9.05387e6	0.574241	0.287121	+	0.957894i	$0.407302\pi$
$758$	−2.01061e6	−0.127103
$759$	−1.16782e7	−0.735818
$760$	0	0
$761$	−7.36583e6	−0.461063	−0.230531	−	0.973065i	$-0.574046\pi$
$761$	−7.36583e6	−0.461063	−0.230531	+	0.973065i	$0.574046\pi$
$762$	−1.29569e7	−0.808375
$763$	2.69679e7	1.67701
$764$	1.18004e6	0.0731413
$765$	0	0
$766$	1.95466e7	1.20365
$767$	1.77226e7	1.08777
$768$	1.74170e7	1.06554
$769$	−1.42854e7	−0.871116	−0.435558	−	0.900161i	$-0.643449\pi$
$769$	−1.42854e7	−0.871116	−0.435558	+	0.900161i	$0.643449\pi$
$770$	0	0
$771$	−5.07960e6	−0.307747
$772$	−2.89915e6	−0.175076
$773$	1.87743e7	1.13009	0.565047	−	0.825058i	$-0.308858\pi$
$773$	1.87743e7	1.13009	0.565047	+	0.825058i	$0.308858\pi$
$774$	−7.50607e6	−0.450361
$775$	0	0
$776$	−946156.	−0.0564038
$777$	2.16426e7	1.28604
$778$	3.90192e7	2.31115
$779$	−496298.	−0.0293021
$780$	0	0
$781$	3.90009e7	2.28795
$782$	2.20031e6	0.128667
$783$	−2.24806e7	−1.31040
$784$	−3.43787e6	−0.199756
$785$	0	0
$786$	−2.12643e7	−1.22770
$787$	−1.21169e7	−0.697355	−0.348678	−	0.937243i	$-0.613369\pi$
$787$	−1.21169e7	−0.697355	−0.348678	+	0.937243i	$0.613369\pi$
$788$	7.97262e6	0.457389
$789$	−1.81160e6	−0.103602
$790$	0	0
$791$	1.46514e7	0.832605
$792$	−538818.	−0.0305231
$793$	2.42679e7	1.37040
$794$	−3.51165e7	−1.97678
$795$	0	0
$796$	1.44858e7	0.810327
$797$	1.46452e6	0.0816676	0.0408338	−	0.999166i	$-0.486999\pi$
$797$	1.46452e6	0.0816676	0.0408338	+	0.999166i	$0.486999\pi$
$798$	−1.39208e6	−0.0773851
$799$	1.74995e6	0.0969749
$800$	0	0
$801$	5.24275e6	0.288721
$802$	−4.20182e7	−2.30676
$803$	−1.92465e7	−1.05332
$804$	−2.94555e7	−1.60704
$805$	0	0
$806$	3.68794e7	1.99961
$807$	2.29528e7	1.24065
$808$	−1.50635e6	−0.0811701
$809$	3.18740e7	1.71224	0.856121	−	0.516775i	$-0.172868\pi$
$809$	3.18740e7	1.71224	0.856121	+	0.516775i	$0.172868\pi$
$810$	0	0
$811$	4.64580e6	0.248032	0.124016	−	0.992280i	$-0.460423\pi$
$811$	4.64580e6	0.248032	0.124016	+	0.992280i	$0.460423\pi$
$812$	−3.65832e7	−1.94711
$813$	3.36913e7	1.78769
$814$	4.67687e7	2.47397
$815$	0	0
$816$	−5.12788e6	−0.269595
$817$	−763497.	−0.0400177
$818$	−2.97601e7	−1.55508
$819$	−8.60288e6	−0.448161
$820$	0	0
$821$	−1.96226e7	−1.01601	−0.508006	−	0.861353i	$-0.669617\pi$
$821$	−1.96226e7	−1.01601	−0.508006	+	0.861353i	$0.669617\pi$
$822$	−2.34002e7	−1.20792
$823$	1.06690e7	0.549067	0.274534	−	0.961577i	$-0.411477\pi$
$823$	1.06690e7	0.549067	0.274534	+	0.961577i	$0.411477\pi$
$824$	−1.88273e6	−0.0965984
$825$	0	0
$826$	2.76069e7	1.40789
$827$	−1.48270e7	−0.753856	−0.376928	−	0.926243i	$-0.623020\pi$
$827$	−1.48270e7	−0.753856	−0.376928	+	0.926243i	$0.623020\pi$
$828$	−2.55896e6	−0.129714
$829$	3.33182e6	0.168382	0.0841908	−	0.996450i	$-0.473169\pi$
$829$	3.33182e6	0.168382	0.0841908	+	0.996450i	$0.473169\pi$
$830$	0	0
$831$	−2.90201e7	−1.45779
$832$	−2.59594e7	−1.30013
$833$	1.00910e6	0.0503874
$834$	−2.85767e6	−0.142264
$835$	0	0
$836$	−1.53152e6	−0.0757891
$837$	1.79727e7	0.886745
$838$	1.49234e6	0.0734107
$839$	−2.10707e7	−1.03341	−0.516707	−	0.856162i	$-0.672842\pi$
$839$	−2.10707e7	−1.03341	−0.516707	+	0.856162i	$0.672842\pi$
$840$	0	0
$841$	3.93494e7	1.91844
$842$	2.80819e7	1.36504
$843$	−3.54669e7	−1.71892
$844$	−4.32136e6	−0.208816
$845$	0	0
$846$	−3.99756e6	−0.192030
$847$	−4.43372e7	−2.12354
$848$	3.45572e7	1.65025
$849$	−1.06561e7	−0.507373
$850$	0	0
$851$	7.94861e6	0.376242
$852$	3.39432e7	1.60197
$853$	−3.97315e6	−0.186966	−0.0934830	−	0.995621i	$-0.529800\pi$
$853$	−3.97315e6	−0.186966	−0.0934830	+	0.995621i	$0.529800\pi$
$854$	3.78027e7	1.77369
$855$	0	0
$856$	1.30091e6	0.0606825
$857$	2.04907e7	0.953026	0.476513	−	0.879167i	$-0.341901\pi$
$857$	2.04907e7	0.953026	0.476513	+	0.879167i	$0.341901\pi$
$858$	−7.38380e7	−3.42422
$859$	−1.18282e6	−0.0546934	−0.0273467	−	0.999626i	$-0.508706\pi$
$859$	−1.18282e6	−0.0546934	−0.0273467	+	0.999626i	$0.508706\pi$
$860$	0	0
$861$	−1.89759e7	−0.872357
$862$	1.63071e7	0.747493
$863$	1.71400e7	0.783402	0.391701	−	0.920093i	$-0.371887\pi$
$863$	1.71400e7	0.783402	0.391701	+	0.920093i	$0.371887\pi$
$864$	−2.39895e7	−1.09329
$865$	0	0
$866$	1.23640e6	0.0560228
$867$	1.50516e6	0.0680040
$868$	2.92473e7	1.31761
$869$	2.77132e7	1.24491
$870$	0	0
$871$	−3.63688e7	−1.62437
$872$	1.81504e6	0.0808343
$873$	8.06814e6	0.358293
$874$	−511267.	−0.0226396
$875$	0	0
$876$	−1.67506e7	−0.737512
$877$	−2.63710e7	−1.15778	−0.578891	−	0.815405i	$-0.696514\pi$
$877$	−2.63710e7	−1.15778	−0.578891	+	0.815405i	$0.696514\pi$
$878$	−3.31582e7	−1.45163
$879$	1.72458e7	0.752853
$880$	0	0
$881$	3.70376e7	1.60769	0.803846	−	0.594838i	$-0.202784\pi$
$881$	3.70376e7	1.60769	0.803846	+	0.594838i	$0.202784\pi$
$882$	−2.30517e6	−0.0997773
$883$	−6.32717e6	−0.273091	−0.136546	−	0.990634i	$-0.543600\pi$
$883$	−6.32717e6	−0.273091	−0.136546	+	0.990634i	$0.543600\pi$
$884$	7.08272e6	0.304838
$885$	0	0
$886$	−1.35040e7	−0.577934
$887$	2.10310e7	0.897535	0.448768	−	0.893648i	$-0.351863\pi$
$887$	2.10310e7	0.897535	0.448768	+	0.893648i	$0.351863\pi$
$888$	1.45663e6	0.0619891
$889$	−1.26872e7	−0.538407
$890$	0	0
$891$	−4.96384e7	−2.09471
$892$	4.23901e7	1.78383
$893$	−406621.	−0.0170632
$894$	−2.29344e7	−0.959716
$895$	0	0
$896$	−2.79601e6	−0.116350
$897$	−1.25492e7	−0.520757
$898$	−4.74738e7	−1.96455
$899$	−4.78570e7	−1.97491
$900$	0	0
$901$	−1.01434e7	−0.416267
$902$	−4.10061e7	−1.67816
$903$	−2.91922e7	−1.19137
$904$	986097.	0.0401327
$905$	0	0
$906$	−1.04253e7	−0.421959
$907$	7.25520e6	0.292840	0.146420	−	0.989222i	$-0.453225\pi$
$907$	7.25520e6	0.292840	0.146420	+	0.989222i	$0.453225\pi$
$908$	3.66230e7	1.47414
$909$	1.28450e7	0.515615
$910$	0	0
$911$	2.53484e7	1.01194	0.505969	−	0.862552i	$-0.331135\pi$
$911$	2.53484e7	1.01194	0.505969	+	0.862552i	$0.331135\pi$
$912$	1.19152e6	0.0474366
$913$	−2.02159e7	−0.802632
$914$	3.18782e7	1.26220
$915$	0	0
$916$	−1.41565e7	−0.557465
$917$	−2.08216e7	−0.817695
$918$	6.77982e6	0.265529
$919$	−2.44762e7	−0.955994	−0.477997	−	0.878361i	$-0.658637\pi$
$919$	−2.44762e7	−0.955994	−0.477997	+	0.878361i	$0.658637\pi$
$920$	0	0
$921$	6.32384e6	0.245658
$922$	−4.60713e7	−1.78486
$923$	4.19098e7	1.61924
$924$	−5.85574e7	−2.25633
$925$	0	0
$926$	2.27638e7	0.872404
$927$	1.60546e7	0.613620
$928$	6.38785e7	2.43492
$929$	−4.39257e7	−1.66986	−0.834928	−	0.550359i	$-0.814491\pi$
$929$	−4.39257e7	−1.66986	−0.834928	+	0.550359i	$0.814491\pi$
$930$	0	0
$931$	−234476.	−0.00886592
$932$	−3.31342e7	−1.24950
$933$	3.61155e7	1.35828
$934$	−2.24970e7	−0.843834
$935$	0	0
$936$	−579006.	−0.0216020
$937$	4.94066e7	1.83838	0.919191	−	0.393811i	$-0.128844\pi$
$937$	4.94066e7	1.83838	0.919191	+	0.393811i	$0.128844\pi$
$938$	−5.66527e7	−2.10239
$939$	−1.75052e7	−0.647894
$940$	0	0
$941$	−3.99442e7	−1.47055	−0.735275	−	0.677769i	$-0.762947\pi$
$941$	−3.99442e7	−1.47055	−0.735275	+	0.677769i	$0.762947\pi$
$942$	−1.94620e7	−0.714596
$943$	−6.96923e6	−0.255215
$944$	−2.36295e7	−0.863027
$945$	0	0
$946$	−6.30831e7	−2.29185
$947$	−5.16182e7	−1.87037	−0.935186	−	0.354158i	$-0.884768\pi$
$947$	−5.16182e7	−1.87037	−0.935186	+	0.354158i	$0.884768\pi$
$948$	2.41193e7	0.871653
$949$	−2.06820e7	−0.745464
$950$	0	0
$951$	−4.33060e7	−1.55273
$952$	394826.	0.0141193
$953$	−6.34618e6	−0.226350	−0.113175	−	0.993575i	$-0.536102\pi$
$953$	−6.34618e6	−0.226350	−0.113175	+	0.993575i	$0.536102\pi$
$954$	2.31714e7	0.824293
$955$	0	0
$956$	2.06346e7	0.730215
$957$	9.58168e7	3.38191
$958$	5.08785e7	1.79110
$959$	−2.29131e7	−0.804521
$960$	0	0
$961$	9.63131e6	0.336416
$962$	5.02570e7	1.75089
$963$	−1.10932e7	−0.385472
$964$	3.83094e7	1.32774
$965$	0	0
$966$	−1.95482e7	−0.674006
$967$	5.14875e6	0.177066	0.0885330	−	0.996073i	$-0.471782\pi$
$967$	5.14875e6	0.177066	0.0885330	+	0.996073i	$0.471782\pi$
$968$	−2.98406e6	−0.102357
$969$	−349741.	−0.0119657
$970$	0	0
$971$	6.52365e6	0.222046	0.111023	−	0.993818i	$-0.464587\pi$
$971$	6.52365e6	0.222046	0.111023	+	0.993818i	$0.464587\pi$
$972$	−1.97687e7	−0.671139
$973$	−2.79818e6	−0.0947532
$974$	5.97007e7	2.01643
$975$	0	0
$976$	−3.23563e7	−1.08726
$977$	1.04754e7	0.351104	0.175552	−	0.984470i	$-0.443829\pi$
$977$	1.04754e7	0.351104	0.175552	+	0.984470i	$0.443829\pi$
$978$	−5.19366e7	−1.73631
$979$	4.40616e7	1.46928
$980$	0	0
$981$	−1.54774e7	−0.513482
$982$	−3.39236e7	−1.12260
$983$	6.70566e6	0.221339	0.110669	−	0.993857i	$-0.464701\pi$
$983$	6.70566e6	0.221339	0.110669	+	0.993857i	$0.464701\pi$
$984$	−1.27715e6	−0.0420488
$985$	0	0
$986$	−1.80531e7	−0.591369
$987$	−1.55471e7	−0.507991
$988$	−1.64575e6	−0.0536378
$989$	−1.07213e7	−0.348545
$990$	0	0
$991$	1.62174e7	0.524563	0.262281	−	0.964991i	$-0.415525\pi$
$991$	1.62174e7	0.524563	0.262281	+	0.964991i	$0.415525\pi$
$992$	−5.10692e7	−1.64771
$993$	−4.25291e7	−1.36871
$994$	6.52839e7	2.09575
$995$	0	0
$996$	−1.75943e7	−0.561984
$997$	4.85435e6	0.154665	0.0773327	−	0.997005i	$-0.475360\pi$
$997$	4.85435e6	0.154665	0.0773327	+	0.997005i	$0.475360\pi$
$998$	5.43917e7	1.72865
$999$	2.44921e7	0.776447

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	425.6.a.k.1.3	yes	15
5.4	even	2		425.6.a.j.1.13	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
425.6.a.j.1.13	✓	15	5.4	even	2
425.6.a.k.1.3	yes	15	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 425 = 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	425.a (trivial)

\(f(q)\)	\(=\)	\(q-8.07389 q^{2} +18.0213 q^{3} +33.1877 q^{4} -145.502 q^{6} -142.473 q^{7} -9.58900 q^{8} +81.7682 q^{9} +687.202 q^{11} +598.086 q^{12} +738.458 q^{13} +1150.31 q^{14} -984.584 q^{16} +289.000 q^{17} -660.187 q^{18} -67.1524 q^{19} -2567.56 q^{21} -5548.39 q^{22} -942.982 q^{23} -172.807 q^{24} -5962.23 q^{26} -2905.61 q^{27} -4728.36 q^{28} +7736.96 q^{29} -6185.50 q^{31} +8256.27 q^{32} +12384.3 q^{33} -2333.35 q^{34} +2713.69 q^{36} -8429.23 q^{37} +542.181 q^{38} +13308.0 q^{39} +7390.63 q^{41} +20730.2 q^{42} +11369.6 q^{43} +22806.6 q^{44} +7613.53 q^{46} +6055.20 q^{47} -17743.5 q^{48} +3491.70 q^{49} +5208.16 q^{51} +24507.7 q^{52} -35098.3 q^{53} +23459.6 q^{54} +1366.18 q^{56} -1210.17 q^{57} -62467.4 q^{58} +23999.4 q^{59} +32862.9 q^{61} +49941.1 q^{62} -11649.8 q^{63} -35153.5 q^{64} -99989.4 q^{66} -49249.7 q^{67} +9591.23 q^{68} -16993.8 q^{69} +56753.1 q^{71} -784.075 q^{72} -28007.0 q^{73} +68056.6 q^{74} -2228.63 q^{76} -97908.1 q^{77} -107447. q^{78} +40327.5 q^{79} -72232.6 q^{81} -59671.1 q^{82} -29417.7 q^{83} -85211.3 q^{84} -91797.0 q^{86} +139430. q^{87} -6589.58 q^{88} +64117.3 q^{89} -105211. q^{91} -31295.4 q^{92} -111471. q^{93} -48889.0 q^{94} +148789. q^{96} +98670.9 q^{97} -28191.6 q^{98} +56191.3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + q^{2} + 9 q^{3} + 277 q^{4} + 169 q^{6} + 181 q^{7} + 753 q^{8} + 1826 q^{9} + 172 q^{11} - 2109 q^{12} - 389 q^{13} + 3635 q^{14} + 6837 q^{16} + 4335 q^{17} + 6742 q^{18} + 5150 q^{19} - 6891 q^{21}+ \cdots - 183214 q^{99}+O(q^{100}) \) 15 * q + q^2 + 9 * q^3 + 277 * q^4 + 169 * q^6 + 181 * q^7 + 753 * q^8 + 1826 * q^9 + 172 * q^11 - 2109 * q^12 - 389 * q^13 + 3635 * q^14 + 6837 * q^16 + 4335 * q^17 + 6742 * q^18 + 5150 * q^19 - 6891 * q^21 - 10522 * q^22 + 3140 * q^23 + 9263 * q^24 + 3181 * q^26 + 1611 * q^27 + 16371 * q^28 + 15214 * q^29 + 18873 * q^31 + 16857 * q^32 - 14700 * q^33 + 289 * q^34 + 83010 * q^36 + 12122 * q^37 - 5302 * q^38 + 17995 * q^39 - 2336 * q^41 - 65201 * q^42 - 660 * q^43 + 40740 * q^44 - 18610 * q^46 + 11494 * q^47 - 48879 * q^48 + 89846 * q^49 + 2601 * q^51 - 71027 * q^52 - 76605 * q^53 - 33675 * q^54 + 71385 * q^56 + 112044 * q^57 - 11270 * q^58 + 7268 * q^59 + 137458 * q^61 + 212145 * q^62 - 25488 * q^63 + 116341 * q^64 + 87900 * q^66 - 5036 * q^67 + 80053 * q^68 - 57176 * q^69 + 180961 * q^71 + 588818 * q^72 - 176062 * q^73 + 11010 * q^74 + 390158 * q^76 + 319456 * q^77 - 339079 * q^78 + 127813 * q^79 + 419847 * q^81 + 200996 * q^82 - 318176 * q^83 - 373745 * q^84 + 214564 * q^86 + 138204 * q^87 - 722070 * q^88 + 97682 * q^89 + 229269 * q^91 + 202080 * q^92 - 514385 * q^93 + 104594 * q^94 + 706465 * q^96 + 381812 * q^97 - 399302 * q^98 - 183214 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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