LMFDB - Embedded newform 425.6.a.k.1.12

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			1.12
Root			$8.32336$ 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)

$f(q)$	$=$	\(q+8.32336 q^{2} -15.0453 q^{3} +37.2783 q^{4} -125.228 q^{6} -79.5600 q^{7} +43.9329 q^{8} -16.6383 q^{9} -577.854 q^{11} -560.864 q^{12} +319.284 q^{13} -662.206 q^{14} -827.235 q^{16} +289.000 q^{17} -138.487 q^{18} +1760.25 q^{19} +1197.01 q^{21} -4809.69 q^{22} +3755.20 q^{23} -660.985 q^{24} +2657.52 q^{26} +3906.34 q^{27} -2965.86 q^{28} +398.563 q^{29} +5762.11 q^{31} -8291.23 q^{32} +8694.01 q^{33} +2405.45 q^{34} -620.248 q^{36} -7138.85 q^{37} +14651.2 q^{38} -4803.73 q^{39} +14410.3 q^{41} +9963.10 q^{42} -18493.9 q^{43} -21541.4 q^{44} +31255.8 q^{46} +1609.02 q^{47} +12446.0 q^{48} -10477.2 q^{49} -4348.10 q^{51} +11902.4 q^{52} +5575.72 q^{53} +32513.9 q^{54} -3495.30 q^{56} -26483.5 q^{57} +3317.38 q^{58} +18038.2 q^{59} +34905.8 q^{61} +47960.1 q^{62} +1323.74 q^{63} -42539.3 q^{64} +72363.3 q^{66} +8036.23 q^{67} +10773.4 q^{68} -56498.1 q^{69} +45301.1 q^{71} -730.970 q^{72} +57817.7 q^{73} -59419.2 q^{74} +65619.1 q^{76} +45974.1 q^{77} -39983.2 q^{78} -41409.2 q^{79} -54729.1 q^{81} +119942. q^{82} +34525.5 q^{83} +44622.3 q^{84} -153931. q^{86} -5996.50 q^{87} -25386.8 q^{88} +39956.1 q^{89} -25402.2 q^{91} +139987. q^{92} -86692.7 q^{93} +13392.5 q^{94} +124744. q^{96} -24119.7 q^{97} -87205.6 q^{98} +9614.52 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	8.32336	1.47138	0.735688	−	0.677321i	$-0.236859\pi$
$2$	8.32336	1.47138	0.735688	+	0.677321i	$0.236859\pi$
$3$	−15.0453	−0.965158	−0.482579	−	0.875853i	$-0.660300\pi$
$3$	−15.0453	−0.965158	−0.482579	+	0.875853i	$0.660300\pi$
$4$	37.2783	1.16495
$5$	0	0
$5$	0	0
$6$	−125.228	−1.42011
$7$	−79.5600	−0.613691	−0.306845	−	0.951759i	$-0.599273\pi$
$7$	−79.5600	−0.613691	−0.306845	+	0.951759i	$0.599273\pi$
$8$	43.9329	0.242697
$9$	−16.6383	−0.0684704
$10$	0	0
$11$	−577.854	−1.43991	−0.719957	−	0.694018i	$-0.755839\pi$
$11$	−577.854	−1.43991	−0.719957	+	0.694018i	$0.755839\pi$
$12$	−560.864	−1.12436
$13$	319.284	0.523985	0.261993	−	0.965070i	$-0.415620\pi$
$13$	319.284	0.523985	0.261993	+	0.965070i	$0.415620\pi$
$14$	−662.206	−0.902970
$15$	0	0
$16$	−827.235	−0.807847
$17$	289.000	0.242536
$17$	289.000	0.242536
$18$	−138.487	−0.100746
$19$	1760.25	1.11864	0.559320	−	0.828952i	$-0.311062\pi$
$19$	1760.25	1.11864	0.559320	+	0.828952i	$0.311062\pi$
$20$	0	0
$21$	1197.01	0.592308
$22$	−4809.69	−2.11866
$23$	3755.20	1.48017	0.740087	−	0.672511i	$-0.234784\pi$
$23$	3755.20	1.48017	0.740087	+	0.672511i	$0.234784\pi$
$24$	−660.985	−0.234241
$25$	0	0
$26$	2657.52	0.770979
$27$	3906.34	1.03124
$28$	−2965.86	−0.714917
$29$	398.563	0.0880038	0.0440019	−	0.999031i	$-0.485989\pi$
$29$	398.563	0.0880038	0.0440019	+	0.999031i	$0.485989\pi$
$30$	0	0
$31$	5762.11	1.07690	0.538452	−	0.842656i	$-0.319009\pi$
$31$	5762.11	1.07690	0.538452	+	0.842656i	$0.319009\pi$
$32$	−8291.23	−1.43134
$33$	8694.01	1.38974
$34$	2405.45	0.356861
$35$	0	0
$36$	−620.248	−0.0797643
$37$	−7138.85	−0.857282	−0.428641	−	0.903475i	$-0.641007\pi$
$37$	−7138.85	−0.857282	−0.428641	+	0.903475i	$0.641007\pi$
$38$	14651.2	1.64594
$39$	−4803.73	−0.505729
$40$	0	0
$41$	14410.3	1.33880	0.669398	−	0.742904i	$-0.266552\pi$
$41$	14410.3	1.33880	0.669398	+	0.742904i	$0.266552\pi$
$42$	9963.10	0.871508
$43$	−18493.9	−1.52531	−0.762654	−	0.646807i	$-0.776104\pi$
$43$	−18493.9	−1.52531	−0.762654	+	0.646807i	$0.776104\pi$
$44$	−21541.4	−1.67742
$45$	0	0
$46$	31255.8	2.17789
$47$	1609.02	0.106247	0.0531236	−	0.998588i	$-0.483082\pi$
$47$	1609.02	0.106247	0.0531236	+	0.998588i	$0.483082\pi$
$48$	12446.0	0.779700
$49$	−10477.2	−0.623384
$50$	0	0
$51$	−4348.10	−0.234085
$52$	11902.4	0.610415
$53$	5575.72	0.272654	0.136327	−	0.990664i	$-0.456470\pi$
$53$	5575.72	0.272654	0.136327	+	0.990664i	$0.456470\pi$
$54$	32513.9	1.51735
$55$	0	0
$56$	−3495.30	−0.148941
$57$	−26483.5	−1.07966
$58$	3317.38	0.129487
$59$	18038.2	0.674626	0.337313	−	0.941392i	$-0.390482\pi$
$59$	18038.2	0.674626	0.337313	+	0.941392i	$0.390482\pi$
$60$	0	0
$61$	34905.8	1.20108	0.600542	−	0.799593i	$-0.294952\pi$
$61$	34905.8	1.20108	0.600542	+	0.799593i	$0.294952\pi$
$62$	47960.1	1.58453
$63$	1323.74	0.0420197
$64$	−42539.3	−1.29820
$65$	0	0
$66$	72363.3	2.04484
$67$	8036.23	0.218708	0.109354	−	0.994003i	$-0.465122\pi$
$67$	8036.23	0.218708	0.109354	+	0.994003i	$0.465122\pi$
$68$	10773.4	0.282541
$69$	−56498.1	−1.42860
$70$	0	0
$71$	45301.1	1.06651	0.533253	−	0.845956i	$-0.320969\pi$
$71$	45301.1	1.06651	0.533253	+	0.845956i	$0.320969\pi$
$72$	−730.970	−0.0166176
$73$	57817.7	1.26985	0.634926	−	0.772573i	$-0.281030\pi$
$73$	57817.7	1.26985	0.634926	+	0.772573i	$0.281030\pi$
$74$	−59419.2	−1.26138
$75$	0	0
$76$	65619.1	1.30316
$77$	45974.1	0.883662
$78$	−39983.2	−0.744117
$79$	−41409.2	−0.746498	−0.373249	−	0.927731i	$-0.621756\pi$
$79$	−41409.2	−0.746498	−0.373249	+	0.927731i	$0.621756\pi$
$80$	0	0
$81$	−54729.1	−0.926841
$82$	119942.	1.96987
$83$	34525.5	0.550104	0.275052	−	0.961429i	$-0.411305\pi$
$83$	34525.5	0.550104	0.275052	+	0.961429i	$0.411305\pi$
$84$	44622.3	0.690007
$85$	0	0
$86$	−153931.	−2.24430
$87$	−5996.50	−0.0849376
$88$	−25386.8	−0.349464
$89$	39956.1	0.534697	0.267349	−	0.963600i	$-0.413852\pi$
$89$	39956.1	0.534697	0.267349	+	0.963600i	$0.413852\pi$
$90$	0	0
$91$	−25402.2	−0.321565
$92$	139987.	1.72432
$93$	−86692.7	−1.03938
$94$	13392.5	0.156330
$95$	0	0
$96$	124744.	1.38147
$97$	−24119.7	−0.260281	−0.130140	−	0.991496i	$-0.541543\pi$
$97$	−24119.7	−0.260281	−0.130140	+	0.991496i	$0.541543\pi$
$98$	−87205.6	−0.917231
$99$	9614.52	0.0985916
$100$	0	0
$101$	−50783.4	−0.495357	−0.247679	−	0.968842i	$-0.579668\pi$
$101$	−50783.4	−0.495357	−0.247679	+	0.968842i	$0.579668\pi$
$102$	−36190.8	−0.344427
$103$	16310.0	0.151482	0.0757408	−	0.997128i	$-0.475868\pi$
$103$	16310.0	0.151482	0.0757408	+	0.997128i	$0.475868\pi$
$104$	14027.1	0.127170
$105$	0	0
$106$	46408.7	0.401176
$107$	−119473.	−1.00881	−0.504406	−	0.863467i	$-0.668289\pi$
$107$	−119473.	−1.00881	−0.504406	+	0.863467i	$0.668289\pi$
$108$	145622.	1.20134
$109$	201437.	1.62395	0.811974	−	0.583694i	$-0.198393\pi$
$109$	201437.	1.62395	0.811974	+	0.583694i	$0.198393\pi$
$110$	0	0
$111$	107406.	0.827412
$112$	65814.8	0.495768
$113$	141994.	1.04610	0.523049	−	0.852303i	$-0.324794\pi$
$113$	141994.	1.04610	0.523049	+	0.852303i	$0.324794\pi$
$114$	−220432.	−1.58859
$115$	0	0
$116$	14857.7	0.102520
$117$	−5312.35	−0.0358775
$118$	150138.	0.992629
$119$	−22992.8	−0.148842
$120$	0	0
$121$	172865.	1.07335
$122$	290534.	1.76724
$123$	−216808.	−1.29215
$124$	214801.	1.25454
$125$	0	0
$126$	11018.0	0.0618267
$127$	−63707.3	−0.350493	−0.175247	−	0.984525i	$-0.556072\pi$
$127$	−63707.3	−0.350493	−0.175247	+	0.984525i	$0.556072\pi$
$128$	−88750.7	−0.478792
$129$	278247.	1.47216
$130$	0	0
$131$	82424.9	0.419643	0.209822	−	0.977740i	$-0.432712\pi$
$131$	82424.9	0.419643	0.209822	+	0.977740i	$0.432712\pi$
$132$	324098.	1.61898
$133$	−140045.	−0.686499
$134$	66888.4	0.321802
$135$	0	0
$136$	12696.6	0.0588628
$137$	−380341.	−1.73130	−0.865649	−	0.500652i	$-0.833094\pi$
$137$	−380341.	−1.73130	−0.865649	+	0.500652i	$0.833094\pi$
$138$	−470254.	−2.10201
$139$	294407.	1.29244	0.646221	−	0.763150i	$-0.276348\pi$
$139$	294407.	1.29244	0.646221	+	0.763150i	$0.276348\pi$
$140$	0	0
$141$	−24208.3	−0.102545
$142$	377058.	1.56923
$143$	−184500.	−0.754494
$144$	13763.8	0.0553136
$145$	0	0
$146$	481237.	1.86843
$147$	157633.	0.601664
$148$	−266124.	−0.998687
$149$	218027.	0.804534	0.402267	−	0.915522i	$-0.368222\pi$
$149$	218027.	0.804534	0.402267	+	0.915522i	$0.368222\pi$
$150$	0	0
$151$	453378.	1.61815	0.809074	−	0.587707i	$-0.199969\pi$
$151$	453378.	1.61815	0.809074	+	0.587707i	$0.199969\pi$
$152$	77332.9	0.271491
$153$	−4808.47	−0.0166065
$154$	382659.	1.30020
$155$	0	0
$156$	−179075.	−0.589146
$157$	−269922.	−0.873956	−0.436978	−	0.899472i	$-0.643951\pi$
$157$	−269922.	−0.873956	−0.436978	+	0.899472i	$0.643951\pi$
$158$	−344663.	−1.09838
$159$	−83888.5	−0.263154
$160$	0	0
$161$	−298763.	−0.908369
$162$	−455529.	−1.36373
$163$	−278526.	−0.821100	−0.410550	−	0.911838i	$-0.634663\pi$
$163$	−278526.	−0.821100	−0.410550	+	0.911838i	$0.634663\pi$
$164$	537192.	1.55962
$165$	0	0
$166$	287368.	0.809410
$167$	407732.	1.13131	0.565657	−	0.824640i	$-0.308623\pi$
$167$	407732.	1.13131	0.565657	+	0.824640i	$0.308623\pi$
$168$	52588.0	0.143752
$169$	−269351.	−0.725439
$170$	0	0
$171$	−29287.6	−0.0765938
$172$	−689421.	−1.77690
$173$	−379209.	−0.963305	−0.481652	−	0.876362i	$-0.659963\pi$
$173$	−379209.	−0.963305	−0.481652	+	0.876362i	$0.659963\pi$
$174$	−49911.0	−0.124975
$175$	0	0
$176$	478022.	1.16323
$177$	−271391.	−0.651121
$178$	332569.	0.786741
$179$	−101490.	−0.236751	−0.118376	−	0.992969i	$-0.537769\pi$
$179$	−101490.	−0.236751	−0.118376	+	0.992969i	$0.537769\pi$
$180$	0	0
$181$	−657822.	−1.49249	−0.746246	−	0.665670i	$-0.768146\pi$
$181$	−657822.	−1.49249	−0.746246	+	0.665670i	$0.768146\pi$
$182$	−211432.	−0.473143
$183$	−525169.	−1.15924
$184$	164977.	0.359235
$185$	0	0
$186$	−721575.	−1.52932
$187$	−167000.	−0.349231
$188$	59981.6	0.123772
$189$	−310788.	−0.632864
$190$	0	0
$191$	−500994.	−0.993685	−0.496843	−	0.867841i	$-0.665507\pi$
$191$	−500994.	−0.993685	−0.496843	+	0.867841i	$0.665507\pi$
$192$	640018.	1.25296
$193$	−518302.	−1.00159	−0.500795	−	0.865566i	$-0.666959\pi$
$193$	−518302.	−1.00159	−0.500795	+	0.865566i	$0.666959\pi$
$194$	−200756.	−0.382970
$195$	0	0
$196$	−390572.	−0.726208
$197$	−516158.	−0.947583	−0.473792	−	0.880637i	$-0.657115\pi$
$197$	−516158.	−0.947583	−0.473792	+	0.880637i	$0.657115\pi$
$198$	80025.1	0.145065
$199$	128960.	0.230847	0.115423	−	0.993316i	$-0.463178\pi$
$199$	128960.	0.230847	0.115423	+	0.993316i	$0.463178\pi$
$200$	0	0
$201$	−120908.	−0.211088
$202$	−422689.	−0.728857
$203$	−31709.6	−0.0540071
$204$	−162090.	−0.272697
$205$	0	0
$206$	135754.	0.222886
$207$	−62480.1	−0.101348
$208$	−264123.	−0.423300
$209$	−1.01717e6	−1.61075
$210$	0	0
$211$	1.14181e6	1.76558	0.882788	−	0.469771i	$-0.155664\pi$
$211$	1.14181e6	1.76558	0.882788	+	0.469771i	$0.155664\pi$
$212$	207853.	0.317627
$213$	−681570.	−1.02935
$214$	−994416.	−1.48434
$215$	0	0
$216$	171617.	0.250280
$217$	−458433.	−0.660886
$218$	1.67663e6	2.38944
$219$	−869885.	−1.22561
$220$	0	0
$221$	92273.1	0.127085
$222$	893980.	1.21743
$223$	700467.	0.943247	0.471623	−	0.881800i	$-0.343668\pi$
$223$	700467.	0.943247	0.471623	+	0.881800i	$0.343668\pi$
$224$	659650.	0.878402
$225$	0	0
$226$	1.18186e6	1.53920
$227$	664426.	0.855820	0.427910	−	0.903821i	$-0.359250\pi$
$227$	664426.	0.855820	0.427910	+	0.903821i	$0.359250\pi$
$228$	−987260.	−1.25775
$229$	−546361.	−0.688480	−0.344240	−	0.938882i	$-0.611863\pi$
$229$	−546361.	−0.688480	−0.344240	+	0.938882i	$0.611863\pi$
$230$	0	0
$231$	−691695.	−0.852874
$232$	17510.0	0.0213583
$233$	1.21671e6	1.46824	0.734120	−	0.679020i	$-0.237595\pi$
$233$	1.21671e6	1.46824	0.734120	+	0.679020i	$0.237595\pi$
$234$	−44216.6	−0.0527893
$235$	0	0
$236$	672433.	0.785903
$237$	623014.	0.720488
$238$	−191378.	−0.219002
$239$	994192.	1.12584	0.562918	−	0.826513i	$-0.309679\pi$
$239$	994192.	1.12584	0.562918	+	0.826513i	$0.309679\pi$
$240$	0	0
$241$	705904.	0.782894	0.391447	−	0.920201i	$-0.371975\pi$
$241$	705904.	0.782894	0.391447	+	0.920201i	$0.371975\pi$
$242$	1.43882e6	1.57931
$243$	−125825.	−0.136694
$244$	1.30123e6	1.39920
$245$	0	0
$246$	−1.80457e6	−1.90124
$247$	562020.	0.586151
$248$	253146.	0.261362
$249$	−519448.	−0.530937
$250$	0	0
$251$	−819519.	−0.821060	−0.410530	−	0.911847i	$-0.634656\pi$
$251$	−819519.	−0.821060	−0.410530	+	0.911847i	$0.634656\pi$
$252$	49346.9	0.0489506
$253$	−2.16996e6	−2.13133
$254$	−530258.	−0.515707
$255$	0	0
$256$	622555.	0.593715
$257$	−1.62712e6	−1.53669	−0.768346	−	0.640035i	$-0.778920\pi$
$257$	−1.62712e6	−1.53669	−0.768346	+	0.640035i	$0.778920\pi$
$258$	2.31595e6	2.16610
$259$	567966.	0.526106
$260$	0	0
$261$	−6631.41	−0.00602566
$262$	686052.	0.617453
$263$	1.69628e6	1.51220	0.756099	−	0.654457i	$-0.227103\pi$
$263$	1.69628e6	1.51220	0.756099	+	0.654457i	$0.227103\pi$
$264$	381953.	0.337288
$265$	0	0
$266$	−1.16565e6	−1.01010
$267$	−601152.	−0.516067
$268$	299577.	0.254783
$269$	383907.	0.323478	0.161739	−	0.986834i	$-0.448290\pi$
$269$	383907.	0.323478	0.161739	+	0.986834i	$0.448290\pi$
$270$	0	0
$271$	2.24637e6	1.85805	0.929027	−	0.370012i	$-0.120646\pi$
$271$	2.24637e6	1.85805	0.929027	+	0.370012i	$0.120646\pi$
$272$	−239071.	−0.195932
$273$	382185.	0.310361
$274$	−3.16571e6	−2.54739
$275$	0	0
$276$	−2.10615e6	−1.66424
$277$	−508909.	−0.398511	−0.199256	−	0.979948i	$-0.563852\pi$
$277$	−508909.	−0.398511	−0.199256	+	0.979948i	$0.563852\pi$
$278$	2.45045e6	1.90167
$279$	−95871.7	−0.0737361
$280$	0	0
$281$	1.23385e6	0.932170	0.466085	−	0.884740i	$-0.345664\pi$
$281$	1.23385e6	0.932170	0.466085	+	0.884740i	$0.345664\pi$
$282$	−201494.	−0.150883
$283$	12391.5	0.00919728	0.00459864	−	0.999989i	$-0.498536\pi$
$283$	12391.5	0.00919728	0.00459864	+	0.999989i	$0.498536\pi$
$284$	1.68875e6	1.24242
$285$	0	0
$286$	−1.53566e6	−1.11014
$287$	−1.14649e6	−0.821606
$288$	137952.	0.0980047
$289$	83521.0	0.0588235
$290$	0	0
$291$	362888.	0.251212
$292$	2.15534e6	1.47931
$293$	241619.	0.164423	0.0822113	−	0.996615i	$-0.473802\pi$
$293$	241619.	0.164423	0.0822113	+	0.996615i	$0.473802\pi$
$294$	1.31204e6	0.885273
$295$	0	0
$296$	−313630.	−0.208060
$297$	−2.25730e6	−1.48490
$298$	1.81472e6	1.18377
$299$	1.19897e6	0.775590
$300$	0	0
$301$	1.47137e6	0.936067
$302$	3.77363e6	2.38090
$303$	764053.	0.478098
$304$	−1.45614e6	−0.903690
$305$	0	0
$306$	−40022.6	−0.0244344
$307$	−627528.	−0.380003	−0.190002	−	0.981784i	$-0.560849\pi$
$307$	−627528.	−0.380003	−0.190002	+	0.981784i	$0.560849\pi$
$308$	1.71383e6	1.02942
$309$	−245388.	−0.146204
$310$	0	0
$311$	−2.03408e6	−1.19252	−0.596262	−	0.802790i	$-0.703348\pi$
$311$	−2.03408e6	−1.19252	−0.596262	+	0.802790i	$0.703348\pi$
$312$	−211042.	−0.122739
$313$	2.92011e6	1.68476	0.842382	−	0.538881i	$-0.181153\pi$
$313$	2.92011e6	1.68476	0.842382	+	0.538881i	$0.181153\pi$
$314$	−2.24666e6	−1.28592
$315$	0	0
$316$	−1.54366e6	−0.869630
$317$	−700960.	−0.391783	−0.195891	−	0.980626i	$-0.562760\pi$
$317$	−700960.	−0.391783	−0.195891	+	0.980626i	$0.562760\pi$
$318$	−698234.	−0.387198
$319$	−230311.	−0.126718
$320$	0	0
$321$	1.79751e6	0.973663
$322$	−2.48671e6	−1.33655
$323$	508712.	0.271310
$324$	−2.04020e6	−1.07972
$325$	0	0
$326$	−2.31827e6	−1.20815
$327$	−3.03068e6	−1.56737
$328$	633088.	0.324922
$329$	−128014.	−0.0652029
$330$	0	0
$331$	1.74718e6	0.876534	0.438267	−	0.898845i	$-0.355592\pi$
$331$	1.74718e6	0.876534	0.438267	+	0.898845i	$0.355592\pi$
$332$	1.28705e6	0.640842
$333$	118778.	0.0586985
$334$	3.39370e6	1.66459
$335$	0	0
$336$	−990205.	−0.478495
$337$	2.10556e6	1.00993	0.504966	−	0.863139i	$-0.331505\pi$
$337$	2.10556e6	1.00993	0.504966	+	0.863139i	$0.331505\pi$
$338$	−2.24190e6	−1.06739
$339$	−2.13634e6	−1.00965
$340$	0	0
$341$	−3.32966e6	−1.55065
$342$	−243771.	−0.112698
$343$	2.17073e6	0.996256
$344$	−812491.	−0.370188
$345$	0	0
$346$	−3.15629e6	−1.41738
$347$	−4.02369e6	−1.79391	−0.896956	−	0.442119i	$-0.854227\pi$
$347$	−4.02369e6	−1.79391	−0.896956	+	0.442119i	$0.854227\pi$
$348$	−223539.	−0.0989477
$349$	3.47512e6	1.52724	0.763618	−	0.645668i	$-0.223421\pi$
$349$	3.47512e6	1.52724	0.763618	+	0.645668i	$0.223421\pi$
$350$	0	0
$351$	1.24723e6	0.540356
$352$	4.79112e6	2.06101
$353$	1.45843e6	0.622943	0.311472	−	0.950255i	$-0.399178\pi$
$353$	1.45843e6	0.622943	0.311472	+	0.950255i	$0.399178\pi$
$354$	−2.25888e6	−0.958043
$355$	0	0
$356$	1.48949e6	0.622894
$357$	345935.	0.143656
$358$	−844740.	−0.348350
$359$	61017.9	0.0249874	0.0124937	−	0.999922i	$-0.496023\pi$
$359$	61017.9	0.0249874	0.0124937	+	0.999922i	$0.496023\pi$
$360$	0	0
$361$	622381.	0.251355
$362$	−5.47529e6	−2.19602
$363$	−2.60081e6	−1.03596
$364$	−946952.	−0.374606
$365$	0	0
$366$	−4.37117e6	−1.70567
$367$	1.26083e6	0.488644	0.244322	−	0.969694i	$-0.421435\pi$
$367$	1.26083e6	0.488644	0.244322	+	0.969694i	$0.421435\pi$
$368$	−3.10643e6	−1.19575
$369$	−239764.	−0.0916679
$370$	0	0
$371$	−443604.	−0.167325
$372$	−3.23176e6	−1.21082
$373$	1.99577e6	0.742741	0.371371	−	0.928485i	$-0.378888\pi$
$373$	1.99577e6	0.742741	0.371371	+	0.928485i	$0.378888\pi$
$374$	−1.39000e6	−0.513849
$375$	0	0
$376$	70689.1	0.0257859
$377$	127255.	0.0461127
$378$	−2.58680e6	−0.931181
$379$	−2.40574e6	−0.860303	−0.430151	−	0.902757i	$-0.641540\pi$
$379$	−2.40574e6	−0.860303	−0.430151	+	0.902757i	$0.641540\pi$
$380$	0	0
$381$	958496.	0.338281
$382$	−4.16995e6	−1.46208
$383$	−106112.	−0.0369632	−0.0184816	−	0.999829i	$-0.505883\pi$
$383$	−106112.	−0.0369632	−0.0184816	+	0.999829i	$0.505883\pi$
$384$	1.33528e6	0.462110
$385$	0	0
$386$	−4.31401e6	−1.47371
$387$	307707.	0.104438
$388$	−899139.	−0.303213
$389$	−2.18277e6	−0.731363	−0.365682	−	0.930740i	$-0.619164\pi$
$389$	−2.18277e6	−0.731363	−0.365682	+	0.930740i	$0.619164\pi$
$390$	0	0
$391$	1.08525e6	0.358995
$392$	−460294.	−0.151294
$393$	−1.24011e6	−0.405022
$394$	−4.29617e6	−1.39425
$395$	0	0
$396$	358413.	0.114854
$397$	−2.04018e6	−0.649669	−0.324835	−	0.945771i	$-0.605309\pi$
$397$	−2.04018e6	−0.649669	−0.324835	+	0.945771i	$0.605309\pi$
$398$	1.07338e6	0.339662
$399$	2.10703e6	0.662580
$400$	0	0
$401$	−1.83789e6	−0.570765	−0.285383	−	0.958414i	$-0.592121\pi$
$401$	−1.83789e6	−0.570765	−0.285383	+	0.958414i	$0.592121\pi$
$402$	−1.00636e6	−0.310590
$403$	1.83975e6	0.564282
$404$	−1.89312e6	−0.577065
$405$	0	0
$406$	−263931.	−0.0794648
$407$	4.12521e6	1.23441
$408$	−191025.	−0.0568119
$409$	3.46310e6	1.02366	0.511831	−	0.859086i	$-0.328968\pi$
$409$	3.46310e6	1.02366	0.511831	+	0.859086i	$0.328968\pi$
$410$	0	0
$411$	5.72235e6	1.67098
$412$	608007.	0.176468
$413$	−1.43512e6	−0.414012
$414$	−520044.	−0.149121
$415$	0	0
$416$	−2.64726e6	−0.750003
$417$	−4.42945e6	−1.24741
$418$	−8.46625e6	−2.37001
$419$	5.79397e6	1.61228	0.806142	−	0.591723i	$-0.201552\pi$
$419$	5.79397e6	1.61228	0.806142	+	0.591723i	$0.201552\pi$
$420$	0	0
$421$	1.69084e6	0.464941	0.232471	−	0.972603i	$-0.425319\pi$
$421$	1.69084e6	0.464941	0.232471	+	0.972603i	$0.425319\pi$
$422$	9.50367e6	2.59783
$423$	−26771.4	−0.00727479
$424$	244958.	0.0661723
$425$	0	0
$426$	−5.67295e6	−1.51456
$427$	−2.77711e6	−0.737094
$428$	−4.45375e6	−1.17521
$429$	2.77586e6	0.728206
$430$	0	0
$431$	−3.93166e6	−1.01949	−0.509744	−	0.860326i	$-0.670260\pi$
$431$	−3.93166e6	−1.01949	−0.509744	+	0.860326i	$0.670260\pi$
$432$	−3.23146e6	−0.833086
$433$	−7.27742e6	−1.86534	−0.932669	−	0.360734i	$-0.882526\pi$
$433$	−7.27742e6	−1.86534	−0.932669	+	0.360734i	$0.882526\pi$
$434$	−3.81570e6	−0.972412
$435$	0	0
$436$	7.50921e6	1.89181
$437$	6.61008e6	1.65578
$438$	−7.24037e6	−1.80333
$439$	−5.21405e6	−1.29126	−0.645631	−	0.763650i	$-0.723405\pi$
$439$	−5.21405e6	−1.29126	−0.645631	+	0.763650i	$0.723405\pi$
$440$	0	0
$441$	174323.	0.0426833
$442$	768022.	0.186990
$443$	4.83492e6	1.17052	0.585261	−	0.810845i	$-0.300992\pi$
$443$	4.83492e6	1.17052	0.585261	+	0.810845i	$0.300992\pi$
$444$	4.00392e6	0.963891
$445$	0	0
$446$	5.83023e6	1.38787
$447$	−3.28029e6	−0.776502
$448$	3.38443e6	0.796692
$449$	1.71605e6	0.401712	0.200856	−	0.979621i	$-0.435628\pi$
$449$	1.71605e6	0.401712	0.200856	+	0.979621i	$0.435628\pi$
$450$	0	0
$451$	−8.32707e6	−1.92775
$452$	5.29327e6	1.21865
$453$	−6.82122e6	−1.56177
$454$	5.53026e6	1.25923
$455$	0	0
$456$	−1.16350e6	−0.262032
$457$	−8.14043e6	−1.82329	−0.911647	−	0.410974i	$-0.865189\pi$
$457$	−8.14043e6	−1.82329	−0.911647	+	0.410974i	$0.865189\pi$
$458$	−4.54756e6	−1.01301
$459$	1.12893e6	0.250113
$460$	0	0
$461$	4.56850e6	1.00120	0.500601	−	0.865678i	$-0.333112\pi$
$461$	4.56850e6	1.00120	0.500601	+	0.865678i	$0.333112\pi$
$462$	−5.75722e6	−1.25490
$463$	−5.79664e6	−1.25668	−0.628339	−	0.777940i	$-0.716265\pi$
$463$	−5.79664e6	−1.25668	−0.628339	+	0.777940i	$0.716265\pi$
$464$	−329705.	−0.0710936
$465$	0	0
$466$	1.01271e7	2.16033
$467$	2.13263e6	0.452505	0.226252	−	0.974069i	$-0.427353\pi$
$467$	2.13263e6	0.452505	0.226252	+	0.974069i	$0.427353\pi$
$468$	−198035.	−0.0417953
$469$	−639362.	−0.134219
$470$	0	0
$471$	4.06107e6	0.843506
$472$	792471.	0.163730
$473$	1.06868e7	2.19631
$474$	5.18557e6	1.06011
$475$	0	0
$476$	−857133.	−0.173393
$477$	−92770.6	−0.0186687
$478$	8.27501e6	1.65653
$479$	−1.43546e6	−0.285859	−0.142930	−	0.989733i	$-0.545652\pi$
$479$	−1.43546e6	−0.285859	−0.142930	+	0.989733i	$0.545652\pi$
$480$	0	0
$481$	−2.27932e6	−0.449203
$482$	5.87549e6	1.15193
$483$	4.49499e6	0.876720
$484$	6.44410e6	1.25040
$485$	0	0
$486$	−1.04728e6	−0.201129
$487$	8.03771e6	1.53571	0.767857	−	0.640622i	$-0.221323\pi$
$487$	8.03771e6	1.53571	0.767857	+	0.640622i	$0.221323\pi$
$488$	1.53351e6	0.291500
$489$	4.19051e6	0.792491
$490$	0	0
$491$	2.15660e6	0.403706	0.201853	−	0.979416i	$-0.435304\pi$
$491$	2.15660e6	0.403706	0.201853	+	0.979416i	$0.435304\pi$
$492$	−8.08223e6	−1.50528
$493$	115185.	0.0213441
$494$	4.67789e6	0.862448
$495$	0	0
$496$	−4.76662e6	−0.869974
$497$	−3.60416e6	−0.654505
$498$	−4.32355e6	−0.781208
$499$	3.89550e6	0.700345	0.350173	−	0.936685i	$-0.386123\pi$
$499$	3.89550e6	0.700345	0.350173	+	0.936685i	$0.386123\pi$
$500$	0	0
$501$	−6.13446e6	−1.09190
$502$	−6.82115e6	−1.20809
$503$	−6.04036e6	−1.06449	−0.532247	−	0.846589i	$-0.678652\pi$
$503$	−6.04036e6	−1.06449	−0.532247	+	0.846589i	$0.678652\pi$
$504$	58155.9	0.0101981
$505$	0	0
$506$	−1.80613e7	−3.13598
$507$	4.05247e6	0.700163
$508$	−2.37490e6	−0.408306
$509$	−3.74827e6	−0.641264	−0.320632	−	0.947204i	$-0.603895\pi$
$509$	−3.74827e6	−0.641264	−0.320632	+	0.947204i	$0.603895\pi$
$510$	0	0
$511$	−4.59997e6	−0.779297
$512$	8.02177e6	1.35237
$513$	6.87614e6	1.15359
$514$	−1.35431e7	−2.26105
$515$	0	0
$516$	1.03726e7	1.71499
$517$	−929781.	−0.152987
$518$	4.72739e6	0.774100
$519$	5.70532e6	0.929741
$520$	0	0
$521$	6.22989e6	1.00551	0.502755	−	0.864429i	$-0.332320\pi$
$521$	6.22989e6	1.00551	0.502755	+	0.864429i	$0.332320\pi$
$522$	−55195.6	−0.00886601
$523$	9.16914e6	1.46580	0.732899	−	0.680337i	$-0.238167\pi$
$523$	9.16914e6	1.46580	0.732899	+	0.680337i	$0.238167\pi$
$524$	3.07266e6	0.488862
$525$	0	0
$526$	1.41188e7	2.22501
$527$	1.66525e6	0.261188
$528$	−7.19199e6	−1.12270
$529$	7.66515e6	1.19092
$530$	0	0
$531$	−300125.	−0.0461920
$532$	−5.22065e6	−0.799734
$533$	4.60099e6	0.701509
$534$	−5.00361e6	−0.759329
$535$	0	0
$536$	353055.	0.0530799
$537$	1.52695e6	0.228502
$538$	3.19539e6	0.475958
$539$	6.05430e6	0.897619
$540$	0	0
$541$	7.04683e6	1.03514	0.517572	−	0.855640i	$-0.326836\pi$
$541$	7.04683e6	1.03514	0.517572	+	0.855640i	$0.326836\pi$
$542$	1.86974e7	2.73390
$543$	9.89714e6	1.44049
$544$	−2.39616e6	−0.347152
$545$	0	0
$546$	3.18106e6	0.456657
$547$	−2.21405e6	−0.316388	−0.158194	−	0.987408i	$-0.550567\pi$
$547$	−2.21405e6	−0.316388	−0.158194	+	0.987408i	$0.550567\pi$
$548$	−1.41785e7	−2.01687
$549$	−580774.	−0.0822387
$550$	0	0
$551$	701570.	0.0984446
$552$	−2.48213e6	−0.346718
$553$	3.29451e6	0.458119
$554$	−4.23583e6	−0.586360
$555$	0	0
$556$	1.09750e7	1.50563
$557$	9.39143e6	1.28261	0.641304	−	0.767287i	$-0.278394\pi$
$557$	9.39143e6	1.28261	0.641304	+	0.767287i	$0.278394\pi$
$558$	−797975.	−0.108493
$559$	−5.90481e6	−0.799239
$560$	0	0
$561$	2.51257e6	0.337063
$562$	1.02697e7	1.37157
$563$	9.33692e6	1.24146	0.620730	−	0.784025i	$-0.286836\pi$
$563$	9.33692e6	1.24146	0.620730	+	0.784025i	$0.286836\pi$
$564$	−902442.	−0.119460
$565$	0	0
$566$	103139.	0.0135326
$567$	4.35424e6	0.568794
$568$	1.99021e6	0.258838
$569$	−1.25243e7	−1.62170	−0.810852	−	0.585252i	$-0.800996\pi$
$569$	−1.25243e7	−1.62170	−0.810852	+	0.585252i	$0.800996\pi$
$570$	0	0
$571$	−7.46956e6	−0.958748	−0.479374	−	0.877611i	$-0.659136\pi$
$571$	−7.46956e6	−0.958748	−0.479374	+	0.877611i	$0.659136\pi$
$572$	−6.87783e6	−0.878945
$573$	7.53761e6	0.959063
$574$	−9.54261e6	−1.20889
$575$	0	0
$576$	707783.	0.0888881
$577$	556852.	0.0696306	0.0348153	−	0.999394i	$-0.488916\pi$
$577$	556852.	0.0696306	0.0348153	+	0.999394i	$0.488916\pi$
$578$	695175.	0.0865515
$579$	7.79802e6	0.966692
$580$	0	0
$581$	−2.74685e6	−0.337594
$582$	3.02045e6	0.369627
$583$	−3.22195e6	−0.392598
$584$	2.54010e6	0.308190
$585$	0	0
$586$	2.01108e6	0.241927
$587$	9.46255e6	1.13348	0.566739	−	0.823898i	$-0.308205\pi$
$587$	9.46255e6	1.13348	0.566739	+	0.823898i	$0.308205\pi$
$588$	5.87628e6	0.700906
$589$	1.01427e7	1.20467
$590$	0	0
$591$	7.76577e6	0.914567
$592$	5.90551e6	0.692553
$593$	−3.49579e6	−0.408234	−0.204117	−	0.978947i	$-0.565432\pi$
$593$	−3.49579e6	−0.408234	−0.204117	+	0.978947i	$0.565432\pi$
$594$	−1.87883e7	−2.18485
$595$	0	0
$596$	8.12767e6	0.937239
$597$	−1.94025e6	−0.222804
$598$	9.97949e6	1.14118
$599$	1.60039e7	1.82246	0.911231	−	0.411895i	$-0.135133\pi$
$599$	1.60039e7	1.82246	0.911231	+	0.411895i	$0.135133\pi$
$600$	0	0
$601$	1.17281e7	1.32447	0.662234	−	0.749297i	$-0.269609\pi$
$601$	1.17281e7	1.32447	0.662234	+	0.749297i	$0.269609\pi$
$602$	1.22468e7	1.37731
$603$	−133709.	−0.0149751
$604$	1.69012e7	1.88505
$605$	0	0
$606$	6.35949e6	0.703462
$607$	1.85236e6	0.204058	0.102029	−	0.994781i	$-0.467467\pi$
$607$	1.85236e6	0.204058	0.102029	+	0.994781i	$0.467467\pi$
$608$	−1.45946e7	−1.60116
$609$	477082.	0.0521254
$610$	0	0
$611$	513736.	0.0556720
$612$	−179252.	−0.0193457
$613$	−1.52983e7	−1.64434	−0.822169	−	0.569244i	$-0.807236\pi$
$613$	−1.52983e7	−1.64434	−0.822169	+	0.569244i	$0.807236\pi$
$614$	−5.22314e6	−0.559127
$615$	0	0
$616$	2.01978e6	0.214463
$617$	−3.48814e6	−0.368876	−0.184438	−	0.982844i	$-0.559047\pi$
$617$	−3.48814e6	−0.368876	−0.184438	+	0.982844i	$0.559047\pi$
$618$	−2.04246e6	−0.215120
$619$	2.45152e6	0.257163	0.128582	−	0.991699i	$-0.458958\pi$
$619$	2.45152e6	0.257163	0.128582	+	0.991699i	$0.458958\pi$
$620$	0	0
$621$	1.46691e7	1.52642
$622$	−1.69304e7	−1.75465
$623$	−3.17891e6	−0.328139
$624$	3.97382e6	0.408551
$625$	0	0
$626$	2.43052e7	2.47892
$627$	1.53036e7	1.55462
$628$	−1.00622e7	−1.01811
$629$	−2.06313e6	−0.207921
$630$	0	0
$631$	−1.72457e7	−1.72428	−0.862139	−	0.506672i	$-0.830876\pi$
$631$	−1.72457e7	−1.72428	−0.862139	+	0.506672i	$0.830876\pi$
$632$	−1.81923e6	−0.181173
$633$	−1.71789e7	−1.70406
$634$	−5.83434e6	−0.576460
$635$	0	0
$636$	−3.12722e6	−0.306560
$637$	−3.34521e6	−0.326644
$638$	−1.91696e6	−0.186450
$639$	−753735.	−0.0730241
$640$	0	0
$641$	2.31048e6	0.222104	0.111052	−	0.993815i	$-0.464578\pi$
$641$	2.31048e6	0.222104	0.111052	+	0.993815i	$0.464578\pi$
$642$	1.49613e7	1.43262
$643$	6.15366e6	0.586956	0.293478	−	0.955966i	$-0.405187\pi$
$643$	6.15366e6	0.586956	0.293478	+	0.955966i	$0.405187\pi$
$644$	−1.11374e7	−1.05820
$645$	0	0
$646$	4.23419e6	0.399199
$647$	−8.01993e6	−0.753199	−0.376600	−	0.926376i	$-0.622907\pi$
$647$	−8.01993e6	−0.753199	−0.376600	+	0.926376i	$0.622907\pi$
$648$	−2.40441e6	−0.224942
$649$	−1.04235e7	−0.971404
$650$	0	0
$651$	6.89727e6	0.637859
$652$	−1.03830e7	−0.956537
$653$	−5.87832e6	−0.539474	−0.269737	−	0.962934i	$-0.586937\pi$
$653$	−5.87832e6	−0.539474	−0.269737	+	0.962934i	$0.586937\pi$
$654$	−2.52254e7	−2.30618
$655$	0	0
$656$	−1.19207e7	−1.08154
$657$	−961988.	−0.0869474
$658$	−1.06550e6	−0.0959380
$659$	−4.40573e6	−0.395188	−0.197594	−	0.980284i	$-0.563313\pi$
$659$	−4.40573e6	−0.395188	−0.197594	+	0.980284i	$0.563313\pi$
$660$	0	0
$661$	8.97238e6	0.798738	0.399369	−	0.916790i	$-0.369229\pi$
$661$	8.97238e6	0.798738	0.399369	+	0.916790i	$0.369229\pi$
$662$	1.45424e7	1.28971
$663$	−1.38828e6	−0.122657
$664$	1.51681e6	0.133509
$665$	0	0
$666$	988635.	0.0863675
$667$	1.49668e6	0.130261
$668$	1.51995e7	1.31792
$669$	−1.05387e7	−0.910382
$670$	0	0
$671$	−2.01705e7	−1.72946
$672$	−9.92464e6	−0.847797
$673$	4.76321e6	0.405380	0.202690	−	0.979243i	$-0.435032\pi$
$673$	4.76321e6	0.405380	0.202690	+	0.979243i	$0.435032\pi$
$674$	1.75253e7	1.48599
$675$	0	0
$676$	−1.00409e7	−0.845098
$677$	485340.	0.0406981	0.0203491	−	0.999793i	$-0.493522\pi$
$677$	485340.	0.0406981	0.0203491	+	0.999793i	$0.493522\pi$
$678$	−1.77815e7	−1.48557
$679$	1.91896e6	0.159732
$680$	0	0
$681$	−9.99651e6	−0.826001
$682$	−2.77139e7	−2.28159
$683$	1.24004e7	1.01715	0.508573	−	0.861019i	$-0.330173\pi$
$683$	1.24004e7	1.01715	0.508573	+	0.861019i	$0.330173\pi$
$684$	−1.09179e6	−0.0892276
$685$	0	0
$686$	1.80678e7	1.46587
$687$	8.22018e6	0.664492
$688$	1.52988e7	1.23221
$689$	1.78024e6	0.142867
$690$	0	0
$691$	205047.	0.0163365	0.00816823	−	0.999967i	$-0.497400\pi$
$691$	205047.	0.0163365	0.00816823	+	0.999967i	$0.497400\pi$
$692$	−1.41363e7	−1.12220
$693$	−764931.	−0.0605047
$694$	−3.34906e7	−2.63952
$695$	0	0
$696$	−263444.	−0.0206141
$697$	4.16458e6	0.324706
$698$	2.89247e7	2.24714
$699$	−1.83058e7	−1.41708
$700$	0	0
$701$	−1.03727e7	−0.797257	−0.398628	−	0.917113i	$-0.630514\pi$
$701$	−1.03727e7	−0.797257	−0.398628	+	0.917113i	$0.630514\pi$
$702$	1.03812e7	0.795067
$703$	−1.25662e7	−0.958990
$704$	2.45815e7	1.86929
$705$	0	0
$706$	1.21390e7	0.916583
$707$	4.04033e6	0.303996
$708$	−1.01170e7	−0.758521
$709$	−9.03252e6	−0.674829	−0.337414	−	0.941356i	$-0.609552\pi$
$709$	−9.03252e6	−0.674829	−0.337414	+	0.941356i	$0.609552\pi$
$710$	0	0
$711$	688978.	0.0511130
$712$	1.75539e6	0.129770
$713$	2.16378e7	1.59401
$714$	2.87934e6	0.211372
$715$	0	0
$716$	−3.78338e6	−0.275802
$717$	−1.49579e7	−1.08661
$718$	507873.	0.0367658
$719$	2.62860e7	1.89628	0.948141	−	0.317849i	$-0.102961\pi$
$719$	2.62860e7	1.89628	0.948141	+	0.317849i	$0.102961\pi$
$720$	0	0
$721$	−1.29762e6	−0.0929628
$722$	5.18030e6	0.369838
$723$	−1.06206e7	−0.755616
$724$	−2.45225e7	−1.73867
$725$	0	0
$726$	−2.16474e7	−1.52428
$727$	−1.65670e7	−1.16254	−0.581271	−	0.813710i	$-0.697444\pi$
$727$	−1.65670e7	−1.16254	−0.581271	+	0.813710i	$0.697444\pi$
$728$	−1.11600e6	−0.0780430
$729$	1.51922e7	1.05877
$730$	0	0
$731$	−5.34474e6	−0.369941
$732$	−1.95774e7	−1.35045
$733$	2.11043e7	1.45081	0.725407	−	0.688321i	$-0.241652\pi$
$733$	2.11043e7	1.45081	0.725407	+	0.688321i	$0.241652\pi$
$734$	1.04944e7	0.718979
$735$	0	0
$736$	−3.11352e7	−2.11864
$737$	−4.64377e6	−0.314921
$738$	−1.99564e6	−0.134878
$739$	2.19360e7	1.47756	0.738780	−	0.673946i	$-0.235402\pi$
$739$	2.19360e7	1.47756	0.738780	+	0.673946i	$0.235402\pi$
$740$	0	0
$741$	−8.45577e6	−0.565728
$742$	−3.69228e6	−0.246198
$743$	1.08537e7	0.721283	0.360642	−	0.932704i	$-0.382558\pi$
$743$	1.08537e7	0.721283	0.360642	+	0.932704i	$0.382558\pi$
$744$	−3.80867e6	−0.252255
$745$	0	0
$746$	1.66115e7	1.09285
$747$	−574446.	−0.0376659
$748$	−6.22547e6	−0.406835
$749$	9.50527e6	0.619099
$750$	0	0
$751$	1.74016e6	0.112587	0.0562935	−	0.998414i	$-0.482072\pi$
$751$	1.74016e6	0.112587	0.0562935	+	0.998414i	$0.482072\pi$
$752$	−1.33104e6	−0.0858315
$753$	1.23299e7	0.792452
$754$	1.05919e6	0.0678491
$755$	0	0
$756$	−1.15857e7	−0.737252
$757$	−2.50864e7	−1.59111	−0.795553	−	0.605884i	$-0.792820\pi$
$757$	−2.50864e7	−1.59111	−0.795553	+	0.605884i	$0.792820\pi$
$758$	−2.00239e7	−1.26583
$759$	3.26477e7	2.05707
$760$	0	0
$761$	−1.37556e7	−0.861032	−0.430516	−	0.902583i	$-0.641668\pi$
$761$	−1.37556e7	−0.861032	−0.430516	+	0.902583i	$0.641668\pi$
$762$	7.97791e6	0.497739
$763$	−1.60263e7	−0.996602
$764$	−1.86762e7	−1.15759
$765$	0	0
$766$	−883212.	−0.0543867
$767$	5.75931e6	0.353494
$768$	−9.36654e6	−0.573028
$769$	2.91944e7	1.78026	0.890131	−	0.455705i	$-0.150613\pi$
$769$	2.91944e7	1.78026	0.890131	+	0.455705i	$0.150613\pi$
$770$	0	0
$771$	2.44805e7	1.48315
$772$	−1.93214e7	−1.16680
$773$	−5.88433e6	−0.354200	−0.177100	−	0.984193i	$-0.556672\pi$
$773$	−5.88433e6	−0.354200	−0.177100	+	0.984193i	$0.556672\pi$
$774$	2.56116e6	0.153668
$775$	0	0
$776$	−1.05965e6	−0.0631694
$777$	−8.54524e6	−0.507775
$778$	−1.81679e7	−1.07611
$779$	2.53658e7	1.49763
$780$	0	0
$781$	−2.61775e7	−1.53568
$782$	9.03294e6	0.528217
$783$	1.55692e6	0.0907533
$784$	8.66712e6	0.503599
$785$	0	0
$786$	−1.03219e7	−0.595939
$787$	9.23431e6	0.531456	0.265728	−	0.964048i	$-0.414388\pi$
$787$	9.23431e6	0.531456	0.265728	+	0.964048i	$0.414388\pi$
$788$	−1.92415e7	−1.10388
$789$	−2.55211e7	−1.45951
$790$	0	0
$791$	−1.12970e7	−0.641981
$792$	422394.	0.0239279
$793$	1.11449e7	0.629350
$794$	−1.69811e7	−0.955907
$795$	0	0
$796$	4.80742e6	0.268924
$797$	4.32222e6	0.241024	0.120512	−	0.992712i	$-0.461546\pi$
$797$	4.32222e6	0.241024	0.120512	+	0.992712i	$0.461546\pi$
$798$	1.75376e7	0.974904
$799$	465007.	0.0257687
$800$	0	0
$801$	−664802.	−0.0366110
$802$	−1.52974e7	−0.839810
$803$	−3.34102e7	−1.82848
$804$	−4.50723e6	−0.245906
$805$	0	0
$806$	1.53129e7	0.830271
$807$	−5.77600e6	−0.312208
$808$	−2.23107e6	−0.120222
$809$	−3.92650e6	−0.210928	−0.105464	−	0.994423i	$-0.533633\pi$
$809$	−3.92650e6	−0.210928	−0.105464	+	0.994423i	$0.533633\pi$
$810$	0	0
$811$	1.86033e7	0.993201	0.496601	−	0.867979i	$-0.334581\pi$
$811$	1.86033e7	0.993201	0.496601	+	0.867979i	$0.334581\pi$
$812$	−1.18208e6	−0.0629154
$813$	−3.37974e7	−1.79332
$814$	3.43356e7	1.81629
$815$	0	0
$816$	3.59690e6	0.189105
$817$	−3.25539e7	−1.70627
$818$	2.88246e7	1.50619
$819$	422651.	0.0220177
$820$	0	0
$821$	−3.35587e7	−1.73759	−0.868795	−	0.495172i	$-0.835105\pi$
$821$	−3.35587e7	−1.73759	−0.868795	+	0.495172i	$0.835105\pi$
$822$	4.76292e7	2.45863
$823$	5.93901e6	0.305643	0.152822	−	0.988254i	$-0.451164\pi$
$823$	5.93901e6	0.305643	0.152822	+	0.988254i	$0.451164\pi$
$824$	716544.	0.0367642
$825$	0	0
$826$	−1.19450e7	−0.609167
$827$	1.25961e7	0.640430	0.320215	−	0.947345i	$-0.396245\pi$
$827$	1.25961e7	0.640430	0.320215	+	0.947345i	$0.396245\pi$
$828$	−2.32915e6	−0.118065
$829$	−2.20005e7	−1.11185	−0.555926	−	0.831232i	$-0.687636\pi$
$829$	−2.20005e7	−1.11185	−0.555926	+	0.831232i	$0.687636\pi$
$830$	0	0
$831$	7.65670e6	0.384626
$832$	−1.35821e7	−0.680236
$833$	−3.02791e6	−0.151193
$834$	−3.68679e7	−1.83541
$835$	0	0
$836$	−3.79183e7	−1.87643
$837$	2.25088e7	1.11055
$838$	4.82253e7	2.37227
$839$	2.54510e7	1.24825	0.624123	−	0.781326i	$-0.285457\pi$
$839$	2.54510e7	1.24825	0.624123	+	0.781326i	$0.285457\pi$
$840$	0	0
$841$	−2.03523e7	−0.992255
$842$	1.40735e7	0.684104
$843$	−1.85636e7	−0.899691
$844$	4.25646e7	2.05680
$845$	0	0
$846$	−222828.	−0.0107039
$847$	−1.37531e7	−0.658708
$848$	−4.61243e6	−0.220262
$849$	−186435.	−0.00887682
$850$	0	0
$851$	−2.68078e7	−1.26893
$852$	−2.54078e7	−1.19913
$853$	4.09671e7	1.92780	0.963901	−	0.266262i	$-0.0857885\pi$
$853$	4.09671e7	1.92780	0.963901	+	0.266262i	$0.0857885\pi$
$854$	−2.31148e7	−1.08454
$855$	0	0
$856$	−5.24880e6	−0.244836
$857$	2.88426e7	1.34147	0.670737	−	0.741695i	$-0.265978\pi$
$857$	2.88426e7	1.34147	0.670737	+	0.741695i	$0.265978\pi$
$858$	2.31045e7	1.07146
$859$	3.52775e6	0.163123	0.0815615	−	0.996668i	$-0.474009\pi$
$859$	3.52775e6	0.163123	0.0815615	+	0.996668i	$0.474009\pi$
$860$	0	0
$861$	1.72492e7	0.792980
$862$	−3.27246e7	−1.50005
$863$	−1.85743e7	−0.848959	−0.424479	−	0.905438i	$-0.639543\pi$
$863$	−1.85743e7	−0.848959	−0.424479	+	0.905438i	$0.639543\pi$
$864$	−3.23884e7	−1.47606
$865$	0	0
$866$	−6.05725e7	−2.74461
$867$	−1.25660e6	−0.0567740
$868$	−1.70896e7	−0.769897
$869$	2.39285e7	1.07489
$870$	0	0
$871$	2.56584e6	0.114600
$872$	8.84970e6	0.394128
$873$	401310.	0.0178215
$874$	5.50181e7	2.43628
$875$	0	0
$876$	−3.24278e7	−1.42777
$877$	−2.95834e6	−0.129882	−0.0649410	−	0.997889i	$-0.520686\pi$
$877$	−2.95834e6	−0.129882	−0.0649410	+	0.997889i	$0.520686\pi$
$878$	−4.33984e7	−1.89993
$879$	−3.63523e6	−0.158694
$880$	0	0
$881$	8.11813e6	0.352384	0.176192	−	0.984356i	$-0.443622\pi$
$881$	8.11813e6	0.352384	0.176192	+	0.984356i	$0.443622\pi$
$882$	1.45095e6	0.0628032
$883$	1.23929e7	0.534897	0.267448	−	0.963572i	$-0.413820\pi$
$883$	1.23929e7	0.534897	0.267448	+	0.963572i	$0.413820\pi$
$884$	3.43978e6	0.148047
$885$	0	0
$886$	4.02427e7	1.72228
$887$	2.51951e7	1.07524	0.537622	−	0.843186i	$-0.319323\pi$
$887$	2.51951e7	1.07524	0.537622	+	0.843186i	$0.319323\pi$
$888$	4.71867e6	0.200811
$889$	5.06855e6	0.215095
$890$	0	0
$891$	3.16254e7	1.33457
$892$	2.61122e7	1.09883
$893$	2.83228e6	0.118852
$894$	−2.73030e7	−1.14253
$895$	0	0
$896$	7.06100e6	0.293830
$897$	−1.80390e7	−0.748566
$898$	1.42833e7	0.591069
$899$	2.29656e6	0.0947717
$900$	0	0
$901$	1.61138e6	0.0661282
$902$	−6.93092e7	−2.83645
$903$	−2.21373e7	−0.903452
$904$	6.23819e6	0.253885
$905$	0	0
$906$	−5.67754e7	−2.29795
$907$	−2.18570e7	−0.882212	−0.441106	−	0.897455i	$-0.645414\pi$
$907$	−2.18570e7	−0.882212	−0.441106	+	0.897455i	$0.645414\pi$
$908$	2.47687e7	0.996984
$909$	844951.	0.0339173
$910$	0	0
$911$	3.57390e7	1.42674	0.713372	−	0.700786i	$-0.247167\pi$
$911$	3.57390e7	1.42674	0.713372	+	0.700786i	$0.247167\pi$
$912$	2.19081e7	0.872203
$913$	−1.99507e7	−0.792103
$914$	−6.77557e7	−2.68275
$915$	0	0
$916$	−2.03674e7	−0.802042
$917$	−6.55772e6	−0.257531
$918$	9.39651e6	0.368010
$919$	−3.39382e7	−1.32556	−0.662781	−	0.748813i	$-0.730624\pi$
$919$	−3.39382e7	−1.32556	−0.662781	+	0.748813i	$0.730624\pi$
$920$	0	0
$921$	9.44136e6	0.366763
$922$	3.80253e7	1.47314
$923$	1.44639e7	0.558834
$924$	−2.57852e7	−0.993552
$925$	0	0
$926$	−4.82475e7	−1.84905
$927$	−271370.	−0.0103720
$928$	−3.30457e6	−0.125964
$929$	−4.83073e7	−1.83643	−0.918213	−	0.396087i	$-0.870368\pi$
$929$	−4.83073e7	−1.83643	−0.918213	+	0.396087i	$0.870368\pi$
$930$	0	0
$931$	−1.84425e7	−0.697342
$932$	4.53568e7	1.71042
$933$	3.06034e7	1.15097
$934$	1.77506e7	0.665804
$935$	0	0
$936$	−233387.	−0.00870738
$937$	−3.02592e7	−1.12592	−0.562961	−	0.826483i	$-0.690338\pi$
$937$	−3.02592e7	−1.12592	−0.562961	+	0.826483i	$0.690338\pi$
$938$	−5.32164e6	−0.197487
$939$	−4.39341e7	−1.62606
$940$	0	0
$941$	−3.94366e6	−0.145186	−0.0725932	−	0.997362i	$-0.523127\pi$
$941$	−3.94366e6	−0.145186	−0.0725932	+	0.997362i	$0.523127\pi$
$942$	3.38017e7	1.24111
$943$	5.41136e7	1.98165
$944$	−1.49218e7	−0.544995
$945$	0	0
$946$	8.89499e7	3.23160
$947$	2.57929e7	0.934599	0.467300	−	0.884099i	$-0.345227\pi$
$947$	2.57929e7	0.934599	0.467300	+	0.884099i	$0.345227\pi$
$948$	2.32249e7	0.839330
$949$	1.84603e7	0.665384
$950$	0	0
$951$	1.05462e7	0.378132
$952$	−1.01014e6	−0.0361235
$953$	8.92590e6	0.318361	0.159181	−	0.987249i	$-0.449115\pi$
$953$	8.92590e6	0.318361	0.159181	+	0.987249i	$0.449115\pi$
$954$	−772163.	−0.0274687
$955$	0	0
$956$	3.70617e7	1.31154
$957$	3.46510e6	0.122303
$958$	−1.19478e7	−0.420606
$959$	3.02599e7	1.06248
$960$	0	0
$961$	4.57272e6	0.159723
$962$	−1.89716e7	−0.660947
$963$	1.98783e6	0.0690738
$964$	2.63149e7	0.912029
$965$	0	0
$966$	3.74134e7	1.28998
$967$	5.25311e7	1.80655	0.903275	−	0.429062i	$-0.141156\pi$
$967$	5.25311e7	1.80655	0.903275	+	0.429062i	$0.141156\pi$
$968$	7.59446e6	0.260500
$969$	−7.65374e6	−0.261857
$970$	0	0
$971$	−2.59937e7	−0.884747	−0.442374	−	0.896831i	$-0.645863\pi$
$971$	−2.59937e7	−0.884747	−0.442374	+	0.896831i	$0.645863\pi$
$972$	−4.69053e6	−0.159242
$973$	−2.34230e7	−0.793160
$974$	6.69008e7	2.25961
$975$	0	0
$976$	−2.88753e7	−0.970292
$977$	−2.87165e7	−0.962487	−0.481244	−	0.876587i	$-0.659815\pi$
$977$	−2.87165e7	−0.962487	−0.481244	+	0.876587i	$0.659815\pi$
$978$	3.48791e7	1.16605
$979$	−2.30888e7	−0.769919
$980$	0	0
$981$	−3.35156e6	−0.111192
$982$	1.79501e7	0.594004
$983$	−5.36618e7	−1.77126	−0.885628	−	0.464395i	$-0.846272\pi$
$983$	−5.36618e7	−1.77126	−0.885628	+	0.464395i	$0.846272\pi$
$984$	−9.52501e6	−0.313601
$985$	0	0
$986$	958722.	0.0314051
$987$	1.92601e6	0.0629311
$988$	2.09511e7	0.682834
$989$	−6.94482e7	−2.25772
$990$	0	0
$991$	−3.36888e7	−1.08969	−0.544843	−	0.838538i	$-0.683411\pi$
$991$	−3.36888e7	−1.08969	−0.544843	+	0.838538i	$0.683411\pi$
$992$	−4.77749e7	−1.54142
$993$	−2.62870e7	−0.845994
$994$	−2.99987e7	−0.963023
$995$	0	0
$996$	−1.93641e7	−0.618513
$997$	7.17769e6	0.228690	0.114345	−	0.993441i	$-0.463523\pi$
$997$	7.17769e6	0.228690	0.114345	+	0.993441i	$0.463523\pi$
$998$	3.24237e7	1.03047
$999$	−2.78868e7	−0.884066

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	425.6.a.k.1.12	yes	15
5.4	even	2		425.6.a.j.1.4	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
425.6.a.j.1.4	✓	15	5.4	even	2
425.6.a.k.1.12	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)

\(f(q)\)	\(=\)	\(q+8.32336 q^{2} -15.0453 q^{3} +37.2783 q^{4} -125.228 q^{6} -79.5600 q^{7} +43.9329 q^{8} -16.6383 q^{9} -577.854 q^{11} -560.864 q^{12} +319.284 q^{13} -662.206 q^{14} -827.235 q^{16} +289.000 q^{17} -138.487 q^{18} +1760.25 q^{19} +1197.01 q^{21} -4809.69 q^{22} +3755.20 q^{23} -660.985 q^{24} +2657.52 q^{26} +3906.34 q^{27} -2965.86 q^{28} +398.563 q^{29} +5762.11 q^{31} -8291.23 q^{32} +8694.01 q^{33} +2405.45 q^{34} -620.248 q^{36} -7138.85 q^{37} +14651.2 q^{38} -4803.73 q^{39} +14410.3 q^{41} +9963.10 q^{42} -18493.9 q^{43} -21541.4 q^{44} +31255.8 q^{46} +1609.02 q^{47} +12446.0 q^{48} -10477.2 q^{49} -4348.10 q^{51} +11902.4 q^{52} +5575.72 q^{53} +32513.9 q^{54} -3495.30 q^{56} -26483.5 q^{57} +3317.38 q^{58} +18038.2 q^{59} +34905.8 q^{61} +47960.1 q^{62} +1323.74 q^{63} -42539.3 q^{64} +72363.3 q^{66} +8036.23 q^{67} +10773.4 q^{68} -56498.1 q^{69} +45301.1 q^{71} -730.970 q^{72} +57817.7 q^{73} -59419.2 q^{74} +65619.1 q^{76} +45974.1 q^{77} -39983.2 q^{78} -41409.2 q^{79} -54729.1 q^{81} +119942. q^{82} +34525.5 q^{83} +44622.3 q^{84} -153931. q^{86} -5996.50 q^{87} -25386.8 q^{88} +39956.1 q^{89} -25402.2 q^{91} +139987. q^{92} -86692.7 q^{93} +13392.5 q^{94} +124744. q^{96} -24119.7 q^{97} -87205.6 q^{98} +9614.52 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + q^{2} + 9 q^{3} + 277 q^{4} + 169 q^{6} + 181 q^{7} + 753 q^{8} + 1826 q^{9} + 172 q^{11} - 2109 q^{12} - 389 q^{13} + 3635 q^{14} + 6837 q^{16} + 4335 q^{17} + 6742 q^{18} + 5150 q^{19} - 6891 q^{21}+ \cdots - 183214 q^{99}+O(q^{100}) \) 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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