LMFDB - Embedded newform 525.4.d.k.274.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [525,4,Mod(274,525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(525, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("525.274");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 525 = 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	525.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$30.9760027530$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 105)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.1
Root			$-1.61803i$ of defining polynomial
Character	$\chi$	$=$	525.274
Dual form			525.4.d.k.274.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.23607i q^{2} -3.00000i q^{3} -9.94427 q^{4} -12.7082 q^{6} -7.00000i q^{7} +8.23607i q^{8} -9.00000 q^{9} -41.5279 q^{11} +29.8328i q^{12} -88.9706i q^{13} -29.6525 q^{14} -44.6656 q^{16} -120.387i q^{17} +38.1246i q^{18} +112.138 q^{19} -21.0000 q^{21} +175.915i q^{22} +115.279i q^{23} +24.7082 q^{24} -376.885 q^{26} +27.0000i q^{27} +69.6099i q^{28} +144.833 q^{29} -258.079 q^{31} +255.095i q^{32} +124.584i q^{33} -509.967 q^{34} +89.4984 q^{36} +48.3344i q^{37} -475.023i q^{38} -266.912 q^{39} +200.885 q^{41} +88.9574i q^{42} +218.217i q^{43} +412.964 q^{44} +488.328 q^{46} +575.659i q^{47} +133.997i q^{48} -49.0000 q^{49} -361.161 q^{51} +884.748i q^{52} +184.302i q^{53} +114.374 q^{54} +57.6525 q^{56} -336.413i q^{57} -613.522i q^{58} +151.502 q^{59} -529.830 q^{61} +1093.24i q^{62} +63.0000i q^{63} +723.276 q^{64} +527.745 q^{66} +1.28485i q^{67} +1197.16i q^{68} +345.836 q^{69} -61.4226 q^{71} -74.1246i q^{72} -484.800i q^{73} +204.748 q^{74} -1115.13 q^{76} +290.695i q^{77} +1130.66i q^{78} -878.257 q^{79} +81.0000 q^{81} -850.964i q^{82} -491.830i q^{83} +208.830 q^{84} +924.381 q^{86} -434.498i q^{87} -342.026i q^{88} +415.560 q^{89} -622.794 q^{91} -1146.36i q^{92} +774.237i q^{93} +2438.53 q^{94} +765.286 q^{96} -1031.70i q^{97} +207.567i q^{98} +373.751 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} - 24 q^{6} - 36 q^{9} - 184 q^{11} - 56 q^{14} + 36 q^{16} + 216 q^{19} - 84 q^{21} + 72 q^{24} - 792 q^{26} + 472 q^{29} - 120 q^{31} - 1056 q^{34} + 36 q^{36} - 48 q^{39} + 88 q^{41} + 24 q^{44}+ \cdots + 1656 q^{99}+O(q^{100}) $ 4 * q - 4 * q^4 - 24 * q^6 - 36 * q^9 - 184 * q^11 - 56 * q^14 + 36 * q^16 + 216 * q^19 - 84 * q^21 + 72 * q^24 - 792 * q^26 + 472 * q^29 - 120 * q^31 - 1056 * q^34 + 36 * q^36 - 48 * q^39 + 88 * q^41 + 24 * q^44 + 880 * q^46 - 196 * q^49 - 264 * q^51 + 216 * q^54 + 168 * q^56 + 928 * q^59 - 1368 * q^61 + 2428 * q^64 + 984 * q^66 + 1920 * q^69 - 1480 * q^71 + 336 * q^74 - 2296 * q^76 + 816 * q^79 + 324 * q^81 + 84 * q^84 + 2016 * q^86 + 2664 * q^89 - 112 * q^91 + 4960 * q^94 + 1800 * q^96 + 1656 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/525\mathbb{Z}\right)^\times$.

$n$	$127$	$176$	$451$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	− 4.23607i	− 1.49768i	−0.662753	−	0.748838i	$-0.730612\pi$
$2$	− 4.23607i	− 1.49768i	0.662753	−	0.748838i	$-0.269388\pi$
$3$	− 3.00000i	− 0.577350i
$3$	− 3.00000i	− 0.577350i
$4$	−9.94427	−1.24303
$5$	0	0
$5$	0	0
$6$	−12.7082	−0.864684
$7$	− 7.00000i	− 0.377964i
$7$	− 7.00000i	− 0.377964i
$8$	8.23607i	0.363986i
$9$	−9.00000	−0.333333
$10$	0	0
$11$	−41.5279	−1.13828	−0.569142	−	0.822239i	$-0.692725\pi$
$11$	−41.5279	−1.13828	−0.569142	+	0.822239i	$0.692725\pi$
$12$	29.8328i	0.717666i
$13$	− 88.9706i	− 1.89815i	−0.315044	−	0.949077i	$-0.602019\pi$
$13$	− 88.9706i	− 1.89815i	0.315044	−	0.949077i	$-0.397981\pi$
$14$	−29.6525	−0.566068
$15$	0	0
$16$	−44.6656	−0.697900
$17$	− 120.387i	− 1.71754i	−0.512364	−	0.858769i	$-0.671230\pi$
$17$	− 120.387i	− 1.71754i	0.512364	−	0.858769i	$-0.328770\pi$
$18$	38.1246i	0.499225i
$19$	112.138	1.35401	0.677004	−	0.735979i	$-0.263278\pi$
$19$	112.138	1.35401	0.677004	+	0.735979i	$0.263278\pi$
$20$	0	0
$21$	−21.0000	−0.218218
$22$	175.915i	1.70478i
$23$	115.279i	1.04510i	0.852609	+	0.522549i	$0.175019\pi$
$23$	115.279i	1.04510i	−0.852609	+	0.522549i	$0.824981\pi$
$24$	24.7082	0.210148
$25$	0	0
$26$	−376.885	−2.84282
$27$	27.0000i	0.192450i
$28$	69.6099i	0.469823i
$29$	144.833	0.927406	0.463703	−	0.885991i	$-0.346520\pi$
$29$	144.833	0.927406	0.463703	+	0.885991i	$0.346520\pi$
$30$	0	0
$31$	−258.079	−1.49524	−0.747618	−	0.664128i	$-0.768803\pi$
$31$	−258.079	−1.49524	−0.747618	+	0.664128i	$0.768803\pi$
$32$	255.095i	1.40922i
$33$	124.584i	0.657188i
$34$	−509.967	−2.57231
$35$	0	0
$36$	89.4984	0.414345
$37$	48.3344i	0.214760i	0.994218	+	0.107380i	$0.0342461\pi$
$37$	48.3344i	0.214760i	−0.994218	+	0.107380i	$0.965754\pi$
$38$	− 475.023i	− 2.02787i
$39$	−266.912	−1.09590
$40$	0	0
$41$	200.885	0.765196	0.382598	−	0.923915i	$-0.375029\pi$
$41$	200.885	0.765196	0.382598	+	0.923915i	$0.375029\pi$
$42$	88.9574i	0.326820i
$43$	218.217i	0.773901i	0.922101	+	0.386950i	$0.126472\pi$
$43$	218.217i	0.773901i	−0.922101	+	0.386950i	$0.873528\pi$
$44$	412.964	1.41493
$45$	0	0
$46$	488.328	1.56522
$47$	575.659i	1.78657i	0.449496	+	0.893283i	$0.351604\pi$
$47$	575.659i	1.78657i	−0.449496	+	0.893283i	$0.648396\pi$
$48$	133.997i	0.402933i
$49$	−49.0000	−0.142857
$50$	0	0
$51$	−361.161	−0.991621
$52$	884.748i	2.35947i
$53$	184.302i	0.477657i	0.971062	+	0.238828i	$0.0767633\pi$
$53$	184.302i	0.477657i	−0.971062	+	0.238828i	$0.923237\pi$
$54$	114.374	0.288228
$55$	0	0
$56$	57.6525	0.137574
$57$	− 336.413i	− 0.781737i
$58$	− 613.522i	− 1.38895i
$59$	151.502	0.334302	0.167151	−	0.985931i	$-0.446543\pi$
$59$	151.502	0.334302	0.167151	+	0.985931i	$0.446543\pi$
$60$	0	0
$61$	−529.830	−1.11209	−0.556047	−	0.831151i	$-0.687683\pi$
$61$	−529.830	−1.11209	−0.556047	+	0.831151i	$0.687683\pi$
$62$	1093.24i	2.23938i
$63$	63.0000i	0.125988i
$64$	723.276	1.41265
$65$	0	0
$66$	527.745	0.984256
$67$	1.28485i	0.00234283i	0.999999	+	0.00117142i	$0.000372873\pi$
$67$	1.28485i	0.00234283i	−0.999999	+	0.00117142i	$0.999627\pi$
$68$	1197.16i	2.13496i
$69$	345.836	0.603388
$70$	0	0
$71$	−61.4226	−0.102669	−0.0513347	−	0.998682i	$-0.516348\pi$
$71$	−61.4226	−0.102669	−0.0513347	+	0.998682i	$0.516348\pi$
$72$	− 74.1246i	− 0.121329i
$73$	− 484.800i	− 0.777282i	−0.921389	−	0.388641i	$-0.872945\pi$
$73$	− 484.800i	− 0.777282i	0.921389	−	0.388641i	$-0.127055\pi$
$74$	204.748	0.321641
$75$	0	0
$76$	−1115.13	−1.68308
$77$	290.695i	0.430231i
$78$	1130.66i	1.64130i
$79$	−878.257	−1.25078	−0.625390	−	0.780312i	$-0.715060\pi$
$79$	−878.257	−1.25078	−0.625390	+	0.780312i	$0.715060\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	− 850.964i	− 1.14602i
$83$	− 491.830i	− 0.650426i	−0.945641	−	0.325213i	$-0.894564\pi$
$83$	− 491.830i	− 0.650426i	0.945641	−	0.325213i	$-0.105436\pi$
$84$	208.830	0.271252
$85$	0	0
$86$	924.381	1.15905
$87$	− 434.498i	− 0.535438i
$88$	− 342.026i	− 0.414320i
$89$	415.560	0.494936	0.247468	−	0.968896i	$-0.420401\pi$
$89$	415.560	0.494936	0.247468	+	0.968896i	$0.420401\pi$
$90$	0	0
$91$	−622.794	−0.717435
$92$	− 1146.36i	− 1.29909i
$93$	774.237i	0.863275i
$94$	2438.53	2.67570
$95$	0	0
$96$	765.286	0.813611
$97$	− 1031.70i	− 1.07993i	−0.841688	−	0.539964i	$-0.818438\pi$
$97$	− 1031.70i	− 1.07993i	0.841688	−	0.539964i	$-0.181562\pi$
$98$	207.567i	0.213954i
$99$	373.751	0.379428
$100$	0	0
$101$	1447.19	1.42576	0.712878	−	0.701288i	$-0.247391\pi$
$101$	1447.19	1.42576	0.712878	+	0.701288i	$0.247391\pi$
$102$	1529.90i	1.48513i
$103$	163.567i	0.156473i	0.996935	+	0.0782364i	$0.0249289\pi$
$103$	163.567i	0.156473i	−0.996935	+	0.0782364i	$0.975071\pi$
$104$	732.768	0.690902
$105$	0	0
$106$	780.715	0.715375
$107$	− 129.653i	− 0.117141i	−0.998283	−	0.0585703i	$-0.981346\pi$
$107$	− 129.653i	− 0.117141i	0.998283	−	0.0585703i	$-0.0186542\pi$
$108$	− 268.495i	− 0.239222i
$109$	−566.681	−0.497965	−0.248983	−	0.968508i	$-0.580096\pi$
$109$	−566.681	−0.497965	−0.248983	+	0.968508i	$0.580096\pi$
$110$	0	0
$111$	145.003	0.123992
$112$	312.659i	0.263782i
$113$	− 809.890i	− 0.674230i	−0.941463	−	0.337115i	$-0.890549\pi$
$113$	− 809.890i	− 0.674230i	0.941463	−	0.337115i	$-0.109451\pi$
$114$	−1425.07	−1.17079
$115$	0	0
$116$	−1440.26	−1.15280
$117$	800.735i	0.632718i
$118$	− 641.771i	− 0.500676i
$119$	−842.709	−0.649168
$120$	0	0
$121$	393.563	0.295690
$122$	2244.39i	1.66556i
$123$	− 602.656i	− 0.441786i
$124$	2566.41	1.85863
$125$	0	0
$126$	266.872	0.188689
$127$	− 2584.25i	− 1.80563i	−0.430028	−	0.902816i	$-0.641496\pi$
$127$	− 2584.25i	− 1.80563i	0.430028	−	0.902816i	$-0.358504\pi$
$128$	− 1023.08i	− 0.706473i
$129$	654.650	0.446812
$130$	0	0
$131$	−1421.10	−0.947804	−0.473902	−	0.880578i	$-0.657155\pi$
$131$	−1421.10	−0.947804	−0.473902	+	0.880578i	$0.657155\pi$
$132$	− 1238.89i	− 0.816908i
$133$	− 784.964i	− 0.511767i
$134$	5.44272	0.00350880
$135$	0	0
$136$	991.515	0.625160
$137$	104.878i	0.0654037i	0.999465	+	0.0327019i	$0.0104112\pi$
$137$	104.878i	0.0654037i	−0.999465	+	0.0327019i	$0.989589\pi$
$138$	− 1464.98i	− 0.903679i
$139$	913.160	0.557217	0.278609	−	0.960405i	$-0.410127\pi$
$139$	913.160	0.557217	0.278609	+	0.960405i	$0.410127\pi$
$140$	0	0
$141$	1726.98	1.03147
$142$	260.190i	0.153765i
$143$	3694.76i	2.16064i
$144$	401.991	0.232633
$145$	0	0
$146$	−2053.65	−1.16412
$147$	147.000i	0.0824786i
$148$	− 480.650i	− 0.266954i
$149$	−1781.45	−0.979476	−0.489738	−	0.871870i	$-0.662908\pi$
$149$	−1781.45	−0.979476	−0.489738	+	0.871870i	$0.662908\pi$
$150$	0	0
$151$	1407.53	0.758564	0.379282	−	0.925281i	$-0.376171\pi$
$151$	1407.53	0.758564	0.379282	+	0.925281i	$0.376171\pi$
$152$	923.574i	0.492841i
$153$	1083.48i	0.572512i
$154$	1231.40	0.644346
$155$	0	0
$156$	2654.24	1.36224
$157$	− 1598.94i	− 0.812798i	−0.913696	−	0.406399i	$-0.866784\pi$
$157$	− 1598.94i	− 0.812798i	0.913696	−	0.406399i	$-0.133216\pi$
$158$	3720.36i	1.87326i
$159$	552.906	0.275775
$160$	0	0
$161$	806.950	0.395010
$162$	− 343.122i	− 0.166408i
$163$	204.892i	0.0984562i	0.998788	+	0.0492281i	$0.0156761\pi$
$163$	204.892i	0.0984562i	−0.998788	+	0.0492281i	$0.984324\pi$
$164$	−1997.66	−0.951165
$165$	0	0
$166$	−2083.42	−0.974127
$167$	− 1165.94i	− 0.540259i	−0.962824	−	0.270129i	$-0.912933\pi$
$167$	− 1165.94i	− 0.540259i	0.962824	−	0.270129i	$-0.0870665\pi$
$168$	− 172.957i	− 0.0794283i
$169$	−5718.76	−2.60299
$170$	0	0
$171$	−1009.24	−0.451336
$172$	− 2170.01i	− 0.961985i
$173$	2538.00i	1.11538i	0.830049	+	0.557690i	$0.188312\pi$
$173$	2538.00i	1.11538i	−0.830049	+	0.557690i	$0.811688\pi$
$174$	−1840.56	−0.801913
$175$	0	0
$176$	1854.87	0.794409
$177$	− 454.505i	− 0.193009i
$178$	− 1760.34i	− 0.741254i
$179$	−392.255	−0.163791	−0.0818954	−	0.996641i	$-0.526097\pi$
$179$	−392.255	−0.163791	−0.0818954	+	0.996641i	$0.526097\pi$
$180$	0	0
$181$	−2978.08	−1.22298	−0.611489	−	0.791253i	$-0.709429\pi$
$181$	−2978.08	−1.22298	−0.611489	+	0.791253i	$0.709429\pi$
$182$	2638.20i	1.07448i
$183$	1589.49i	0.642068i
$184$	−949.443	−0.380401
$185$	0	0
$186$	3279.72	1.29291
$187$	4999.41i	1.95504i
$188$	− 5724.51i	− 2.22076i
$189$	189.000	0.0727393
$190$	0	0
$191$	−1097.37	−0.415722	−0.207861	−	0.978158i	$-0.566650\pi$
$191$	−1097.37	−0.415722	−0.207861	+	0.978158i	$0.566650\pi$
$192$	− 2169.83i	− 0.815592i
$193$	− 3500.31i	− 1.30548i	−0.757582	−	0.652740i	$-0.773619\pi$
$193$	− 3500.31i	− 1.30548i	0.757582	−	0.652740i	$-0.226381\pi$
$194$	−4370.34	−1.61738
$195$	0	0
$196$	487.269	0.177576
$197$	1573.96i	0.569237i	0.958641	+	0.284618i	$0.0918669\pi$
$197$	1573.96i	0.569237i	−0.958641	+	0.284618i	$0.908133\pi$
$198$	− 1583.23i	− 0.568260i
$199$	3396.62	1.20995	0.604976	−	0.796244i	$-0.293183\pi$
$199$	3396.62	1.20995	0.604976	+	0.796244i	$0.293183\pi$
$200$	0	0
$201$	3.85456	0.00135263
$202$	− 6130.42i	− 2.13532i
$203$	− 1013.83i	− 0.350527i
$204$	3591.48	1.23262
$205$	0	0
$206$	692.879	0.234346
$207$	− 1037.51i	− 0.348366i
$208$	3973.93i	1.32472i
$209$	−4656.84	−1.54125
$210$	0	0
$211$	3337.81	1.08903	0.544513	−	0.838753i	$-0.316715\pi$
$211$	3337.81	1.08903	0.544513	+	0.838753i	$0.316715\pi$
$212$	− 1832.75i	− 0.593744i
$213$	184.268i	0.0592762i
$214$	−549.220	−0.175439
$215$	0	0
$216$	−222.374	−0.0700492
$217$	1806.55i	0.565146i
$218$	2400.50i	0.745791i
$219$	−1454.40	−0.448764
$220$	0	0
$221$	−10710.9	−3.26015
$222$	− 614.243i	− 0.185700i
$223$	− 127.328i	− 0.0382356i	−0.999817	−	0.0191178i	$-0.993914\pi$
$223$	− 127.328i	− 0.0382356i	0.999817	−	0.0191178i	$-0.00608575\pi$
$224$	1785.67	0.532633
$225$	0	0
$226$	−3430.75	−1.00978
$227$	3844.12i	1.12398i	0.827144	+	0.561990i	$0.189964\pi$
$227$	3844.12i	1.12398i	−0.827144	+	0.561990i	$0.810036\pi$
$228$	3345.39i	0.971726i
$229$	−2536.95	−0.732080	−0.366040	−	0.930599i	$-0.619287\pi$
$229$	−2536.95	−0.732080	−0.366040	+	0.930599i	$0.619287\pi$
$230$	0	0
$231$	872.085	0.248394
$232$	1192.85i	0.337563i
$233$	− 3987.44i	− 1.12114i	−0.828107	−	0.560570i	$-0.810582\pi$
$233$	− 3987.44i	− 1.12114i	0.828107	−	0.560570i	$-0.189418\pi$
$234$	3391.97	0.947607
$235$	0	0
$236$	−1506.57	−0.415549
$237$	2634.77i	0.722138i
$238$	3569.77i	0.972244i
$239$	3367.18	0.911317	0.455659	−	0.890155i	$-0.349404\pi$
$239$	3367.18	0.911317	0.455659	+	0.890155i	$0.349404\pi$
$240$	0	0
$241$	−939.551	−0.251128	−0.125564	−	0.992086i	$-0.540074\pi$
$241$	−939.551	−0.251128	−0.125564	+	0.992086i	$0.540074\pi$
$242$	− 1667.16i	− 0.442848i
$243$	− 243.000i	− 0.0641500i
$244$	5268.77	1.38237
$245$	0	0
$246$	−2552.89	−0.661653
$247$	− 9976.96i	− 2.57012i
$248$	− 2125.56i	− 0.544246i
$249$	−1475.49	−0.375523
$250$	0	0
$251$	−1403.96	−0.353056	−0.176528	−	0.984296i	$-0.556487\pi$
$251$	−1403.96	−0.353056	−0.176528	+	0.984296i	$0.556487\pi$
$252$	− 626.489i	− 0.156608i
$253$	− 4787.28i	− 1.18962i
$254$	−10947.1	−2.70425
$255$	0	0
$256$	1452.36	0.354579
$257$	− 1964.86i	− 0.476905i	−0.971154	−	0.238453i	$-0.923360\pi$
$257$	− 1964.86i	− 0.476905i	0.971154	−	0.238453i	$-0.0766402\pi$
$258$	− 2773.14i	− 0.669179i
$259$	338.341	0.0811717
$260$	0	0
$261$	−1303.50	−0.309135
$262$	6019.89i	1.41950i
$263$	393.821i	0.0923347i	0.998934	+	0.0461673i	$0.0147007\pi$
$263$	393.821i	0.0923347i	−0.998934	+	0.0461673i	$0.985299\pi$
$264$	−1026.08	−0.239208
$265$	0	0
$266$	−3325.16	−0.766462
$267$	− 1246.68i	− 0.285751i
$268$	− 12.7769i	− 0.00291222i
$269$	1877.03	0.425444	0.212722	−	0.977113i	$-0.431767\pi$
$269$	1877.03	0.425444	0.212722	+	0.977113i	$0.431767\pi$
$270$	0	0
$271$	−689.909	−0.154646	−0.0773228	−	0.997006i	$-0.524637\pi$
$271$	−689.909	−0.154646	−0.0773228	+	0.997006i	$0.524637\pi$
$272$	5377.16i	1.19867i
$273$	1868.38i	0.414211i
$274$	444.269	0.0979536
$275$	0	0
$276$	−3439.09	−0.750031
$277$	6289.13i	1.36418i	0.731270	+	0.682088i	$0.238928\pi$
$277$	6289.13i	1.36418i	−0.731270	+	0.682088i	$0.761072\pi$
$278$	− 3868.21i	− 0.834531i
$279$	2322.71	0.498412
$280$	0	0
$281$	−1954.87	−0.415010	−0.207505	−	0.978234i	$-0.566534\pi$
$281$	−1954.87	−0.415010	−0.207505	+	0.978234i	$0.566534\pi$
$282$	− 7315.60i	− 1.54481i
$283$	− 5033.96i	− 1.05738i	−0.848816	−	0.528688i	$-0.822684\pi$
$283$	− 5033.96i	− 1.05738i	0.848816	−	0.528688i	$-0.177316\pi$
$284$	610.803	0.127621
$285$	0	0
$286$	15651.2	3.23594
$287$	− 1406.20i	− 0.289217i
$288$	− 2295.86i	− 0.469738i
$289$	−9580.03	−1.94993
$290$	0	0
$291$	−3095.09	−0.623497
$292$	4820.99i	0.966188i
$293$	− 6369.12i	− 1.26993i	−0.772543	−	0.634963i	$-0.781015\pi$
$293$	− 6369.12i	− 1.26993i	0.772543	−	0.634963i	$-0.218985\pi$
$294$	622.702	0.123526
$295$	0	0
$296$	−398.085	−0.0781697
$297$	− 1121.25i	− 0.219063i
$298$	7546.34i	1.46694i
$299$	10256.4	1.98376
$300$	0	0
$301$	1527.52	0.292507
$302$	− 5962.39i	− 1.13608i
$303$	− 4341.58i	− 0.823160i
$304$	−5008.70	−0.944963
$305$	0	0
$306$	4589.71	0.857438
$307$	− 6619.83i	− 1.23066i	−0.788268	−	0.615332i	$-0.789022\pi$
$307$	− 6619.83i	− 1.23066i	0.788268	−	0.615332i	$-0.210978\pi$
$308$	− 2890.75i	− 0.534792i
$309$	490.700	0.0903396
$310$	0	0
$311$	−9909.22	−1.80675	−0.903377	−	0.428848i	$-0.858920\pi$
$311$	−9909.22	−1.80675	−0.903377	+	0.428848i	$0.858920\pi$
$312$	− 2198.30i	− 0.398892i
$313$	422.336i	0.0762678i	0.999273	+	0.0381339i	$0.0121413\pi$
$313$	422.336i	0.0762678i	−0.999273	+	0.0381339i	$0.987859\pi$
$314$	−6773.22	−1.21731
$315$	0	0
$316$	8733.63	1.55476
$317$	− 4902.78i	− 0.868668i	−0.900752	−	0.434334i	$-0.856984\pi$
$317$	− 4902.78i	− 0.868668i	0.900752	−	0.434334i	$-0.143016\pi$
$318$	− 2342.15i	− 0.413022i
$319$	−6014.60	−1.05565
$320$	0	0
$321$	−388.960	−0.0676312
$322$	− 3418.30i	− 0.591597i
$323$	− 13499.9i	− 2.32556i
$324$	−805.486	−0.138115
$325$	0	0
$326$	867.935	0.147455
$327$	1700.04i	0.287500i
$328$	1654.51i	0.278521i
$329$	4029.62	0.675258
$330$	0	0
$331$	−5281.74	−0.877071	−0.438535	−	0.898714i	$-0.644503\pi$
$331$	−5281.74	−0.877071	−0.438535	+	0.898714i	$0.644503\pi$
$332$	4890.89i	0.808501i
$333$	− 435.009i	− 0.0715867i
$334$	−4939.01	−0.809133
$335$	0	0
$336$	937.978	0.152294
$337$	4459.60i	0.720860i	0.932786	+	0.360430i	$0.117370\pi$
$337$	4459.60i	0.720860i	−0.932786	+	0.360430i	$0.882630\pi$
$338$	24225.1i	3.89843i
$339$	−2429.67	−0.389267
$340$	0	0
$341$	10717.5	1.70200
$342$	4275.21i	0.675956i
$343$	343.000i	0.0539949i
$344$	−1797.25	−0.281689
$345$	0	0
$346$	10751.2	1.67048
$347$	5261.97i	0.814056i	0.913416	+	0.407028i	$0.133435\pi$
$347$	5261.97i	0.814056i	−0.913416	+	0.407028i	$0.866565\pi$
$348$	4320.77i	0.665568i
$349$	−960.325	−0.147292	−0.0736461	−	0.997284i	$-0.523464\pi$
$349$	−960.325	−0.147292	−0.0736461	+	0.997284i	$0.523464\pi$
$350$	0	0
$351$	2402.21	0.365300
$352$	− 10593.6i	− 1.60409i
$353$	8925.80i	1.34581i	0.739727	+	0.672907i	$0.234955\pi$
$353$	8925.80i	1.34581i	−0.739727	+	0.672907i	$0.765045\pi$
$354$	−1925.31	−0.289066
$355$	0	0
$356$	−4132.45	−0.615222
$357$	2528.13i	0.374797i
$358$	1661.62i	0.245306i
$359$	3056.27	0.449314	0.224657	−	0.974438i	$-0.427874\pi$
$359$	3056.27	0.449314	0.224657	+	0.974438i	$0.427874\pi$
$360$	0	0
$361$	5715.88	0.833340
$362$	12615.4i	1.83162i
$363$	− 1180.69i	− 0.170717i
$364$	6193.23	0.891796
$365$	0	0
$366$	6733.18	0.961610
$367$	− 1813.52i	− 0.257943i	−0.991648	−	0.128971i	$-0.958832\pi$
$367$	− 1813.52i	− 0.257943i	0.991648	−	0.128971i	$-0.0411675\pi$
$368$	− 5148.99i	− 0.729375i
$369$	−1807.97	−0.255065
$370$	0	0
$371$	1290.11	0.180537
$372$	− 7699.22i	− 1.07308i
$373$	4517.48i	0.627094i	0.949573	+	0.313547i	$0.101517\pi$
$373$	4517.48i	0.627094i	−0.949573	+	0.313547i	$0.898483\pi$
$374$	21177.9	2.92802
$375$	0	0
$376$	−4741.17	−0.650285
$377$	− 12885.9i	− 1.76036i
$378$	− 800.617i	− 0.108940i
$379$	4931.24	0.668340	0.334170	−	0.942513i	$-0.391544\pi$
$379$	4931.24	0.668340	0.334170	+	0.942513i	$0.391544\pi$
$380$	0	0
$381$	−7752.75	−1.04248
$382$	4648.53i	0.622617i
$383$	1482.37i	0.197770i	0.995099	+	0.0988849i	$0.0315276\pi$
$383$	1482.37i	0.197770i	−0.995099	+	0.0988849i	$0.968472\pi$
$384$	−3069.25	−0.407883
$385$	0	0
$386$	−14827.5	−1.95519
$387$	− 1963.95i	− 0.257967i
$388$	10259.5i	1.34239i
$389$	5448.98	0.710217	0.355109	−	0.934825i	$-0.384444\pi$
$389$	5448.98	0.710217	0.355109	+	0.934825i	$0.384444\pi$
$390$	0	0
$391$	13878.0	1.79500
$392$	− 403.567i	− 0.0519980i
$393$	4263.31i	0.547215i
$394$	6667.38	0.852532
$395$	0	0
$396$	−3716.68	−0.471642
$397$	− 13675.9i	− 1.72891i	−0.502713	−	0.864453i	$-0.667665\pi$
$397$	− 13675.9i	− 1.72891i	0.502713	−	0.864453i	$-0.332335\pi$
$398$	− 14388.3i	− 1.81212i
$399$	−2354.89	−0.295469
$400$	0	0
$401$	14109.9	1.75714	0.878570	−	0.477613i	$-0.158498\pi$
$401$	14109.9	1.75714	0.878570	+	0.477613i	$0.158498\pi$
$402$	− 16.3282i	− 0.00202581i
$403$	22961.4i	2.83819i
$404$	−14391.3	−1.77226
$405$	0	0
$406$	−4294.65	−0.524975
$407$	− 2007.22i	− 0.244458i
$408$	− 2974.55i	− 0.360936i
$409$	13995.6	1.69203	0.846015	−	0.533159i	$-0.178995\pi$
$409$	13995.6	1.69203	0.846015	+	0.533159i	$0.178995\pi$
$410$	0	0
$411$	314.633	0.0377609
$412$	− 1626.55i	− 0.194501i
$413$	− 1060.51i	− 0.126354i
$414$	−4394.95	−0.521740
$415$	0	0
$416$	22696.0	2.67491
$417$	− 2739.48i	− 0.321709i
$418$	19726.7i	2.30829i
$419$	9840.61	1.14736	0.573682	−	0.819078i	$-0.305515\pi$
$419$	9840.61	1.14736	0.573682	+	0.819078i	$0.305515\pi$
$420$	0	0
$421$	−12660.5	−1.46564	−0.732822	−	0.680420i	$-0.761797\pi$
$421$	−12660.5	−1.46564	−0.732822	+	0.680420i	$0.761797\pi$
$422$	− 14139.2i	− 1.63101i
$423$	− 5180.93i	− 0.595522i
$424$	−1517.92	−0.173860
$425$	0	0
$426$	780.571	0.0887765
$427$	3708.81i	0.420332i
$428$	1289.31i	0.145610i
$429$	11084.3	1.24744
$430$	0	0
$431$	−4578.91	−0.511736	−0.255868	−	0.966712i	$-0.582361\pi$
$431$	−4578.91	−0.511736	−0.255868	+	0.966712i	$0.582361\pi$
$432$	− 1205.97i	− 0.134311i
$433$	3279.88i	0.364020i	0.983297	+	0.182010i	$0.0582604\pi$
$433$	3279.88i	0.364020i	−0.983297	+	0.182010i	$0.941740\pi$
$434$	7652.68	0.846406
$435$	0	0
$436$	5635.23	0.618988
$437$	12927.1i	1.41507i
$438$	6160.94i	0.672103i
$439$	427.807	0.0465105	0.0232552	−	0.999730i	$-0.492597\pi$
$439$	427.807	0.0465105	0.0232552	+	0.999730i	$0.492597\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	45372.1i	4.88265i
$443$	− 15441.2i	− 1.65605i	−0.560688	−	0.828027i	$-0.689463\pi$
$443$	− 15441.2i	− 1.65605i	0.560688	−	0.828027i	$-0.310537\pi$
$444$	−1441.95	−0.154126
$445$	0	0
$446$	−539.371	−0.0572645
$447$	5344.35i	0.565501i
$448$	− 5062.93i	− 0.533931i
$449$	−9382.02	−0.986113	−0.493057	−	0.869997i	$-0.664120\pi$
$449$	−9382.02	−0.986113	−0.493057	+	0.869997i	$0.664120\pi$
$450$	0	0
$451$	−8342.34	−0.871010
$452$	8053.77i	0.838091i
$453$	− 4222.59i	− 0.437957i
$454$	16284.0	1.68336
$455$	0	0
$456$	2770.72	0.284542
$457$	13570.4i	1.38905i	0.719469	+	0.694524i	$0.244385\pi$
$457$	13570.4i	1.38905i	−0.719469	+	0.694524i	$0.755615\pi$
$458$	10746.7i	1.09642i
$459$	3250.45	0.330540
$460$	0	0
$461$	1251.88	0.126477	0.0632386	−	0.997998i	$-0.479857\pi$
$461$	1251.88	0.126477	0.0632386	+	0.997998i	$0.479857\pi$
$462$	− 3694.21i	− 0.372014i
$463$	− 7934.36i	− 0.796417i	−0.917295	−	0.398209i	$-0.869632\pi$
$463$	− 7934.36i	− 0.796417i	0.917295	−	0.398209i	$-0.130368\pi$
$464$	−6469.05	−0.647237
$465$	0	0
$466$	−16891.1	−1.67911
$467$	7583.76i	0.751466i	0.926728	+	0.375733i	$0.122609\pi$
$467$	7583.76i	0.751466i	−0.926728	+	0.375733i	$0.877391\pi$
$468$	− 7962.73i	− 0.786490i
$469$	8.99396	0.000885507 0
$470$	0	0
$471$	−4796.82	−0.469269
$472$	1247.78i	0.121681i
$473$	− 9062.07i	− 0.880919i
$474$	11161.1	1.08153
$475$	0	0
$476$	8380.13	0.806938
$477$	− 1658.72i	− 0.159219i
$478$	− 14263.6i	− 1.36486i
$479$	5829.34	0.556053	0.278027	−	0.960573i	$-0.410320\pi$
$479$	5829.34	0.556053	0.278027	+	0.960573i	$0.410320\pi$
$480$	0	0
$481$	4300.34	0.407648
$482$	3980.00i	0.376108i
$483$	− 2420.85i	− 0.228059i
$484$	−3913.70	−0.367553
$485$	0	0
$486$	−1029.36	−0.0960760
$487$	− 19902.1i	− 1.85185i	−0.377708	−	0.925925i	$-0.623288\pi$
$487$	− 19902.1i	− 1.85185i	0.377708	−	0.925925i	$-0.376712\pi$
$488$	− 4363.71i	− 0.404787i
$489$	614.675	0.0568437
$490$	0	0
$491$	−16821.6	−1.54613	−0.773065	−	0.634327i	$-0.781277\pi$
$491$	−16821.6	−1.54613	−0.773065	+	0.634327i	$0.781277\pi$
$492$	5992.98i	0.549155i
$493$	− 17436.0i	− 1.59285i
$494$	−42263.1	−3.84920
$495$	0	0
$496$	11527.3	1.04353
$497$	429.958i	0.0388054i
$498$	6250.27i	0.562412i
$499$	−6031.83	−0.541126	−0.270563	−	0.962702i	$-0.587210\pi$
$499$	−6031.83	−0.541126	−0.270563	+	0.962702i	$0.587210\pi$
$500$	0	0
$501$	−3497.82	−0.311919
$502$	5947.27i	0.528764i
$503$	17176.4i	1.52258i	0.648412	+	0.761290i	$0.275434\pi$
$503$	17176.4i	1.52258i	−0.648412	+	0.761290i	$0.724566\pi$
$504$	−518.872	−0.0458580
$505$	0	0
$506$	−20279.2	−1.78166
$507$	17156.3i	1.50284i
$508$	25698.5i	2.24446i
$509$	−4706.59	−0.409854	−0.204927	−	0.978777i	$-0.565696\pi$
$509$	−4706.59	−0.409854	−0.204927	+	0.978777i	$0.565696\pi$
$510$	0	0
$511$	−3393.60	−0.293785
$512$	− 14336.9i	− 1.23752i
$513$	3027.72i	0.260579i
$514$	−8323.28	−0.714250
$515$	0	0
$516$	−6510.02	−0.555402
$517$	− 23905.9i	− 2.03362i
$518$	− 1433.23i	− 0.121569i
$519$	7614.01	0.643965
$520$	0	0
$521$	−8557.18	−0.719572	−0.359786	−	0.933035i	$-0.617150\pi$
$521$	−8557.18	−0.719572	−0.359786	+	0.933035i	$0.617150\pi$
$522$	5521.69i	0.462985i
$523$	− 18248.5i	− 1.52572i	−0.646566	−	0.762858i	$-0.723796\pi$
$523$	− 18248.5i	− 1.52572i	0.646566	−	0.762858i	$-0.276204\pi$
$524$	14131.8	1.17815
$525$	0	0
$526$	1668.25	0.138287
$527$	31069.3i	2.56813i
$528$	− 5564.60i	− 0.458652i
$529$	−1122.16	−0.0922302
$530$	0	0
$531$	−1363.51	−0.111434
$532$	7805.90i	0.636144i
$533$	− 17872.9i	− 1.45246i
$534$	−5281.03	−0.427963
$535$	0	0
$536$	−10.5821	−0.000852758 0
$537$	1176.77i	0.0945646i
$538$	− 7951.22i	− 0.637177i
$539$	2034.87	0.162612
$540$	0	0
$541$	−5734.17	−0.455696	−0.227848	−	0.973697i	$-0.573169\pi$
$541$	−5734.17	−0.455696	−0.227848	+	0.973697i	$0.573169\pi$
$542$	2922.50i	0.231609i
$543$	8934.24i	0.706087i
$544$	30710.1	2.42038
$545$	0	0
$546$	7914.59	0.620354
$547$	− 8002.52i	− 0.625527i	−0.949831	−	0.312763i	$-0.898745\pi$
$547$	− 8002.52i	− 0.625527i	0.949831	−	0.312763i	$-0.101255\pi$
$548$	− 1042.93i	− 0.0812991i
$549$	4768.47	0.370698
$550$	0	0
$551$	16241.2	1.25572
$552$	2848.33i	0.219625i
$553$	6147.80i	0.472750i
$554$	26641.2	2.04310
$555$	0	0
$556$	−9080.71	−0.692640
$557$	1276.82i	0.0971289i	0.998820	+	0.0485644i	$0.0154646\pi$
$557$	1276.82i	0.0971289i	−0.998820	+	0.0485644i	$0.984535\pi$
$558$	− 9839.16i	− 0.746460i
$559$	19414.9	1.46898
$560$	0	0
$561$	14998.2	1.12875
$562$	8280.96i	0.621550i
$563$	− 11027.7i	− 0.825507i	−0.910843	−	0.412753i	$-0.864567\pi$
$563$	− 11027.7i	− 0.825507i	0.910843	−	0.412753i	$-0.135433\pi$
$564$	−17173.5	−1.28216
$565$	0	0
$566$	−21324.2	−1.58361
$567$	− 567.000i	− 0.0419961i
$568$	− 505.881i	− 0.0373702i
$569$	4519.03	0.332948	0.166474	−	0.986046i	$-0.446762\pi$
$569$	4519.03	0.332948	0.166474	+	0.986046i	$0.446762\pi$
$570$	0	0
$571$	3598.81	0.263758	0.131879	−	0.991266i	$-0.457899\pi$
$571$	3598.81	0.263758	0.131879	+	0.991266i	$0.457899\pi$
$572$	− 36741.7i	− 2.68575i
$573$	3292.11i	0.240017i
$574$	−5956.75	−0.433153
$575$	0	0
$576$	−6509.48	−0.470883
$577$	− 3439.23i	− 0.248140i	−0.992273	−	0.124070i	$-0.960405\pi$
$577$	− 3439.23i	− 0.248140i	0.992273	−	0.124070i	$-0.0395948\pi$
$578$	40581.6i	2.92037i
$579$	−10500.9	−0.753720
$580$	0	0
$581$	−3442.81	−0.245838
$582$	13111.0i	0.933797i
$583$	− 7653.66i	− 0.543709i
$584$	3992.85	0.282920
$585$	0	0
$586$	−26980.0	−1.90194
$587$	21285.2i	1.49665i	0.663330	+	0.748327i	$0.269143\pi$
$587$	21285.2i	1.49665i	−0.663330	+	0.748327i	$0.730857\pi$
$588$	− 1461.81i	− 0.102524i
$589$	−28940.4	−2.02456
$590$	0	0
$591$	4721.87	0.328649
$592$	− 2158.89i	− 0.149881i
$593$	14200.8i	0.983404i	0.870764	+	0.491702i	$0.163625\pi$
$593$	14200.8i	0.983404i	−0.870764	+	0.491702i	$0.836375\pi$
$594$	−4749.70	−0.328085
$595$	0	0
$596$	17715.2	1.21752
$597$	− 10189.9i	− 0.698566i
$598$	− 43446.8i	− 2.97103i
$599$	8885.05	0.606065	0.303033	−	0.952980i	$-0.402001\pi$
$599$	8885.05	0.606065	0.303033	+	0.952980i	$0.402001\pi$
$600$	0	0
$601$	−2052.89	−0.139333	−0.0696664	−	0.997570i	$-0.522193\pi$
$601$	−2052.89	−0.139333	−0.0696664	+	0.997570i	$0.522193\pi$
$602$	− 6470.67i	− 0.438081i
$603$	− 11.5637i	0 0.000780943i
$604$	−13996.9	−0.942920
$605$	0	0
$606$	−18391.2	−1.23283
$607$	10280.0i	0.687404i	0.939079	+	0.343702i	$0.111681\pi$
$607$	10280.0i	0.687404i	−0.939079	+	0.343702i	$0.888319\pi$
$608$	28605.8i	1.90809i
$609$	−3041.49	−0.202377
$610$	0	0
$611$	51216.8	3.39118
$612$	− 10774.4i	− 0.711652i
$613$	− 23409.5i	− 1.54242i	−0.636584	−	0.771208i	$-0.719653\pi$
$613$	− 23409.5i	− 1.54242i	0.636584	−	0.771208i	$-0.280347\pi$
$614$	−28042.0	−1.84314
$615$	0	0
$616$	−2394.18	−0.156598
$617$	− 6632.75i	− 0.432779i	−0.976307	−	0.216389i	$-0.930572\pi$
$617$	− 6632.75i	− 0.432779i	0.976307	−	0.216389i	$-0.0694281\pi$
$618$	− 2078.64i	− 0.135299i
$619$	−10734.0	−0.696990	−0.348495	−	0.937311i	$-0.613307\pi$
$619$	−10734.0	−0.696990	−0.348495	+	0.937311i	$0.613307\pi$
$620$	0	0
$621$	−3112.52	−0.201129
$622$	41976.1i	2.70593i
$623$	− 2908.92i	− 0.187068i
$624$	11921.8	0.764829
$625$	0	0
$626$	1789.04	0.114225
$627$	13970.5i	0.889839i
$628$	15900.3i	1.01034i
$629$	5818.83	0.368858
$630$	0	0
$631$	−17071.0	−1.07700	−0.538499	−	0.842626i	$-0.681008\pi$
$631$	−17071.0	−1.07700	−0.538499	+	0.842626i	$0.681008\pi$
$632$	− 7233.38i	− 0.455267i
$633$	− 10013.4i	− 0.628749i
$634$	−20768.5	−1.30098
$635$	0	0
$636$	−5498.24	−0.342798
$637$	4359.56i	0.271165i
$638$	25478.2i	1.58102i
$639$	552.804	0.0342231
$640$	0	0
$641$	−19389.7	−1.19477	−0.597386	−	0.801954i	$-0.703794\pi$
$641$	−19389.7	−1.19477	−0.597386	+	0.801954i	$0.703794\pi$
$642$	1647.66i	0.101290i
$643$	25409.3i	1.55839i	0.626780	+	0.779196i	$0.284372\pi$
$643$	25409.3i	1.55839i	−0.626780	+	0.779196i	$0.715628\pi$
$644$	−8024.54	−0.491011
$645$	0	0
$646$	−57186.6	−3.48294
$647$	− 6039.08i	− 0.366956i	−0.983024	−	0.183478i	$-0.941264\pi$
$647$	− 6039.08i	− 0.366956i	0.983024	−	0.183478i	$-0.0587357\pi$
$648$	667.122i	0.0404429i
$649$	−6291.54	−0.380531
$650$	0	0
$651$	5419.66	0.326287
$652$	− 2037.50i	− 0.122384i
$653$	30666.2i	1.83776i	0.394532	+	0.918882i	$0.370907\pi$
$653$	30666.2i	1.83776i	−0.394532	+	0.918882i	$0.629093\pi$
$654$	7201.50	0.430582
$655$	0	0
$656$	−8972.67	−0.534031
$657$	4363.20i	0.259094i
$658$	− 17069.7i	− 1.01132i
$659$	2765.96	0.163500	0.0817500	−	0.996653i	$-0.473949\pi$
$659$	2765.96	0.163500	0.0817500	+	0.996653i	$0.473949\pi$
$660$	0	0
$661$	27261.8	1.60418	0.802089	−	0.597204i	$-0.203722\pi$
$661$	27261.8	1.60418	0.802089	+	0.597204i	$0.203722\pi$
$662$	22373.8i	1.31357i
$663$	32132.7i	1.88225i
$664$	4050.74	0.236746
$665$	0	0
$666$	−1842.73	−0.107214
$667$	16696.1i	0.969230i
$668$	11594.4i	0.671560i
$669$	−381.985	−0.0220753
$670$	0	0
$671$	22002.7	1.26588
$672$	− 5357.00i	− 0.307516i
$673$	1048.17i	0.0600356i	0.999549	+	0.0300178i	$0.00955640\pi$
$673$	1048.17i	0.0600356i	−0.999549	+	0.0300178i	$0.990444\pi$
$674$	18891.2	1.07961
$675$	0	0
$676$	56869.0	3.23560
$677$	− 34554.7i	− 1.96166i	−0.194860	−	0.980831i	$-0.562425\pi$
$677$	− 34554.7i	− 1.96166i	0.194860	−	0.980831i	$-0.437575\pi$
$678$	10292.2i	0.582996i
$679$	−7221.89	−0.408175
$680$	0	0
$681$	11532.4	0.648930
$682$	− 45399.9i	− 2.54905i
$683$	− 14711.6i	− 0.824192i	−0.911140	−	0.412096i	$-0.864797\pi$
$683$	− 14711.6i	− 0.824192i	0.911140	−	0.412096i	$-0.135203\pi$
$684$	10036.2	0.561026
$685$	0	0
$686$	1452.97	0.0808669
$687$	7610.85i	0.422667i
$688$	− 9746.79i	− 0.540106i
$689$	16397.4	0.906666
$690$	0	0
$691$	−24522.6	−1.35005	−0.675024	−	0.737796i	$-0.735867\pi$
$691$	−24522.6	−1.35005	−0.675024	+	0.737796i	$0.735867\pi$
$692$	− 25238.6i	− 1.38646i
$693$	− 2616.26i	− 0.143410i
$694$	22290.1	1.21919
$695$	0	0
$696$	3578.56	0.194892
$697$	− 24184.0i	− 1.31425i
$698$	4068.00i	0.220596i
$699$	−11962.3	−0.647291
$700$	0	0
$701$	19912.2	1.07286	0.536429	−	0.843946i	$-0.319773\pi$
$701$	19912.2	1.07286	0.536429	+	0.843946i	$0.319773\pi$
$702$	− 10175.9i	− 0.547101i
$703$	5420.11i	0.290787i
$704$	−30036.1	−1.60799
$705$	0	0
$706$	37810.3	2.01559
$707$	− 10130.4i	− 0.538885i
$708$	4519.72i	0.239917i
$709$	−6208.79	−0.328880	−0.164440	−	0.986387i	$-0.552582\pi$
$709$	−6208.79	−0.328880	−0.164440	+	0.986387i	$0.552582\pi$
$710$	0	0
$711$	7904.31	0.416927
$712$	3422.58i	0.180150i
$713$	− 29751.0i	− 1.56267i
$714$	10709.3	0.561325
$715$	0	0
$716$	3900.69	0.203597
$717$	− 10101.5i	− 0.526149i
$718$	− 12946.6i	− 0.672927i
$719$	−13063.6	−0.677593	−0.338797	−	0.940860i	$-0.610020\pi$
$719$	−13063.6	−0.677593	−0.338797	+	0.940860i	$0.610020\pi$
$720$	0	0
$721$	1144.97	0.0591411
$722$	− 24212.9i	− 1.24807i
$723$	2818.65i	0.144989i
$724$	29614.8	1.52020
$725$	0	0
$726$	−5001.49	−0.255678
$727$	− 12897.0i	− 0.657940i	−0.944340	−	0.328970i	$-0.893298\pi$
$727$	− 12897.0i	− 0.657940i	0.944340	−	0.328970i	$-0.106702\pi$
$728$	− 5129.37i	− 0.261136i
$729$	−729.000	−0.0370370
$730$	0	0
$731$	26270.5	1.32920
$732$	− 15806.3i	− 0.798112i
$733$	− 11699.6i	− 0.589540i	−0.955568	−	0.294770i	$-0.904757\pi$
$733$	− 11699.6i	− 0.589540i	0.955568	−	0.294770i	$-0.0952430\pi$
$734$	−7682.19	−0.386315
$735$	0	0
$736$	−29407.0	−1.47277
$737$	− 53.3571i	− 0.00266681i
$738$	7658.68i	0.382005i
$739$	−14974.0	−0.745368	−0.372684	−	0.927958i	$-0.621562\pi$
$739$	−14974.0	−0.745368	−0.372684	+	0.927958i	$0.621562\pi$
$740$	0	0
$741$	−29930.9	−1.48386
$742$	− 5465.01i	− 0.270386i
$743$	− 18500.7i	− 0.913492i	−0.889597	−	0.456746i	$-0.849015\pi$
$743$	− 18500.7i	− 0.913492i	0.889597	−	0.456746i	$-0.150985\pi$
$744$	−6376.67	−0.314220
$745$	0	0
$746$	19136.3	0.939184
$747$	4426.47i	0.216809i
$748$	− 49715.5i	− 2.43019i
$749$	−907.572	−0.0442750
$750$	0	0
$751$	−26348.4	−1.28025	−0.640125	−	0.768271i	$-0.721117\pi$
$751$	−26348.4	−1.28025	−0.640125	+	0.768271i	$0.721117\pi$
$752$	− 25712.2i	− 1.24684i
$753$	4211.88i	0.203837i
$754$	−54585.4	−2.63645
$755$	0	0
$756$	−1879.47	−0.0904174
$757$	− 28061.7i	− 1.34732i	−0.739042	−	0.673659i	$-0.764722\pi$
$757$	− 28061.7i	− 1.34732i	0.739042	−	0.673659i	$-0.235278\pi$
$758$	− 20889.1i	− 1.00096i
$759$	−14361.8	−0.686826
$760$	0	0
$761$	−3579.22	−0.170495	−0.0852476	−	0.996360i	$-0.527168\pi$
$761$	−3579.22	−0.170495	−0.0852476	+	0.996360i	$0.527168\pi$
$762$	32841.2i	1.56130i
$763$	3966.77i	0.188213i
$764$	10912.5	0.516757
$765$	0	0
$766$	6279.44	0.296195
$767$	− 13479.2i	− 0.634557i
$768$	− 4357.07i	− 0.204716i
$769$	−4339.61	−0.203499	−0.101749	−	0.994810i	$-0.532444\pi$
$769$	−4339.61	−0.203499	−0.101749	+	0.994810i	$0.532444\pi$
$770$	0	0
$771$	−5894.58	−0.275341
$772$	34808.0i	1.62276i
$773$	− 10005.1i	− 0.465537i	−0.972532	−	0.232769i	$-0.925222\pi$
$773$	− 10005.1i	− 0.465537i	0.972532	−	0.232769i	$-0.0747785\pi$
$774$	−8319.43	−0.386351
$775$	0	0
$776$	8497.14	0.393079
$777$	− 1015.02i	− 0.0468645i
$778$	− 23082.3i	− 1.06368i
$779$	22526.8	1.03608
$780$	0	0
$781$	2550.75	0.116867
$782$	− 58788.4i	− 2.68832i
$783$	3910.49i	0.178479i
$784$	2188.62	0.0997001
$785$	0	0
$786$	18059.7	0.819550
$787$	17826.8i	0.807443i	0.914882	+	0.403721i	$0.132284\pi$
$787$	17826.8i	0.807443i	−0.914882	+	0.403721i	$0.867716\pi$
$788$	− 15651.8i	− 0.707581i
$789$	1181.46	0.0533094
$790$	0	0
$791$	−5669.23	−0.254835
$792$	3078.24i	0.138107i
$793$	47139.3i	2.11093i
$794$	−57932.2	−2.58934
$795$	0	0
$796$	−33777.0	−1.50401
$797$	36723.0i	1.63211i	0.577971	+	0.816057i	$0.303845\pi$
$797$	36723.0i	1.63211i	−0.577971	+	0.816057i	$0.696155\pi$
$798$	9975.49i	0.442517i
$799$	69301.9	3.06849
$800$	0	0
$801$	−3740.04	−0.164979
$802$	− 59770.4i	− 2.63163i
$803$	20132.7i	0.884767i
$804$	−38.3307	−0.00168137
$805$	0	0
$806$	97266.2	4.25069
$807$	− 5631.08i	− 0.245630i
$808$	11919.2i	0.518955i
$809$	−5657.55	−0.245870	−0.122935	−	0.992415i	$-0.539231\pi$
$809$	−5657.55	−0.245870	−0.122935	+	0.992415i	$0.539231\pi$
$810$	0	0
$811$	7532.41	0.326139	0.163070	−	0.986615i	$-0.447861\pi$
$811$	7532.41	0.326139	0.163070	+	0.986615i	$0.447861\pi$
$812$	10081.8i	0.435716i
$813$	2069.73i	0.0892847i
$814$	−8502.73	−0.366119
$815$	0	0
$816$	16131.5	0.692053
$817$	24470.3i	1.04787i
$818$	− 59286.5i	− 2.53411i
$819$	5605.15	0.239145
$820$	0	0
$821$	−6489.25	−0.275854	−0.137927	−	0.990442i	$-0.544044\pi$
$821$	−6489.25	−0.275854	−0.137927	+	0.990442i	$0.544044\pi$
$822$	− 1332.81i	− 0.0565535i
$823$	− 7901.57i	− 0.334668i	−0.985900	−	0.167334i	$-0.946484\pi$
$823$	− 7901.57i	− 0.334668i	0.985900	−	0.167334i	$-0.0535157\pi$
$824$	−1347.15	−0.0569539
$825$	0	0
$826$	−4492.40	−0.189238
$827$	− 37815.8i	− 1.59007i	−0.606566	−	0.795033i	$-0.707453\pi$
$827$	− 37815.8i	− 1.59007i	0.606566	−	0.795033i	$-0.292547\pi$
$828$	10317.3i	0.433031i
$829$	−26073.5	−1.09236	−0.546182	−	0.837667i	$-0.683919\pi$
$829$	−26073.5	−1.09236	−0.546182	+	0.837667i	$0.683919\pi$
$830$	0	0
$831$	18867.4	0.787608
$832$	− 64350.2i	− 2.68142i
$833$	5898.96i	0.245362i
$834$	−11604.6	−0.481817
$835$	0	0
$836$	46308.9	1.91582
$837$	− 6968.13i	− 0.287758i
$838$	− 41685.5i	− 1.71838i
$839$	15590.3	0.641523	0.320762	−	0.947160i	$-0.396061\pi$
$839$	15590.3	0.641523	0.320762	+	0.947160i	$0.396061\pi$
$840$	0	0
$841$	−3412.46	−0.139918
$842$	53630.8i	2.19506i
$843$	5864.61i	0.239606i
$844$	−33192.1	−1.35370
$845$	0	0
$846$	−21946.8	−0.891899
$847$	− 2754.94i	− 0.111760i
$848$	− 8231.96i	− 0.333357i
$849$	−15101.9	−0.610477
$850$	0	0
$851$	−5571.92	−0.224445
$852$	− 1832.41i	− 0.0736823i
$853$	17476.1i	0.701488i	0.936471	+	0.350744i	$0.114071\pi$
$853$	17476.1i	0.701488i	−0.936471	+	0.350744i	$0.885929\pi$
$854$	15710.8	0.629521
$855$	0	0
$856$	1067.83	0.0426376
$857$	− 5694.54i	− 0.226980i	−0.993539	−	0.113490i	$-0.963797\pi$
$857$	− 5694.54i	− 0.226980i	0.993539	−	0.113490i	$-0.0362030\pi$
$858$	− 46953.7i	− 1.86827i
$859$	−27313.6	−1.08490	−0.542448	−	0.840089i	$-0.682503\pi$
$859$	−27313.6	−1.08490	−0.542448	+	0.840089i	$0.682503\pi$
$860$	0	0
$861$	−4218.59	−0.166979
$862$	19396.6i	0.766415i
$863$	9046.07i	0.356815i	0.983957	+	0.178408i	$0.0570946\pi$
$863$	9046.07i	0.356815i	−0.983957	+	0.178408i	$0.942905\pi$
$864$	−6887.57	−0.271204
$865$	0	0
$866$	13893.8	0.545185
$867$	28740.1i	1.12580i
$868$	− 17964.8i	− 0.702496i
$869$	36472.1	1.42374
$870$	0	0
$871$	114.314	0.00444705
$872$	− 4667.22i	− 0.181252i
$873$	9285.28i	0.359976i
$874$	54760.0	2.11932
$875$	0	0
$876$	14463.0	0.557829
$877$	− 2104.29i	− 0.0810224i	−0.999179	−	0.0405112i	$-0.987101\pi$
$877$	− 2104.29i	− 0.0810224i	0.999179	−	0.0405112i	$-0.0128986\pi$
$878$	− 1812.22i	− 0.0696576i
$879$	−19107.4	−0.733192
$880$	0	0
$881$	−22589.6	−0.863861	−0.431931	−	0.901907i	$-0.642167\pi$
$881$	−22589.6	−0.863861	−0.431931	+	0.901907i	$0.642167\pi$
$882$	− 1868.11i	− 0.0713179i
$883$	2419.71i	0.0922193i	0.998936	+	0.0461096i	$0.0146824\pi$
$883$	2419.71i	0.0922193i	−0.998936	+	0.0461096i	$0.985318\pi$
$884$	106512.	4.05248
$885$	0	0
$886$	−65409.8	−2.48023
$887$	13177.0i	0.498806i	0.968400	+	0.249403i	$0.0802344\pi$
$887$	13177.0i	0.498806i	−0.968400	+	0.249403i	$0.919766\pi$
$888$	1194.26i	0.0451313i
$889$	−18089.8	−0.682464
$890$	0	0
$891$	−3363.76	−0.126476
$892$	1266.19i	0.0475281i
$893$	64553.2i	2.41902i
$894$	22639.0	0.846937
$895$	0	0
$896$	−7161.58	−0.267022
$897$	− 30769.2i	− 1.14532i
$898$	39742.9i	1.47688i
$899$	−37378.3	−1.38669
$900$	0	0
$901$	22187.5	0.820393
$902$	35338.7i	1.30449i
$903$	− 4582.55i	− 0.168879i
$904$	6670.31	0.245411
$905$	0	0
$906$	−17887.2	−0.655918
$907$	9189.14i	0.336406i	0.985752	+	0.168203i	$0.0537964\pi$
$907$	9189.14i	0.336406i	−0.985752	+	0.168203i	$0.946204\pi$
$908$	− 38227.0i	− 1.39714i
$909$	−13024.8	−0.475252
$910$	0	0
$911$	−17045.8	−0.619928	−0.309964	−	0.950748i	$-0.600317\pi$
$911$	−17045.8	−0.619928	−0.309964	+	0.950748i	$0.600317\pi$
$912$	15026.1i	0.545575i
$913$	20424.6i	0.740369i
$914$	57485.0	2.08034
$915$	0	0
$916$	25228.1	0.910001
$917$	9947.71i	0.358236i
$918$	− 13769.1i	− 0.495042i
$919$	30825.0	1.10645	0.553223	−	0.833033i	$-0.313398\pi$
$919$	30825.0	1.10645	0.553223	+	0.833033i	$0.313398\pi$
$920$	0	0
$921$	−19859.5	−0.710524
$922$	− 5303.06i	− 0.189422i
$923$	5464.81i	0.194882i
$924$	−8672.25	−0.308762
$925$	0	0
$926$	−33610.5	−1.19277
$927$	− 1472.10i	− 0.0521576i
$928$	36946.2i	1.30691i
$929$	5785.88	0.204336	0.102168	−	0.994767i	$-0.467422\pi$
$929$	5785.88	0.204336	0.102168	+	0.994767i	$0.467422\pi$
$930$	0	0
$931$	−5494.75	−0.193430
$932$	39652.2i	1.39362i
$933$	29727.7i	1.04313i
$934$	32125.3	1.12545
$935$	0	0
$936$	−6594.91	−0.230301
$937$	− 13680.9i	− 0.476986i	−0.971144	−	0.238493i	$-0.923347\pi$
$937$	− 13680.9i	− 0.476986i	0.971144	−	0.238493i	$-0.0766534\pi$
$938$	− 38.0990i	− 0.00132620i
$939$	1267.01	0.0440333
$940$	0	0
$941$	−45448.8	−1.57448	−0.787242	−	0.616644i	$-0.788492\pi$
$941$	−45448.8	−1.57448	−0.787242	+	0.616644i	$0.788492\pi$
$942$	20319.6i	0.702813i
$943$	23157.8i	0.799705i
$944$	−6766.91	−0.233310
$945$	0	0
$946$	−38387.6	−1.31933
$947$	− 7788.45i	− 0.267255i	−0.991032	−	0.133628i	$-0.957337\pi$
$947$	− 7788.45i	− 0.267255i	0.991032	−	0.133628i	$-0.0426626\pi$
$948$	− 26200.9i	− 0.897642i
$949$	−43133.0	−1.47540
$950$	0	0
$951$	−14708.4	−0.501526
$952$	− 6940.61i	− 0.236288i
$953$	− 6149.43i	− 0.209024i	−0.994524	−	0.104512i	$-0.966672\pi$
$953$	− 6149.43i	− 0.209024i	0.994524	−	0.104512i	$-0.0333280\pi$
$954$	−7026.44	−0.238458
$955$	0	0
$956$	−33484.2	−1.13280
$957$	18043.8i	0.609481i
$958$	− 24693.5i	− 0.832788i
$959$	734.144	0.0247203
$960$	0	0
$961$	36813.7	1.23573
$962$	− 18216.5i	− 0.610524i
$963$	1166.88i	0.0390469i
$964$	9343.15	0.312160
$965$	0	0
$966$	−10254.9	−0.341559
$967$	23902.9i	0.794896i	0.917625	+	0.397448i	$0.130104\pi$
$967$	23902.9i	0.794896i	−0.917625	+	0.397448i	$0.869896\pi$
$968$	3241.42i	0.107627i
$969$	−40499.8	−1.34266
$970$	0	0
$971$	−8015.06	−0.264898	−0.132449	−	0.991190i	$-0.542284\pi$
$971$	−8015.06	−0.264898	−0.132449	+	0.991190i	$0.542284\pi$
$972$	2416.46i	0.0797407i
$973$	− 6392.12i	− 0.210608i
$974$	−84306.7	−2.77347
$975$	0	0
$976$	23665.2	0.776131
$977$	− 34861.1i	− 1.14156i	−0.821103	−	0.570780i	$-0.806641\pi$
$977$	− 34861.1i	− 1.14156i	0.821103	−	0.570780i	$-0.193359\pi$
$978$	− 2603.80i	− 0.0851334i
$979$	−17257.3	−0.563378
$980$	0	0
$981$	5100.13	0.165988
$982$	71257.6i	2.31560i
$983$	− 6620.83i	− 0.214824i	−0.994215	−	0.107412i	$-0.965744\pi$
$983$	− 6620.83i	− 0.214824i	0.994215	−	0.107412i	$-0.0342563\pi$
$984$	4963.52	0.160804
$985$	0	0
$986$	−73860.0	−2.38558
$987$	− 12088.8i	− 0.389860i
$988$	99213.6i	3.19474i
$989$	−25155.7	−0.808802
$990$	0	0
$991$	10360.1	0.332089	0.166045	−	0.986118i	$-0.446900\pi$
$991$	10360.1	0.332089	0.166045	+	0.986118i	$0.446900\pi$
$992$	− 65834.7i	− 2.10711i
$993$	15845.2i	0.506377i
$994$	1821.33	0.0581179
$995$	0	0
$996$	14672.7	0.466788
$997$	− 40309.3i	− 1.28045i	−0.768188	−	0.640225i	$-0.778841\pi$
$997$	− 40309.3i	− 1.28045i	0.768188	−	0.640225i	$-0.221159\pi$
$998$	25551.3i	0.810432i
$999$	−1305.03	−0.0413306

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.4.d.k.274.1		4
5.2	odd	4		525.4.a.o.1.2		2
5.3	odd	4		105.4.a.d.1.1	✓	2
5.4	even	2	inner	525.4.d.k.274.4		4
15.2	even	4		1575.4.a.n.1.1		2
15.8	even	4		315.4.a.l.1.2		2
20.3	even	4		1680.4.a.bd.1.1		2
35.13	even	4		735.4.a.m.1.1		2
105.83	odd	4		2205.4.a.be.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.a.d.1.1	✓	2	5.3	odd	4
315.4.a.l.1.2		2	15.8	even	4
525.4.a.o.1.2		2	5.2	odd	4
525.4.d.k.274.1		4	1.1	even	1	trivial
525.4.d.k.274.4		4	5.4	even	2	inner
735.4.a.m.1.1		2	35.13	even	4
1575.4.a.n.1.1		2	15.2	even	4
1680.4.a.bd.1.1		2	20.3	even	4
2205.4.a.be.1.2		2	105.83	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	525.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.23607i q^{2} -3.00000i q^{3} -9.94427 q^{4} -12.7082 q^{6} -7.00000i q^{7} +8.23607i q^{8} -9.00000 q^{9} -41.5279 q^{11} +29.8328i q^{12} -88.9706i q^{13} -29.6525 q^{14} -44.6656 q^{16} -120.387i q^{17} +38.1246i q^{18} +112.138 q^{19} -21.0000 q^{21} +175.915i q^{22} +115.279i q^{23} +24.7082 q^{24} -376.885 q^{26} +27.0000i q^{27} +69.6099i q^{28} +144.833 q^{29} -258.079 q^{31} +255.095i q^{32} +124.584i q^{33} -509.967 q^{34} +89.4984 q^{36} +48.3344i q^{37} -475.023i q^{38} -266.912 q^{39} +200.885 q^{41} +88.9574i q^{42} +218.217i q^{43} +412.964 q^{44} +488.328 q^{46} +575.659i q^{47} +133.997i q^{48} -49.0000 q^{49} -361.161 q^{51} +884.748i q^{52} +184.302i q^{53} +114.374 q^{54} +57.6525 q^{56} -336.413i q^{57} -613.522i q^{58} +151.502 q^{59} -529.830 q^{61} +1093.24i q^{62} +63.0000i q^{63} +723.276 q^{64} +527.745 q^{66} +1.28485i q^{67} +1197.16i q^{68} +345.836 q^{69} -61.4226 q^{71} -74.1246i q^{72} -484.800i q^{73} +204.748 q^{74} -1115.13 q^{76} +290.695i q^{77} +1130.66i q^{78} -878.257 q^{79} +81.0000 q^{81} -850.964i q^{82} -491.830i q^{83} +208.830 q^{84} +924.381 q^{86} -434.498i q^{87} -342.026i q^{88} +415.560 q^{89} -622.794 q^{91} -1146.36i q^{92} +774.237i q^{93} +2438.53 q^{94} +765.286 q^{96} -1031.70i q^{97} +207.567i q^{98} +373.751 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 24 q^{6} - 36 q^{9} - 184 q^{11} - 56 q^{14} + 36 q^{16} + 216 q^{19} - 84 q^{21} + 72 q^{24} - 792 q^{26} + 472 q^{29} - 120 q^{31} - 1056 q^{34} + 36 q^{36} - 48 q^{39} + 88 q^{41} + 24 q^{44}+ \cdots + 1656 q^{99}+O(q^{100}) \) 4 * q - 4 * q^4 - 24 * q^6 - 36 * q^9 - 184 * q^11 - 56 * q^14 + 36 * q^16 + 216 * q^19 - 84 * q^21 + 72 * q^24 - 792 * q^26 + 472 * q^29 - 120 * q^31 - 1056 * q^34 + 36 * q^36 - 48 * q^39 + 88 * q^41 + 24 * q^44 + 880 * q^46 - 196 * q^49 - 264 * q^51 + 216 * q^54 + 168 * q^56 + 928 * q^59 - 1368 * q^61 + 2428 * q^64 + 984 * q^66 + 1920 * q^69 - 1480 * q^71 + 336 * q^74 - 2296 * q^76 + 816 * q^79 + 324 * q^81 + 84 * q^84 + 2016 * q^86 + 2664 * q^89 - 112 * q^91 + 4960 * q^94 + 1800 * q^96 + 1656 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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