LMFDB - Embedded newform 425.4.b.e.324.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [425,4,Mod(324,425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(425, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("425.324");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 425 = 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	425.b (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:	$25.0758117524$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 85)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			324.1
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	425.324
Dual form			425.4.b.e.324.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-3.73205i q^{2} +2.73205i q^{3} -5.92820 q^{4} +10.1962 q^{6} +16.5885i q^{7} -7.73205i q^{8} +19.5359 q^{9} -41.5167 q^{11} -16.1962i q^{12} -8.49742i q^{13} +61.9090 q^{14} -76.2820 q^{16} -17.0000i q^{17} -72.9090i q^{18} +100.392 q^{19} -45.3205 q^{21} +154.942i q^{22} +195.937i q^{23} +21.1244 q^{24} -31.7128 q^{26} +127.138i q^{27} -98.3397i q^{28} +187.674 q^{29} +279.219 q^{31} +222.832i q^{32} -113.426i q^{33} -63.4449 q^{34} -115.813 q^{36} -7.59739i q^{37} -374.669i q^{38} +23.2154 q^{39} -73.9512 q^{41} +169.138i q^{42} +331.520i q^{43} +246.119 q^{44} +731.247 q^{46} -187.856i q^{47} -208.406i q^{48} +67.8231 q^{49} +46.4449 q^{51} +50.3744i q^{52} +192.851i q^{53} +474.487 q^{54} +128.263 q^{56} +274.277i q^{57} -700.410i q^{58} +603.685 q^{59} +426.190 q^{61} -1042.06i q^{62} +324.070i q^{63} +221.364 q^{64} -423.310 q^{66} +511.472i q^{67} +100.779i q^{68} -535.310 q^{69} -548.886 q^{71} -151.053i q^{72} -575.177i q^{73} -28.3538 q^{74} -595.146 q^{76} -688.697i q^{77} -86.6410i q^{78} +1235.91 q^{79} +180.121 q^{81} +275.990i q^{82} -536.228i q^{83} +268.669 q^{84} +1237.25 q^{86} +512.736i q^{87} +321.009i q^{88} -1183.18 q^{89} +140.959 q^{91} -1161.56i q^{92} +762.841i q^{93} -701.090 q^{94} -608.788 q^{96} +1261.17i q^{97} -253.119i q^{98} -811.065 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4} + 20 q^{6} + 92 q^{9} - 76 q^{11} + 116 q^{14} - 28 q^{16} + 360 q^{19} - 112 q^{21} + 36 q^{24} - 16 q^{26} + 432 q^{29} + 140 q^{31} - 136 q^{34} + 188 q^{36} + 176 q^{39} + 688 q^{41} + 548 q^{44}+ \cdots - 1436 q^{99}+O(q^{100}) $ 4 * q + 4 * q^4 + 20 * q^6 + 92 * q^9 - 76 * q^11 + 116 * q^14 - 28 * q^16 + 360 * q^19 - 112 * q^21 + 36 * q^24 - 16 * q^26 + 432 * q^29 + 140 * q^31 - 136 * q^34 + 188 * q^36 + 176 * q^39 + 688 * q^41 + 548 * q^44 + 1380 * q^46 + 396 * q^49 + 68 * q^51 + 928 * q^54 + 132 * q^56 + 2872 * q^59 - 152 * q^61 + 1412 * q^64 - 848 * q^66 - 1296 * q^69 - 3020 * q^71 + 136 * q^74 + 72 * q^76 + 1916 * q^79 + 1732 * q^81 + 368 * q^84 + 2344 * q^86 + 408 * q^89 + 2864 * q^91 - 1488 * q^94 - 1292 * q^96 - 1436 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/425\mathbb{Z}\right)^\times$.

$n$	$52$	$326$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 3.73205i	− 1.31948i	−0.751494	−	0.659740i	$-0.770667\pi$
$2$	− 3.73205i	− 1.31948i	0.751494	−	0.659740i	$-0.229333\pi$
$3$	2.73205i	0.525783i	0.964825	+	0.262892i	$0.0846762\pi$
$3$	2.73205i	0.525783i	−0.964825	+	0.262892i	$0.915324\pi$
$4$	−5.92820	−0.741025
$5$	0	0
$5$	0	0
$6$	10.1962	0.693760
$7$	16.5885i	0.895693i	0.894111	+	0.447846i	$0.147809\pi$
$7$	16.5885i	0.895693i	−0.894111	+	0.447846i	$0.852191\pi$
$8$	− 7.73205i	− 0.341712i
$9$	19.5359	0.723552
$10$	0	0
$11$	−41.5167	−1.13798	−0.568988	−	0.822346i	$-0.692665\pi$
$11$	−41.5167	−1.13798	−0.568988	+	0.822346i	$0.692665\pi$
$12$	− 16.1962i	− 0.389619i
$13$	− 8.49742i	− 0.181289i	−0.995883	−	0.0906447i	$-0.971107\pi$
$13$	− 8.49742i	− 0.181289i	0.995883	−	0.0906447i	$-0.0288927\pi$
$14$	61.9090	1.18185
$15$	0	0
$16$	−76.2820	−1.19191
$17$	− 17.0000i	− 0.242536i
$17$	− 17.0000i	− 0.242536i
$18$	− 72.9090i	− 0.954712i
$19$	100.392	1.21219	0.606094	−	0.795393i	$-0.292735\pi$
$19$	100.392	1.21219	0.606094	+	0.795393i	$0.292735\pi$
$20$	0	0
$21$	−45.3205	−0.470940
$22$	154.942i	1.50154i
$23$	195.937i	1.77634i	0.459519	+	0.888168i	$0.348022\pi$
$23$	195.937i	1.77634i	−0.459519	+	0.888168i	$0.651978\pi$
$24$	21.1244	0.179666
$25$	0	0
$26$	−31.7128	−0.239207
$27$	127.138i	0.906215i
$28$	− 98.3397i	− 0.663731i
$29$	187.674	1.20173	0.600866	−	0.799349i	$-0.294822\pi$
$29$	187.674	1.20173	0.600866	+	0.799349i	$0.294822\pi$
$30$	0	0
$31$	279.219	1.61772	0.808859	−	0.588003i	$-0.200086\pi$
$31$	279.219	1.61772	0.808859	+	0.588003i	$0.200086\pi$
$32$	222.832i	1.23098i
$33$	− 113.426i	− 0.598329i
$34$	−63.4449	−0.320021
$35$	0	0
$36$	−115.813	−0.536170
$37$	− 7.59739i	− 0.0337568i	−0.999858	−	0.0168784i	$-0.994627\pi$
$37$	− 7.59739i	− 0.0337568i	0.999858	−	0.0168784i	$-0.00537282\pi$
$38$	− 374.669i	− 1.59946i
$39$	23.2154	0.0953189
$40$	0	0
$41$	−73.9512	−0.281689	−0.140844	−	0.990032i	$-0.544982\pi$
$41$	−73.9512	−0.281689	−0.140844	+	0.990032i	$0.544982\pi$
$42$	169.138i	0.621396i
$43$	331.520i	1.17573i	0.808959	+	0.587865i	$0.200031\pi$
$43$	331.520i	1.17573i	−0.808959	+	0.587865i	$0.799969\pi$
$44$	246.119	0.843270
$45$	0	0
$46$	731.247	2.34384
$47$	− 187.856i	− 0.583014i	−0.956569	−	0.291507i	$-0.905843\pi$
$47$	− 187.856i	− 0.583014i	0.956569	−	0.291507i	$-0.0941567\pi$
$48$	− 208.406i	− 0.626685i
$49$	67.8231	0.197735
$50$	0	0
$51$	46.4449	0.127521
$52$	50.3744i	0.134340i
$53$	192.851i	0.499814i	0.968270	+	0.249907i	$0.0804001\pi$
$53$	192.851i	0.499814i	−0.968270	+	0.249907i	$0.919600\pi$
$54$	474.487	1.19573
$55$	0	0
$56$	128.263	0.306069
$57$	274.277i	0.637348i
$58$	− 700.410i	− 1.58566i
$59$	603.685	1.33209	0.666043	−	0.745914i	$-0.267987\pi$
$59$	603.685	1.33209	0.666043	+	0.745914i	$0.267987\pi$
$60$	0	0
$61$	426.190	0.894557	0.447279	−	0.894395i	$-0.352393\pi$
$61$	426.190	0.894557	0.447279	+	0.894395i	$0.352393\pi$
$62$	− 1042.06i	− 2.13454i
$63$	324.070i	0.648080i
$64$	221.364	0.432352
$65$	0	0
$66$	−423.310	−0.789483
$67$	511.472i	0.932630i	0.884619	+	0.466315i	$0.154419\pi$
$67$	511.472i	0.932630i	−0.884619	+	0.466315i	$0.845581\pi$
$68$	100.779i	0.179725i
$69$	−535.310	−0.933968
$70$	0	0
$71$	−548.886	−0.917476	−0.458738	−	0.888572i	$-0.651698\pi$
$71$	−548.886	−0.917476	−0.458738	+	0.888572i	$0.651698\pi$
$72$	− 151.053i	− 0.247246i
$73$	− 575.177i	− 0.922183i	−0.887353	−	0.461092i	$-0.847458\pi$
$73$	− 575.177i	− 0.922183i	0.887353	−	0.461092i	$-0.152542\pi$
$74$	−28.3538	−0.0445414
$75$	0	0
$76$	−595.146	−0.898262
$77$	− 688.697i	− 1.01928i
$78$	− 86.6410i	− 0.125771i
$79$	1235.91	1.76013	0.880065	−	0.474853i	$-0.157499\pi$
$79$	1235.91	1.76013	0.880065	+	0.474853i	$0.157499\pi$
$80$	0	0
$81$	180.121	0.247079
$82$	275.990i	0.371682i
$83$	− 536.228i	− 0.709141i	−0.935029	−	0.354570i	$-0.884627\pi$
$83$	− 536.228i	− 0.709141i	0.935029	−	0.354570i	$-0.115373\pi$
$84$	268.669	0.348979
$85$	0	0
$86$	1237.25	1.55135
$87$	512.736i	0.631851i
$88$	321.009i	0.388860i
$89$	−1183.18	−1.40918	−0.704590	−	0.709615i	$-0.748869\pi$
$89$	−1183.18	−1.40918	−0.704590	+	0.709615i	$0.748869\pi$
$90$	0	0
$91$	140.959	0.162379
$92$	− 1161.56i	− 1.31631i
$93$	762.841i	0.850569i
$94$	−701.090	−0.769275
$95$	0	0
$96$	−608.788	−0.647231
$97$	1261.17i	1.32013i	0.751207	+	0.660066i	$0.229472\pi$
$97$	1261.17i	1.32013i	−0.751207	+	0.660066i	$0.770528\pi$
$98$	− 253.119i	− 0.260907i
$99$	−811.065	−0.823385
$100$	0	0
$101$	−428.141	−0.421798	−0.210899	−	0.977508i	$-0.567639\pi$
$101$	−428.141	−0.421798	−0.210899	+	0.977508i	$0.567639\pi$
$102$	− 173.335i	− 0.168262i
$103$	− 619.856i	− 0.592973i	−0.955037	−	0.296487i	$-0.904185\pi$
$103$	− 619.856i	− 0.592973i	0.955037	−	0.296487i	$-0.0958151\pi$
$104$	−65.7025	−0.0619487
$105$	0	0
$106$	719.731	0.659495
$107$	− 51.4960i	− 0.0465263i	−0.999729	−	0.0232631i	$-0.992594\pi$
$107$	− 51.4960i	− 0.0465263i	0.999729	−	0.0232631i	$-0.00740555\pi$
$108$	− 753.703i	− 0.671528i
$109$	−990.841	−0.870691	−0.435346	−	0.900263i	$-0.643374\pi$
$109$	−990.841	−0.870691	−0.435346	+	0.900263i	$0.643374\pi$
$110$	0	0
$111$	20.7564	0.0177488
$112$	− 1265.40i	− 1.06758i
$113$	− 865.413i	− 0.720453i	−0.932865	−	0.360226i	$-0.882699\pi$
$113$	− 865.413i	− 0.720453i	0.932865	−	0.360226i	$-0.117301\pi$
$114$	1023.62	0.840968
$115$	0	0
$116$	−1112.57	−0.890514
$117$	− 166.005i	− 0.131172i
$118$	− 2252.98i	− 1.75766i
$119$	282.004	0.217237
$120$	0	0
$121$	392.633	0.294991
$122$	− 1590.56i	− 1.18035i
$123$	− 202.038i	− 0.148107i
$124$	−1655.27	−1.19877
$125$	0	0
$126$	1209.45	0.855128
$127$	− 2052.21i	− 1.43389i	−0.697130	−	0.716945i	$-0.745540\pi$
$127$	− 2052.21i	− 1.43389i	0.697130	−	0.716945i	$-0.254460\pi$
$128$	956.514i	0.660505i
$129$	−905.731	−0.618179
$130$	0	0
$131$	−77.4603	−0.0516621	−0.0258311	−	0.999666i	$-0.508223\pi$
$131$	−77.4603	−0.0516621	−0.0258311	+	0.999666i	$0.508223\pi$
$132$	672.410i	0.443377i
$133$	1665.35i	1.08575i
$134$	1908.84	1.23059
$135$	0	0
$136$	−131.445	−0.0828772
$137$	1936.63i	1.20772i	0.797091	+	0.603860i	$0.206371\pi$
$137$	1936.63i	1.20772i	−0.797091	+	0.603860i	$0.793629\pi$
$138$	1997.80i	1.23235i
$139$	1687.63	1.02980	0.514901	−	0.857249i	$-0.327828\pi$
$139$	1687.63	1.02980	0.514901	+	0.857249i	$0.327828\pi$
$140$	0	0
$141$	513.233	0.306539
$142$	2048.47i	1.21059i
$143$	352.785i	0.206303i
$144$	−1490.24	−0.862406
$145$	0	0
$146$	−2146.59	−1.21680
$147$	185.296i	0.103966i
$148$	45.0388i	0.0250147i
$149$	−488.052	−0.268341	−0.134170	−	0.990958i	$-0.542837\pi$
$149$	−488.052	−0.268341	−0.134170	+	0.990958i	$0.542837\pi$
$150$	0	0
$151$	1459.95	0.786813	0.393406	−	0.919365i	$-0.371297\pi$
$151$	1459.95	0.786813	0.393406	+	0.919365i	$0.371297\pi$
$152$	− 776.238i	− 0.414219i
$153$	− 332.110i	− 0.175487i
$154$	−2570.25	−1.34492
$155$	0	0
$156$	−137.626	−0.0706337
$157$	35.7286i	0.0181621i	0.999959	+	0.00908107i	$0.00289063\pi$
$157$	35.7286i	0.0181621i	−0.999959	+	0.00908107i	$0.997109\pi$
$158$	− 4612.46i	− 2.32246i
$159$	−526.879	−0.262794
$160$	0	0
$161$	−3250.29	−1.59105
$162$	− 672.219i	− 0.326016i
$163$	3305.37i	1.58832i	0.607706	+	0.794162i	$0.292090\pi$
$163$	3305.37i	1.58832i	−0.607706	+	0.794162i	$0.707910\pi$
$164$	438.398	0.208739
$165$	0	0
$166$	−2001.23	−0.935696
$167$	3228.66i	1.49606i	0.663667	+	0.748028i	$0.268999\pi$
$167$	3228.66i	1.49606i	−0.663667	+	0.748028i	$0.731001\pi$
$168$	350.420i	0.160926i
$169$	2124.79	0.967134
$170$	0	0
$171$	1961.25	0.877081
$172$	− 1965.32i	− 0.871246i
$173$	958.357i	0.421171i	0.977575	+	0.210585i	$0.0675370\pi$
$173$	958.357i	0.421171i	−0.977575	+	0.210585i	$0.932463\pi$
$174$	1913.56	0.833714
$175$	0	0
$176$	3166.98	1.35636
$177$	1649.30i	0.700388i
$178$	4415.69i	1.85938i
$179$	−1705.71	−0.712239	−0.356119	−	0.934440i	$-0.615900\pi$
$179$	−1705.71	−0.712239	−0.356119	+	0.934440i	$0.615900\pi$
$180$	0	0
$181$	600.566	0.246628	0.123314	−	0.992368i	$-0.460648\pi$
$181$	600.566	0.246628	0.123314	+	0.992368i	$0.460648\pi$
$182$	− 526.067i	− 0.214256i
$183$	1164.37i	0.470343i
$184$	1515.00	0.606994
$185$	0	0
$186$	2846.96	1.12231
$187$	705.783i	0.276000i
$188$	1113.65i	0.432028i
$189$	−2109.03	−0.811690
$190$	0	0
$191$	−3994.31	−1.51318	−0.756591	−	0.653888i	$-0.773137\pi$
$191$	−3994.31	−1.51318	−0.756591	+	0.653888i	$0.773137\pi$
$192$	604.778i	0.227323i
$193$	1586.31i	0.591633i	0.955245	+	0.295816i	$0.0955916\pi$
$193$	1586.31i	0.591633i	−0.955245	+	0.295816i	$0.904408\pi$
$194$	4706.77	1.74189
$195$	0	0
$196$	−402.069	−0.146527
$197$	− 3011.98i	− 1.08931i	−0.838660	−	0.544656i	$-0.816660\pi$
$197$	− 3011.98i	− 1.08931i	0.838660	−	0.544656i	$-0.183340\pi$
$198$	3026.94i	1.08644i
$199$	2065.77	0.735872	0.367936	−	0.929851i	$-0.380065\pi$
$199$	2065.77	0.735872	0.367936	+	0.929851i	$0.380065\pi$
$200$	0	0
$201$	−1397.37	−0.490361
$202$	1597.84i	0.556554i
$203$	3113.23i	1.07638i
$204$	−275.335	−0.0944965
$205$	0	0
$206$	−2313.34	−0.782416
$207$	3827.81i	1.28527i
$208$	648.201i	0.216080i
$209$	−4167.95	−1.37944
$210$	0	0
$211$	−1691.46	−0.551873	−0.275936	−	0.961176i	$-0.588988\pi$
$211$	−1691.46	−0.551873	−0.275936	+	0.961176i	$0.588988\pi$
$212$	− 1143.26i	− 0.370375i
$213$	− 1499.58i	− 0.482394i
$214$	−192.186	−0.0613904
$215$	0	0
$216$	983.041	0.309664
$217$	4631.82i	1.44898i
$218$	3697.87i	1.14886i
$219$	1571.41	0.484869
$220$	0	0
$221$	−144.456	−0.0439691
$222$	− 77.4641i	− 0.0234191i
$223$	1034.01i	0.310505i	0.987875	+	0.155253i	$0.0496191\pi$
$223$	1034.01i	0.310505i	−0.987875	+	0.155253i	$0.950381\pi$
$224$	−3696.44	−1.10258
$225$	0	0
$226$	−3229.76	−0.950622
$227$	− 202.321i	− 0.0591565i	−0.999562	−	0.0295783i	$-0.990584\pi$
$227$	− 202.321i	− 0.0591565i	0.999562	−	0.0295783i	$-0.00941642\pi$
$228$	− 1625.97i	− 0.472291i
$229$	−6346.43	−1.83137	−0.915685	−	0.401897i	$-0.868351\pi$
$229$	−6346.43	−1.83137	−0.915685	+	0.401897i	$0.868351\pi$
$230$	0	0
$231$	1881.56	0.535919
$232$	− 1451.11i	− 0.410646i
$233$	− 3768.52i	− 1.05959i	−0.848126	−	0.529794i	$-0.822269\pi$
$233$	− 3768.52i	− 1.05959i	0.848126	−	0.529794i	$-0.177731\pi$
$234$	−619.538	−0.173079
$235$	0	0
$236$	−3578.77	−0.987109
$237$	3376.56i	0.925447i
$238$	− 1052.45i	− 0.286640i
$239$	−5761.72	−1.55939	−0.779696	−	0.626159i	$-0.784626\pi$
$239$	−5761.72	−1.55939	−0.779696	+	0.626159i	$0.784626\pi$
$240$	0	0
$241$	−2015.26	−0.538650	−0.269325	−	0.963049i	$-0.586800\pi$
$241$	−2015.26	−0.538650	−0.269325	+	0.963049i	$0.586800\pi$
$242$	− 1465.33i	− 0.389235i
$243$	3924.84i	1.03612i
$244$	−2526.54	−0.662890
$245$	0	0
$246$	−754.018	−0.195424
$247$	− 853.076i	− 0.219757i
$248$	− 2158.94i	− 0.552793i
$249$	1465.00	0.372854
$250$	0	0
$251$	808.179	0.203234	0.101617	−	0.994824i	$-0.467598\pi$
$251$	808.179	0.203234	0.101617	+	0.994824i	$0.467598\pi$
$252$	− 1921.16i	− 0.480244i
$253$	− 8134.66i	− 2.02143i
$254$	−7658.94	−1.89199
$255$	0	0
$256$	5340.67	1.30387
$257$	1911.01i	0.463836i	0.972735	+	0.231918i	$0.0745001\pi$
$257$	1911.01i	0.463836i	−0.972735	+	0.231918i	$0.925500\pi$
$258$	3380.23i	0.815675i
$259$	126.029	0.0302357
$260$	0	0
$261$	3666.39	0.869516
$262$	289.086i	0.0681671i
$263$	− 314.792i	− 0.0738058i	−0.999319	−	0.0369029i	$-0.988251\pi$
$263$	− 314.792i	− 0.0738058i	0.999319	−	0.0369029i	$-0.0117492\pi$
$264$	−877.013	−0.204456
$265$	0	0
$266$	6215.18	1.43262
$267$	− 3232.51i	− 0.740923i
$268$	− 3032.11i	− 0.691102i
$269$	7100.88	1.60947	0.804736	−	0.593633i	$-0.202307\pi$
$269$	7100.88	1.60947	0.804736	+	0.593633i	$0.202307\pi$
$270$	0	0
$271$	−3238.60	−0.725944	−0.362972	−	0.931800i	$-0.618238\pi$
$271$	−3238.60	−0.725944	−0.362972	+	0.931800i	$0.618238\pi$
$272$	1296.79i	0.289080i
$273$	385.108i	0.0853764i
$274$	7227.60	1.59356
$275$	0	0
$276$	3173.43	0.692094
$277$	− 4682.33i	− 1.01565i	−0.861461	−	0.507823i	$-0.830450\pi$
$277$	− 4682.33i	− 1.01565i	0.861461	−	0.507823i	$-0.169550\pi$
$278$	− 6298.31i	− 1.35880i
$279$	5454.80	1.17050
$280$	0	0
$281$	5914.75	1.25567	0.627837	−	0.778345i	$-0.283940\pi$
$281$	5914.75	1.25567	0.627837	+	0.778345i	$0.283940\pi$
$282$	− 1915.41i	− 0.404472i
$283$	− 8154.53i	− 1.71285i	−0.516272	−	0.856425i	$-0.672681\pi$
$283$	− 8154.53i	− 1.71285i	0.516272	−	0.856425i	$-0.327319\pi$
$284$	3253.91	0.679873
$285$	0	0
$286$	1316.61	0.272213
$287$	− 1226.74i	− 0.252307i
$288$	4353.22i	0.890681i
$289$	−289.000	−0.0588235
$290$	0	0
$291$	−3445.59	−0.694104
$292$	3409.77i	0.683361i
$293$	− 3005.77i	− 0.599314i	−0.954047	−	0.299657i	$-0.903128\pi$
$293$	− 3005.77i	− 0.599314i	0.954047	−	0.299657i	$-0.0968722\pi$
$294$	691.535	0.137181
$295$	0	0
$296$	−58.7434	−0.0115351
$297$	− 5278.36i	− 1.03125i
$298$	1821.43i	0.354070i
$299$	1664.96	0.322031
$300$	0	0
$301$	−5499.41	−1.05309
$302$	− 5448.59i	− 1.03818i
$303$	− 1169.70i	− 0.221774i
$304$	−7658.13	−1.44482
$305$	0	0
$306$	−1239.45	−0.231552
$307$	− 4911.16i	− 0.913013i	−0.889720	−	0.456506i	$-0.849101\pi$
$307$	− 4911.16i	− 0.913013i	0.889720	−	0.456506i	$-0.150899\pi$
$308$	4082.74i	0.755310i
$309$	1693.48	0.311776
$310$	0	0
$311$	−5449.43	−0.993597	−0.496799	−	0.867866i	$-0.665491\pi$
$311$	−5449.43	−0.993597	−0.496799	+	0.867866i	$0.665491\pi$
$312$	− 179.503i	− 0.0325716i
$313$	5273.09i	0.952246i	0.879379	+	0.476123i	$0.157958\pi$
$313$	5273.09i	0.952246i	−0.879379	+	0.476123i	$0.842042\pi$
$314$	133.341	0.0239646
$315$	0	0
$316$	−7326.70	−1.30430
$317$	− 10737.6i	− 1.90246i	−0.308477	−	0.951232i	$-0.599819\pi$
$317$	− 10737.6i	− 1.90246i	0.308477	−	0.951232i	$-0.400181\pi$
$318$	1966.34i	0.346751i
$319$	−7791.61	−1.36754
$320$	0	0
$321$	140.690	0.0244627
$322$	12130.3i	2.09936i
$323$	− 1706.67i	− 0.293999i
$324$	−1067.79	−0.183092
$325$	0	0
$326$	12335.8	2.09576
$327$	− 2707.03i	− 0.457795i
$328$	571.795i	0.0962563i
$329$	3116.25	0.522202
$330$	0	0
$331$	−3868.26	−0.642354	−0.321177	−	0.947019i	$-0.604078\pi$
$331$	−3868.26	−0.642354	−0.321177	+	0.947019i	$0.604078\pi$
$332$	3178.87i	0.525491i
$333$	− 148.422i	− 0.0244248i
$334$	12049.5	1.97401
$335$	0	0
$336$	3457.14	0.561317
$337$	881.122i	0.142427i	0.997461	+	0.0712133i	$0.0226871\pi$
$337$	881.122i	0.142427i	−0.997461	+	0.0712133i	$0.977313\pi$
$338$	− 7929.84i	− 1.27611i
$339$	2364.35	0.378802
$340$	0	0
$341$	−11592.2	−1.84092
$342$	− 7319.50i	− 1.15729i
$343$	6814.92i	1.07280i
$344$	2563.33	0.401761
$345$	0	0
$346$	3576.64	0.555726
$347$	− 722.460i	− 0.111769i	−0.998437	−	0.0558843i	$-0.982202\pi$
$347$	− 722.460i	− 0.111769i	0.998437	−	0.0558843i	$-0.0177978\pi$
$348$	− 3039.60i	− 0.468218i
$349$	−10730.8	−1.64587	−0.822935	−	0.568136i	$-0.807665\pi$
$349$	−10730.8	−1.64587	−0.822935	+	0.568136i	$0.807665\pi$
$350$	0	0
$351$	1080.35	0.164287
$352$	− 9251.24i	− 1.40083i
$353$	− 6601.89i	− 0.995420i	−0.867344	−	0.497710i	$-0.834174\pi$
$353$	− 6601.89i	− 0.995420i	0.867344	−	0.497710i	$-0.165826\pi$
$354$	6155.26	0.924148
$355$	0	0
$356$	7014.14	1.04424
$357$	770.449i	0.114220i
$358$	6365.80i	0.939784i
$359$	−3836.05	−0.563952	−0.281976	−	0.959421i	$-0.590990\pi$
$359$	−3836.05	−0.563952	−0.281976	+	0.959421i	$0.590990\pi$
$360$	0	0
$361$	3219.61	0.469400
$362$	− 2241.34i	− 0.325421i
$363$	1072.69i	0.155101i
$364$	−835.634	−0.120327
$365$	0	0
$366$	4345.49	0.620608
$367$	10269.6i	1.46068i	0.683084	+	0.730340i	$0.260638\pi$
$367$	10269.6i	1.46068i	−0.683084	+	0.730340i	$0.739362\pi$
$368$	− 14946.5i	− 2.11723i
$369$	−1444.70	−0.203816
$370$	0	0
$371$	−3199.10	−0.447680
$372$	− 4522.28i	− 0.630293i
$373$	− 2920.52i	− 0.405412i	−0.979240	−	0.202706i	$-0.935026\pi$
$373$	− 2920.52i	− 0.405412i	0.979240	−	0.202706i	$-0.0649735\pi$
$374$	2634.02	0.364176
$375$	0	0
$376$	−1452.52	−0.199223
$377$	− 1594.75i	− 0.217861i
$378$	7871.01i	1.07101i
$379$	4683.11	0.634711	0.317355	−	0.948307i	$-0.397205\pi$
$379$	4683.11	0.634711	0.317355	+	0.948307i	$0.397205\pi$
$380$	0	0
$381$	5606.74	0.753915
$382$	14907.0i	1.99661i
$383$	− 9524.24i	− 1.27067i	−0.772237	−	0.635334i	$-0.780862\pi$
$383$	− 9524.24i	− 1.27067i	0.772237	−	0.635334i	$-0.219138\pi$
$384$	−2613.24	−0.347283
$385$	0	0
$386$	5920.19	0.780647
$387$	6476.55i	0.850702i
$388$	− 7476.50i	− 0.978252i
$389$	12389.7	1.61486	0.807430	−	0.589964i	$-0.200858\pi$
$389$	12389.7	1.61486	0.807430	+	0.589964i	$0.200858\pi$
$390$	0	0
$391$	3330.93	0.430825
$392$	− 524.412i	− 0.0675683i
$393$	− 211.626i	− 0.0271631i
$394$	−11240.8	−1.43732
$395$	0	0
$396$	4808.16	0.610149
$397$	2797.07i	0.353604i	0.984246	+	0.176802i	$0.0565752\pi$
$397$	2797.07i	0.353604i	−0.984246	+	0.176802i	$0.943425\pi$
$398$	− 7709.56i	− 0.970968i
$399$	−4549.83	−0.570868
$400$	0	0
$401$	−10776.6	−1.34204	−0.671021	−	0.741438i	$-0.734144\pi$
$401$	−10776.6	−1.34204	−0.671021	+	0.741438i	$0.734144\pi$
$402$	5215.04i	0.647022i
$403$	− 2372.64i	− 0.293275i
$404$	2538.11	0.312563
$405$	0	0
$406$	11618.7	1.42027
$407$	315.418i	0.0384145i
$408$	− 359.114i	− 0.0435755i
$409$	1225.28	0.148133	0.0740663	−	0.997253i	$-0.476402\pi$
$409$	1225.28	0.148133	0.0740663	+	0.997253i	$0.476402\pi$
$410$	0	0
$411$	−5290.97	−0.634999
$412$	3674.63i	0.439408i
$413$	10014.2i	1.19314i
$414$	14285.6	1.69589
$415$	0	0
$416$	1893.50	0.223164
$417$	4610.68i	0.541453i
$418$	15555.0i	1.82014i
$419$	7621.82	0.888664	0.444332	−	0.895862i	$-0.353441\pi$
$419$	7621.82	0.888664	0.444332	+	0.895862i	$0.353441\pi$
$420$	0	0
$421$	−15961.2	−1.84775	−0.923874	−	0.382696i	$-0.874996\pi$
$421$	−15961.2	−1.84775	−0.923874	+	0.382696i	$0.874996\pi$
$422$	6312.63i	0.728184i
$423$	− 3669.94i	− 0.421841i
$424$	1491.14	0.170792
$425$	0	0
$426$	−5596.52	−0.636508
$427$	7069.83i	0.801248i
$428$	305.279i	0.0344771i
$429$	−963.825	−0.108471
$430$	0	0
$431$	−1956.16	−0.218619	−0.109309	−	0.994008i	$-0.534864\pi$
$431$	−1956.16	−0.218619	−0.109309	+	0.994008i	$0.534864\pi$
$432$	− 9698.38i	− 1.08012i
$433$	4825.52i	0.535565i	0.963479	+	0.267783i	$0.0862909\pi$
$433$	4825.52i	0.535565i	−0.963479	+	0.267783i	$0.913709\pi$
$434$	17286.2	1.91190
$435$	0	0
$436$	5873.91	0.645204
$437$	19670.6i	2.15325i
$438$	− 5864.59i	− 0.639774i
$439$	9766.60	1.06181	0.530905	−	0.847432i	$-0.321852\pi$
$439$	9766.60	1.06181	0.530905	+	0.847432i	$0.321852\pi$
$440$	0	0
$441$	1324.98	0.143071
$442$	539.118i	0.0580163i
$443$	− 11455.1i	− 1.22855i	−0.789091	−	0.614276i	$-0.789448\pi$
$443$	− 11455.1i	− 1.22855i	0.789091	−	0.614276i	$-0.210552\pi$
$444$	−123.048	−0.0131523
$445$	0	0
$446$	3858.99	0.409705
$447$	− 1333.38i	− 0.141089i
$448$	3672.09i	0.387254i
$449$	9716.46	1.02126	0.510632	−	0.859799i	$-0.329411\pi$
$449$	9716.46	1.02126	0.510632	+	0.859799i	$0.329411\pi$
$450$	0	0
$451$	3070.21	0.320555
$452$	5130.34i	0.533874i
$453$	3988.65i	0.413693i
$454$	−755.073	−0.0780558
$455$	0	0
$456$	2120.72	0.217789
$457$	10499.8i	1.07475i	0.843344	+	0.537374i	$0.180584\pi$
$457$	10499.8i	1.07475i	−0.843344	+	0.537374i	$0.819416\pi$
$458$	23685.2i	2.41645i
$459$	2161.35	0.219789
$460$	0	0
$461$	18649.9	1.88419	0.942095	−	0.335345i	$-0.108853\pi$
$461$	18649.9	1.88419	0.942095	+	0.335345i	$0.108853\pi$
$462$	− 7022.06i	− 0.707134i
$463$	− 11011.7i	− 1.10530i	−0.833413	−	0.552651i	$-0.813616\pi$
$463$	− 11011.7i	− 1.10530i	0.833413	−	0.552651i	$-0.186384\pi$
$464$	−14316.2	−1.43235
$465$	0	0
$466$	−14064.3	−1.39811
$467$	− 14194.7i	− 1.40654i	−0.710924	−	0.703269i	$-0.751723\pi$
$467$	− 14194.7i	− 1.40654i	0.710924	−	0.703269i	$-0.248277\pi$
$468$	984.110i	0.0972019i
$469$	−8484.53	−0.835350
$470$	0	0
$471$	−97.6125	−0.00954935
$472$	− 4667.72i	− 0.455189i
$473$	− 13763.6i	− 1.33795i
$474$	12601.5	1.22111
$475$	0	0
$476$	−1671.78	−0.160978
$477$	3767.52i	0.361642i
$478$	21503.0i	2.05758i
$479$	−14553.9	−1.38828	−0.694139	−	0.719841i	$-0.744215\pi$
$479$	−14553.9	−1.38828	−0.694139	+	0.719841i	$0.744215\pi$
$480$	0	0
$481$	−64.5582	−0.00611975
$482$	7521.07i	0.710737i
$483$	− 8879.97i	− 0.836548i
$484$	−2327.61	−0.218596
$485$	0	0
$486$	14647.7	1.36715
$487$	8047.78i	0.748829i	0.927261	+	0.374414i	$0.122156\pi$
$487$	8047.78i	0.748829i	−0.927261	+	0.374414i	$0.877844\pi$
$488$	− 3295.32i	− 0.305681i
$489$	−9030.45	−0.835114
$490$	0	0
$491$	6917.64	0.635822	0.317911	−	0.948121i	$-0.397019\pi$
$491$	6917.64	0.635822	0.317911	+	0.948121i	$0.397019\pi$
$492$	1197.73i	0.109751i
$493$	− 3190.46i	− 0.291463i
$494$	−3183.72	−0.289964
$495$	0	0
$496$	−21299.4	−1.92817
$497$	− 9105.17i	− 0.821776i
$498$	− 5467.46i	− 0.491974i
$499$	20501.4	1.83922	0.919608	−	0.392836i	$-0.128506\pi$
$499$	20501.4	1.83922	0.919608	+	0.392836i	$0.128506\pi$
$500$	0	0
$501$	−8820.87	−0.786601
$502$	− 3016.16i	− 0.268163i
$503$	12448.0i	1.10344i	0.834029	+	0.551720i	$0.186028\pi$
$503$	12448.0i	1.10344i	−0.834029	+	0.551720i	$0.813972\pi$
$504$	2505.73	0.221456
$505$	0	0
$506$	−30358.9	−2.66723
$507$	5805.04i	0.508503i
$508$	12165.9i	1.06255i
$509$	5349.99	0.465882	0.232941	−	0.972491i	$-0.425165\pi$
$509$	5349.99	0.465882	0.232941	+	0.972491i	$0.425165\pi$
$510$	0	0
$511$	9541.30	0.825992
$512$	− 12279.5i	− 1.05993i
$513$	12763.7i	1.09850i
$514$	7132.00	0.612022
$515$	0	0
$516$	5369.36	0.458087
$517$	7799.17i	0.663457i
$518$	− 470.346i	− 0.0398954i
$519$	−2618.28	−0.221444
$520$	0	0
$521$	2375.79	0.199780	0.0998900	−	0.994998i	$-0.468151\pi$
$521$	2375.79	0.199780	0.0998900	+	0.994998i	$0.468151\pi$
$522$	− 13683.1i	− 1.14731i
$523$	1983.51i	0.165837i	0.996556	+	0.0829186i	$0.0264241\pi$
$523$	1983.51i	0.165837i	−0.996556	+	0.0829186i	$0.973576\pi$
$524$	459.201	0.0382830
$525$	0	0
$526$	−1174.82	−0.0973852
$527$	− 4746.73i	− 0.392354i
$528$	8652.34i	0.713153i
$529$	−26224.4	−2.15537
$530$	0	0
$531$	11793.5	0.963833
$532$	− 9872.55i	− 0.804567i
$533$	628.395i	0.0510672i
$534$	−12063.9	−0.977633
$535$	0	0
$536$	3954.72	0.318690
$537$	− 4660.09i	− 0.374483i
$538$	− 26500.8i	− 2.12367i
$539$	−2815.79	−0.225018
$540$	0	0
$541$	−19924.8	−1.58343	−0.791713	−	0.610893i	$-0.790811\pi$
$541$	−19924.8	−1.58343	−0.791713	+	0.610893i	$0.790811\pi$
$542$	12086.6i	0.957869i
$543$	1640.78i	0.129673i
$544$	3788.14	0.298558
$545$	0	0
$546$	1437.24	0.112652
$547$	− 360.478i	− 0.0281772i	−0.999901	−	0.0140886i	$-0.995515\pi$
$547$	− 360.478i	− 0.0281772i	0.999901	−	0.0140886i	$-0.00448469\pi$
$548$	− 11480.7i	− 0.894951i
$549$	8326.00	0.647258
$550$	0	0
$551$	18841.1	1.45673
$552$	4139.05i	0.319148i
$553$	20501.8i	1.57654i
$554$	−17474.7	−1.34012
$555$	0	0
$556$	−10004.6	−0.763110
$557$	− 18433.0i	− 1.40221i	−0.713059	−	0.701104i	$-0.752691\pi$
$557$	− 18433.0i	− 1.40221i	0.713059	−	0.701104i	$-0.247309\pi$
$558$	− 20357.6i	− 1.54445i
$559$	2817.07	0.213147
$560$	0	0
$561$	−1928.24	−0.145116
$562$	− 22074.2i	− 1.65684i
$563$	4574.39i	0.342429i	0.985234	+	0.171215i	$0.0547692\pi$
$563$	4574.39i	0.342429i	−0.985234	+	0.171215i	$0.945231\pi$
$564$	−3042.55	−0.227153
$565$	0	0
$566$	−30433.1	−2.26007
$567$	2987.92i	0.221307i
$568$	4244.01i	0.313512i
$569$	−24577.3	−1.81078	−0.905390	−	0.424580i	$-0.860422\pi$
$569$	−24577.3	−1.81078	−0.905390	+	0.424580i	$0.860422\pi$
$570$	0	0
$571$	18058.2	1.32349	0.661745	−	0.749729i	$-0.269816\pi$
$571$	18058.2	1.32349	0.661745	+	0.749729i	$0.269816\pi$
$572$	− 2091.38i	− 0.152876i
$573$	− 10912.6i	− 0.795606i
$574$	−4578.24	−0.332913
$575$	0	0
$576$	4324.55	0.312829
$577$	− 25619.8i	− 1.84847i	−0.381828	−	0.924233i	$-0.624705\pi$
$577$	− 25619.8i	− 1.84847i	0.381828	−	0.924233i	$-0.375295\pi$
$578$	1078.56i	0.0776164i
$579$	−4333.88	−0.311071
$580$	0	0
$581$	8895.20	0.635172
$582$	12859.1i	0.915855i
$583$	− 8006.54i	− 0.568777i
$584$	−4447.30	−0.315121
$585$	0	0
$586$	−11217.7	−0.790782
$587$	− 23487.4i	− 1.65150i	−0.564039	−	0.825748i	$-0.690753\pi$
$587$	− 23487.4i	− 1.65150i	0.564039	−	0.825748i	$-0.309247\pi$
$588$	− 1098.47i	− 0.0770413i
$589$	28031.5	1.96098
$590$	0	0
$591$	8228.87	0.572742
$592$	579.544i	0.0402350i
$593$	22914.9i	1.58685i	0.608669	+	0.793424i	$0.291704\pi$
$593$	22914.9i	1.58685i	−0.608669	+	0.793424i	$0.708296\pi$
$594$	−19699.1	−1.36071
$595$	0	0
$596$	2893.27	0.198847
$597$	5643.79i	0.386909i
$598$	− 6213.72i	− 0.424913i
$599$	16868.5	1.15063	0.575315	−	0.817932i	$-0.304880\pi$
$599$	16868.5	1.15063	0.575315	+	0.817932i	$0.304880\pi$
$600$	0	0
$601$	−3820.28	−0.259289	−0.129644	−	0.991561i	$-0.541384\pi$
$601$	−3820.28	−0.259289	−0.129644	+	0.991561i	$0.541384\pi$
$602$	20524.1i	1.38953i
$603$	9992.06i	0.674806i
$604$	−8654.85	−0.583048
$605$	0	0
$606$	−4365.39	−0.292627
$607$	− 739.230i	− 0.0494307i	−0.999695	−	0.0247153i	$-0.992132\pi$
$607$	− 739.230i	− 0.0494307i	0.999695	−	0.0247153i	$-0.00786794\pi$
$608$	22370.6i	1.49218i
$609$	−8505.50	−0.565944
$610$	0	0
$611$	−1596.30	−0.105694
$612$	1968.82i	0.130040i
$613$	20527.3i	1.35251i	0.736666	+	0.676257i	$0.236399\pi$
$613$	20527.3i	1.35251i	−0.736666	+	0.676257i	$0.763601\pi$
$614$	−18328.7	−1.20470
$615$	0	0
$616$	−5325.04	−0.348299
$617$	6823.03i	0.445194i	0.974911	+	0.222597i	$0.0714534\pi$
$617$	6823.03i	0.445194i	−0.974911	+	0.222597i	$0.928547\pi$
$618$	− 6320.15i	− 0.411381i
$619$	−2964.75	−0.192510	−0.0962548	−	0.995357i	$-0.530686\pi$
$619$	−2964.75	−0.192510	−0.0962548	+	0.995357i	$0.530686\pi$
$620$	0	0
$621$	−24911.1	−1.60974
$622$	20337.5i	1.31103i
$623$	− 19627.2i	− 1.26219i
$624$	−1770.92	−0.113611
$625$	0	0
$626$	19679.5	1.25647
$627$	− 11387.1i	− 0.725288i
$628$	− 211.807i	− 0.0134586i
$629$	−129.156	−0.00818723
$630$	0	0
$631$	25883.6	1.63298	0.816491	−	0.577358i	$-0.195916\pi$
$631$	25883.6	1.63298	0.816491	+	0.577358i	$0.195916\pi$
$632$	− 9556.09i	− 0.601457i
$633$	− 4621.16i	− 0.290165i
$634$	−40073.1	−2.51026
$635$	0	0
$636$	3123.45	0.194737
$637$	− 576.321i	− 0.0358472i
$638$	29078.7i	1.80445i
$639$	−10723.0	−0.663841
$640$	0	0
$641$	−7806.60	−0.481033	−0.240517	−	0.970645i	$-0.577317\pi$
$641$	−7806.60	−0.481033	−0.240517	+	0.970645i	$0.577317\pi$
$642$	− 525.061i	− 0.0322781i
$643$	− 15373.9i	− 0.942902i	−0.881892	−	0.471451i	$-0.843730\pi$
$643$	− 15373.9i	− 0.942902i	0.881892	−	0.471451i	$-0.156270\pi$
$644$	19268.4	1.17901
$645$	0	0
$646$	−6369.38	−0.387925
$647$	14583.3i	0.886135i	0.896488	+	0.443068i	$0.146110\pi$
$647$	14583.3i	0.886135i	−0.896488	+	0.443068i	$0.853890\pi$
$648$	− 1392.70i	− 0.0844298i
$649$	−25063.0	−1.51588
$650$	0	0
$651$	−12654.4	−0.761848
$652$	− 19594.9i	− 1.17699i
$653$	− 7477.60i	− 0.448118i	−0.974576	−	0.224059i	$-0.928069\pi$
$653$	− 7477.60i	− 0.448118i	0.974576	−	0.224059i	$-0.0719308\pi$
$654$	−10102.8	−0.604051
$655$	0	0
$656$	5641.15	0.335747
$657$	− 11236.6i	− 0.667247i
$658$	− 11630.0i	− 0.689034i
$659$	18061.0	1.06761	0.533805	−	0.845607i	$-0.320762\pi$
$659$	18061.0	1.06761	0.533805	+	0.845607i	$0.320762\pi$
$660$	0	0
$661$	−3386.54	−0.199276	−0.0996379	−	0.995024i	$-0.531768\pi$
$661$	−3386.54	−0.199276	−0.0996379	+	0.995024i	$0.531768\pi$
$662$	14436.6i	0.847572i
$663$	− 394.662i	− 0.0231182i
$664$	−4146.14	−0.242322
$665$	0	0
$666$	−553.918	−0.0322280
$667$	36772.4i	2.13468i
$668$	− 19140.2i	− 1.10862i
$669$	−2824.98	−0.163258
$670$	0	0
$671$	−17694.0	−1.01799
$672$	− 10098.9i	− 0.579720i
$673$	− 5873.50i	− 0.336414i	−0.985752	−	0.168207i	$-0.946202\pi$
$673$	− 5873.50i	− 0.336414i	0.985752	−	0.168207i	$-0.0537977\pi$
$674$	3288.39	0.187929
$675$	0	0
$676$	−12596.2	−0.716671
$677$	− 21147.3i	− 1.20053i	−0.799803	−	0.600263i	$-0.795063\pi$
$677$	− 21147.3i	− 1.20053i	0.799803	−	0.600263i	$-0.204937\pi$
$678$	− 8823.88i	− 0.499822i
$679$	−20920.9	−1.18243
$680$	0	0
$681$	552.752	0.0311035
$682$	43262.9i	2.42906i
$683$	25556.1i	1.43174i	0.698234	+	0.715870i	$0.253970\pi$
$683$	25556.1i	1.43174i	−0.698234	+	0.715870i	$0.746030\pi$
$684$	−11626.7	−0.649939
$685$	0	0
$686$	25433.6	1.41554
$687$	− 17338.8i	− 0.962904i
$688$	− 25289.1i	− 1.40136i
$689$	1638.74	0.0906110
$690$	0	0
$691$	−2255.04	−0.124147	−0.0620737	−	0.998072i	$-0.519771\pi$
$691$	−2255.04	−0.124147	−0.0620737	+	0.998072i	$0.519771\pi$
$692$	− 5681.33i	− 0.312098i
$693$	− 13454.3i	− 0.737500i
$694$	−2696.26	−0.147476
$695$	0	0
$696$	3964.50	0.215911
$697$	1257.17i	0.0683196i
$698$	40048.0i	2.17169i
$699$	10295.8	0.557114
$700$	0	0
$701$	26613.7	1.43393	0.716966	−	0.697108i	$-0.245530\pi$
$701$	26613.7	1.43393	0.716966	+	0.697108i	$0.245530\pi$
$702$	− 4031.92i	− 0.216773i
$703$	− 762.719i	− 0.0409196i
$704$	−9190.30	−0.492006
$705$	0	0
$706$	−24638.6	−1.31344
$707$	− 7102.20i	− 0.377801i
$708$	− 9777.37i	− 0.519006i
$709$	12957.1	0.686339	0.343169	−	0.939274i	$-0.388500\pi$
$709$	12957.1	0.686339	0.343169	+	0.939274i	$0.388500\pi$
$710$	0	0
$711$	24144.5	1.27355
$712$	9148.42i	0.481533i
$713$	54709.4i	2.87361i
$714$	2875.35	0.150711
$715$	0	0
$716$	10111.8	0.527787
$717$	− 15741.3i	− 0.819902i
$718$	14316.3i	0.744123i
$719$	−2156.18	−0.111839	−0.0559194	−	0.998435i	$-0.517809\pi$
$719$	−2156.18	−0.111839	−0.0559194	+	0.998435i	$0.517809\pi$
$720$	0	0
$721$	10282.5	0.531122
$722$	− 12015.8i	− 0.619364i
$723$	− 5505.80i	− 0.283213i
$724$	−3560.28	−0.182758
$725$	0	0
$726$	4003.35	0.204653
$727$	20197.3i	1.03037i	0.857080	+	0.515183i	$0.172276\pi$
$727$	20197.3i	1.03037i	−0.857080	+	0.515183i	$0.827724\pi$
$728$	− 1089.90i	− 0.0554869i
$729$	−5859.60	−0.297698
$730$	0	0
$731$	5635.85	0.285156
$732$	− 6902.63i	− 0.348536i
$733$	31546.6i	1.58963i	0.606850	+	0.794816i	$0.292433\pi$
$733$	31546.6i	1.58963i	−0.606850	+	0.794816i	$0.707567\pi$
$734$	38326.7	1.92734
$735$	0	0
$736$	−43661.1	−2.18664
$737$	− 21234.6i	− 1.06131i
$738$	5391.71i	0.268932i
$739$	6903.36	0.343632	0.171816	−	0.985129i	$-0.445037\pi$
$739$	6903.36	0.343632	0.171816	+	0.985129i	$0.445037\pi$
$740$	0	0
$741$	2330.65	0.115544
$742$	11939.2i	0.590704i
$743$	− 17697.6i	− 0.873837i	−0.899501	−	0.436919i	$-0.856070\pi$
$743$	− 17697.6i	− 0.873837i	0.899501	−	0.436919i	$-0.143930\pi$
$744$	5898.32	0.290649
$745$	0	0
$746$	−10899.5	−0.534932
$747$	− 10475.7i	− 0.513100i
$748$	− 4184.03i	− 0.204523i
$749$	854.240	0.0416732
$750$	0	0
$751$	−15547.9	−0.755459	−0.377729	−	0.925916i	$-0.623295\pi$
$751$	−15547.9	−0.755459	−0.377729	+	0.925916i	$0.623295\pi$
$752$	14330.1i	0.694899i
$753$	2207.98i	0.106857i
$754$	−5951.68	−0.287463
$755$	0	0
$756$	12502.8	0.601483
$757$	− 9321.80i	− 0.447565i	−0.974639	−	0.223782i	$-0.928159\pi$
$757$	− 9321.80i	− 0.447565i	0.974639	−	0.223782i	$-0.0718405\pi$
$758$	− 17477.6i	− 0.837487i
$759$	22224.3	1.06283
$760$	0	0
$761$	3553.19	0.169255	0.0846275	−	0.996413i	$-0.473030\pi$
$761$	3553.19	0.169255	0.0846275	+	0.996413i	$0.473030\pi$
$762$	− 20924.6i	− 0.994776i
$763$	− 16436.5i	− 0.779872i
$764$	23679.1	1.12131
$765$	0	0
$766$	−35545.0	−1.67662
$767$	− 5129.76i	− 0.241493i
$768$	14591.0i	0.685556i
$769$	16346.5	0.766539	0.383269	−	0.923637i	$-0.374798\pi$
$769$	16346.5	0.766539	0.383269	+	0.923637i	$0.374798\pi$
$770$	0	0
$771$	−5220.99	−0.243877
$772$	− 9403.97i	− 0.438415i
$773$	25978.1i	1.20875i	0.796699	+	0.604376i	$0.206578\pi$
$773$	25978.1i	1.20875i	−0.796699	+	0.604376i	$0.793422\pi$
$774$	24170.8	1.12248
$775$	0	0
$776$	9751.46	0.451105
$777$	344.317i	0.0158974i
$778$	− 46238.8i	− 2.13077i
$779$	−7424.13	−0.341460
$780$	0	0
$781$	22787.9	1.04407
$782$	− 12431.2i	− 0.568464i
$783$	23860.6i	1.08903i
$784$	−5173.68	−0.235682
$785$	0	0
$786$	−789.797	−0.0358411
$787$	− 28832.9i	− 1.30595i	−0.757381	−	0.652973i	$-0.773521\pi$
$787$	− 28832.9i	− 1.30595i	0.757381	−	0.652973i	$-0.226479\pi$
$788$	17855.6i	0.807208i
$789$	860.028	0.0388058
$790$	0	0
$791$	14355.9	0.645304
$792$	6271.20i	0.281360i
$793$	− 3621.51i	− 0.162174i
$794$	10438.8	0.466573
$795$	0	0
$796$	−12246.3	−0.545300
$797$	1944.04i	0.0864010i	0.999066	+	0.0432005i	$0.0137554\pi$
$797$	1944.04i	0.0864010i	−0.999066	+	0.0432005i	$0.986245\pi$
$798$	16980.2i	0.753249i
$799$	−3193.56	−0.141402
$800$	0	0
$801$	−23114.5	−1.01961
$802$	40218.9i	1.77080i
$803$	23879.4i	1.04942i
$804$	8283.87	0.363370
$805$	0	0
$806$	−8854.83	−0.386970
$807$	19400.0i	0.846234i
$808$	3310.41i	0.144133i
$809$	3649.63	0.158608	0.0793042	−	0.996850i	$-0.474730\pi$
$809$	3649.63	0.158608	0.0793042	+	0.996850i	$0.474730\pi$
$810$	0	0
$811$	3446.54	0.149228	0.0746142	−	0.997212i	$-0.476227\pi$
$811$	3446.54	0.149228	0.0746142	+	0.997212i	$0.476227\pi$
$812$	− 18455.8i	− 0.797627i
$813$	− 8848.02i	− 0.381690i
$814$	1177.16	0.0506871
$815$	0	0
$816$	−3542.91	−0.151993
$817$	33282.1i	1.42521i
$818$	− 4572.81i	− 0.195458i
$819$	2753.76	0.117490
$820$	0	0
$821$	23999.4	1.02020	0.510101	−	0.860114i	$-0.329608\pi$
$821$	23999.4	1.02020	0.510101	+	0.860114i	$0.329608\pi$
$822$	19746.2i	0.837868i
$823$	− 45868.4i	− 1.94274i	−0.237578	−	0.971368i	$-0.576354\pi$
$823$	− 45868.4i	− 1.94274i	0.237578	−	0.971368i	$-0.423646\pi$
$824$	−4792.76	−0.202626
$825$	0	0
$826$	37373.5	1.57432
$827$	25703.9i	1.08079i	0.841412	+	0.540394i	$0.181725\pi$
$827$	25703.9i	1.08079i	−0.841412	+	0.540394i	$0.818275\pi$
$828$	− 22692.0i	− 0.952418i
$829$	−5582.56	−0.233884	−0.116942	−	0.993139i	$-0.537309\pi$
$829$	−5582.56	−0.233884	−0.116942	+	0.993139i	$0.537309\pi$
$830$	0	0
$831$	12792.4	0.534010
$832$	− 1881.02i	− 0.0783808i
$833$	− 1152.99i	− 0.0479578i
$834$	17207.3	0.714436
$835$	0	0
$836$	24708.5	1.02220
$837$	35499.5i	1.46600i
$838$	− 28445.0i	− 1.17257i
$839$	11733.1	0.482805	0.241402	−	0.970425i	$-0.422393\pi$
$839$	11733.1	0.482805	0.241402	+	0.970425i	$0.422393\pi$
$840$	0	0
$841$	10832.7	0.444162
$842$	59568.1i	2.43807i
$843$	16159.4i	0.660213i
$844$	10027.3	0.408952
$845$	0	0
$846$	−13696.4	−0.556611
$847$	6513.18i	0.264221i
$848$	− 14711.1i	− 0.595732i
$849$	22278.6	0.900588
$850$	0	0
$851$	1488.61	0.0599634
$852$	8889.84i	0.357466i
$853$	− 23737.8i	− 0.952834i	−0.879219	−	0.476417i	$-0.841935\pi$
$853$	− 23737.8i	− 0.952834i	0.879219	−	0.476417i	$-0.158065\pi$
$854$	26385.0	1.05723
$855$	0	0
$856$	−398.170	−0.0158986
$857$	− 13090.6i	− 0.521780i	−0.965369	−	0.260890i	$-0.915984\pi$
$857$	− 13090.6i	− 0.521780i	0.965369	−	0.260890i	$-0.0840160\pi$
$858$	3597.05i	0.143125i
$859$	14982.0	0.595086	0.297543	−	0.954708i	$-0.403833\pi$
$859$	14982.0	0.595086	0.297543	+	0.954708i	$0.403833\pi$
$860$	0	0
$861$	3351.51	0.132659
$862$	7300.47i	0.288463i
$863$	− 2666.74i	− 0.105188i	−0.998616	−	0.0525938i	$-0.983251\pi$
$863$	− 2666.74i	− 0.105188i	0.998616	−	0.0525938i	$-0.0167489\pi$
$864$	−28330.5	−1.11554
$865$	0	0
$866$	18009.1	0.706667
$867$	− 789.563i	− 0.0309284i
$868$	− 27458.3i	− 1.07373i
$869$	−51310.7	−2.00299
$870$	0	0
$871$	4346.19	0.169076
$872$	7661.23i	0.297525i
$873$	24638.2i	0.955184i
$874$	73411.6	2.84117
$875$	0	0
$876$	−9315.65	−0.359300
$877$	23168.6i	0.892074i	0.895014	+	0.446037i	$0.147165\pi$
$877$	23168.6i	0.892074i	−0.895014	+	0.446037i	$0.852835\pi$
$878$	− 36449.4i	− 1.40104i
$879$	8211.91	0.315109
$880$	0	0
$881$	−8797.34	−0.336424	−0.168212	−	0.985751i	$-0.553799\pi$
$881$	−8797.34	−0.336424	−0.168212	+	0.985751i	$0.553799\pi$
$882$	− 4944.91i	− 0.188780i
$883$	− 26761.1i	− 1.01991i	−0.860201	−	0.509955i	$-0.829662\pi$
$883$	− 26761.1i	− 1.01991i	0.860201	−	0.509955i	$-0.170338\pi$
$884$	856.366	0.0325822
$885$	0	0
$886$	−42751.0	−1.62105
$887$	24934.6i	0.943881i	0.881630	+	0.471940i	$0.156446\pi$
$887$	24934.6i	0.943881i	−0.881630	+	0.471940i	$0.843554\pi$
$888$	− 160.490i	− 0.00606496i
$889$	34043.0	1.28432
$890$	0	0
$891$	−7478.01	−0.281170
$892$	− 6129.84i	− 0.230092i
$893$	− 18859.3i	− 0.706723i
$894$	−4976.25	−0.186164
$895$	0	0
$896$	−15867.1	−0.591610
$897$	4548.76i	0.169318i
$898$	− 36262.3i	− 1.34754i
$899$	52402.3	1.94406
$900$	0	0
$901$	3278.47	0.121223
$902$	− 11458.2i	− 0.422966i
$903$	− 15024.7i	− 0.553699i
$904$	−6691.41	−0.246187
$905$	0	0
$906$	14885.8	0.545859
$907$	5006.91i	0.183298i	0.995791	+	0.0916492i	$0.0292139\pi$
$907$	5006.91i	0.183298i	−0.995791	+	0.0916492i	$0.970786\pi$
$908$	1199.40i	0.0438365i
$909$	−8364.12	−0.305193
$910$	0	0
$911$	1890.74	0.0687628	0.0343814	−	0.999409i	$-0.489054\pi$
$911$	1890.74	0.0687628	0.0343814	+	0.999409i	$0.489054\pi$
$912$	− 20922.4i	− 0.759660i
$913$	22262.4i	0.806986i
$914$	39185.8	1.41811
$915$	0	0
$916$	37622.9	1.35709
$917$	− 1284.95i	− 0.0462734i
$918$	− 8066.28i	− 0.290008i
$919$	12849.0	0.461206	0.230603	−	0.973048i	$-0.425930\pi$
$919$	12849.0	0.461206	0.230603	+	0.973048i	$0.425930\pi$
$920$	0	0
$921$	13417.5	0.480047
$922$	− 69602.3i	− 2.48615i
$923$	4664.12i	0.166329i
$924$	−11154.2	−0.397130
$925$	0	0
$926$	−41096.1	−1.45842
$927$	− 12109.5i	− 0.429047i
$928$	41819.9i	1.47931i
$929$	−49273.8	−1.74017	−0.870087	−	0.492899i	$-0.835937\pi$
$929$	−49273.8	−1.74017	−0.870087	+	0.492899i	$0.835937\pi$
$930$	0	0
$931$	6808.92	0.239692
$932$	22340.6i	0.785182i
$933$	− 14888.1i	− 0.522417i
$934$	−52975.4	−1.85590
$935$	0	0
$936$	−1283.56	−0.0448231
$937$	− 29936.4i	− 1.04373i	−0.853027	−	0.521867i	$-0.825236\pi$
$937$	− 29936.4i	− 1.04373i	0.853027	−	0.521867i	$-0.174764\pi$
$938$	31664.7i	1.10223i
$939$	−14406.4	−0.500675
$940$	0	0
$941$	8120.33	0.281313	0.140656	−	0.990058i	$-0.455079\pi$
$941$	8120.33	0.281313	0.140656	+	0.990058i	$0.455079\pi$
$942$	364.295i	0.0126002i
$943$	− 14489.8i	− 0.500374i
$944$	−46050.3	−1.58772
$945$	0	0
$946$	−51366.5	−1.76540
$947$	− 32978.0i	− 1.13162i	−0.824536	−	0.565809i	$-0.808564\pi$
$947$	− 32978.0i	− 1.13162i	0.824536	−	0.565809i	$-0.191436\pi$
$948$	− 20016.9i	− 0.685780i
$949$	−4887.52	−0.167182
$950$	0	0
$951$	29335.5	1.00028
$952$	− 2180.47i	− 0.0742325i
$953$	− 37722.3i	− 1.28221i	−0.767454	−	0.641104i	$-0.778477\pi$
$953$	− 37722.3i	− 1.28221i	0.767454	−	0.641104i	$-0.221523\pi$
$954$	14060.6	0.477178
$955$	0	0
$956$	34156.6	1.15555
$957$	− 21287.1i	− 0.719032i
$958$	54316.0i	1.83181i
$959$	−32125.7	−1.08175
$960$	0	0
$961$	48172.3	1.61701
$962$	240.934i	0.00807488i
$963$	− 1006.02i	− 0.0336642i
$964$	11946.9	0.399153
$965$	0	0
$966$	−33140.5	−1.10381
$967$	− 16615.2i	− 0.552542i	−0.961080	−	0.276271i	$-0.910901\pi$
$967$	− 16615.2i	− 0.552542i	0.961080	−	0.276271i	$-0.0890987\pi$
$968$	− 3035.86i	− 0.100802i
$969$	4662.71	0.154580
$970$	0	0
$971$	17678.7	0.584279	0.292140	−	0.956376i	$-0.405633\pi$
$971$	17678.7	0.584279	0.292140	+	0.956376i	$0.405633\pi$
$972$	− 23267.2i	− 0.767795i
$973$	27995.1i	0.922387i
$974$	30034.7	0.988064
$975$	0	0
$976$	−32510.6	−1.06623
$977$	− 15575.2i	− 0.510024i	−0.966938	−	0.255012i	$-0.917920\pi$
$977$	− 15575.2i	− 0.510024i	0.966938	−	0.255012i	$-0.0820795\pi$
$978$	33702.1i	1.10192i
$979$	49121.8	1.60361
$980$	0	0
$981$	−19357.0	−0.629990
$982$	− 25817.0i	− 0.838954i
$983$	51898.9i	1.68395i	0.539520	+	0.841973i	$0.318606\pi$
$983$	51898.9i	1.68395i	−0.539520	+	0.841973i	$0.681394\pi$
$984$	−1562.17	−0.0506100
$985$	0	0
$986$	−11907.0	−0.384579
$987$	8513.75i	0.274565i
$988$	5057.21i	0.162845i
$989$	−64957.2	−2.08849
$990$	0	0
$991$	−60599.6	−1.94249	−0.971246	−	0.238080i	$-0.923482\pi$
$991$	−60599.6	−1.94249	−0.971246	+	0.238080i	$0.923482\pi$
$992$	62219.0i	1.99139i
$993$	− 10568.3i	− 0.337739i
$994$	−33981.0	−1.08432
$995$	0	0
$996$	−8684.83	−0.276295
$997$	− 18476.9i	− 0.586929i	−0.955970	−	0.293464i	$-0.905192\pi$
$997$	− 18476.9i	− 0.586929i	0.955970	−	0.293464i	$-0.0948082\pi$
$998$	− 76512.3i	− 2.42681i
$999$	965.920	0.0305909

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	425.4.b.e.324.1		4
5.2	odd	4		425.4.a.e.1.2		2
5.3	odd	4		85.4.a.d.1.1	✓	2
5.4	even	2	inner	425.4.b.e.324.4		4
15.8	even	4		765.4.a.i.1.2		2
20.3	even	4		1360.4.a.m.1.2		2
85.33	odd	4		1445.4.a.i.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.4.a.d.1.1	✓	2	5.3	odd	4
425.4.a.e.1.2		2	5.2	odd	4
425.4.b.e.324.1		4	1.1	even	1	trivial
425.4.b.e.324.4		4	5.4	even	2	inner
765.4.a.i.1.2		2	15.8	even	4
1360.4.a.m.1.2		2	20.3	even	4
1445.4.a.i.1.1		2	85.33	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:	\( N \)	\(=\)	\( 425 = 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	425.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.73205i q^{2} +2.73205i q^{3} -5.92820 q^{4} +10.1962 q^{6} +16.5885i q^{7} -7.73205i q^{8} +19.5359 q^{9} -41.5167 q^{11} -16.1962i q^{12} -8.49742i q^{13} +61.9090 q^{14} -76.2820 q^{16} -17.0000i q^{17} -72.9090i q^{18} +100.392 q^{19} -45.3205 q^{21} +154.942i q^{22} +195.937i q^{23} +21.1244 q^{24} -31.7128 q^{26} +127.138i q^{27} -98.3397i q^{28} +187.674 q^{29} +279.219 q^{31} +222.832i q^{32} -113.426i q^{33} -63.4449 q^{34} -115.813 q^{36} -7.59739i q^{37} -374.669i q^{38} +23.2154 q^{39} -73.9512 q^{41} +169.138i q^{42} +331.520i q^{43} +246.119 q^{44} +731.247 q^{46} -187.856i q^{47} -208.406i q^{48} +67.8231 q^{49} +46.4449 q^{51} +50.3744i q^{52} +192.851i q^{53} +474.487 q^{54} +128.263 q^{56} +274.277i q^{57} -700.410i q^{58} +603.685 q^{59} +426.190 q^{61} -1042.06i q^{62} +324.070i q^{63} +221.364 q^{64} -423.310 q^{66} +511.472i q^{67} +100.779i q^{68} -535.310 q^{69} -548.886 q^{71} -151.053i q^{72} -575.177i q^{73} -28.3538 q^{74} -595.146 q^{76} -688.697i q^{77} -86.6410i q^{78} +1235.91 q^{79} +180.121 q^{81} +275.990i q^{82} -536.228i q^{83} +268.669 q^{84} +1237.25 q^{86} +512.736i q^{87} +321.009i q^{88} -1183.18 q^{89} +140.959 q^{91} -1161.56i q^{92} +762.841i q^{93} -701.090 q^{94} -608.788 q^{96} +1261.17i q^{97} -253.119i q^{98} -811.065 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4} + 20 q^{6} + 92 q^{9} - 76 q^{11} + 116 q^{14} - 28 q^{16} + 360 q^{19} - 112 q^{21} + 36 q^{24} - 16 q^{26} + 432 q^{29} + 140 q^{31} - 136 q^{34} + 188 q^{36} + 176 q^{39} + 688 q^{41} + 548 q^{44}+ \cdots - 1436 q^{99}+O(q^{100}) \) 4 * q + 4 * q^4 + 20 * q^6 + 92 * q^9 - 76 * q^11 + 116 * q^14 - 28 * q^16 + 360 * q^19 - 112 * q^21 + 36 * q^24 - 16 * q^26 + 432 * q^29 + 140 * q^31 - 136 * q^34 + 188 * q^36 + 176 * q^39 + 688 * q^41 + 548 * q^44 + 1380 * q^46 + 396 * q^49 + 68 * q^51 + 928 * q^54 + 132 * q^56 + 2872 * q^59 - 152 * q^61 + 1412 * q^64 - 848 * q^66 - 1296 * q^69 - 3020 * q^71 + 136 * q^74 + 72 * q^76 + 1916 * q^79 + 1732 * q^81 + 368 * q^84 + 2344 * q^86 + 408 * q^89 + 2864 * q^91 - 1488 * q^94 - 1292 * q^96 - 1436 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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