LMFDB - Embedded newform 425.4.b.g.324.4

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:	$6$
Coefficient field:	6.0.27206656.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 4x^{3} + 25x^{2} - 30x + 18 $ x^6 - 2x^5 + 2x^4 + 4x^3 + 25x^2 - 30*x + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 85)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			324.4
Root			$1.90022 - 1.90022i$ of defining polynomial
Character	$\chi$	$=$	425.324
Dual form			425.4.b.g.324.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.22168i q^{2} +1.15753i q^{3} +6.50750 q^{4} -1.41413 q^{6} +14.6743i q^{7} +17.7235i q^{8} +25.6601 q^{9} -71.3743 q^{11} +7.53262i q^{12} +28.4035i q^{13} -17.9273 q^{14} +30.4076 q^{16} +17.0000i q^{17} +31.3484i q^{18} -85.9592 q^{19} -16.9860 q^{21} -87.1964i q^{22} +7.17872i q^{23} -20.5154 q^{24} -34.7000 q^{26} +60.9556i q^{27} +95.4934i q^{28} +27.1610 q^{29} -15.6170 q^{31} +178.936i q^{32} -82.6178i q^{33} -20.7685 q^{34} +166.983 q^{36} -67.0036i q^{37} -105.014i q^{38} -32.8779 q^{39} -38.5701 q^{41} -20.7514i q^{42} +251.495i q^{43} -464.469 q^{44} -8.77008 q^{46} +57.9121i q^{47} +35.1977i q^{48} +127.663 q^{49} -19.6780 q^{51} +184.836i q^{52} +677.807i q^{53} -74.4680 q^{54} -260.081 q^{56} -99.5002i q^{57} +33.1819i q^{58} -598.645 q^{59} +346.063 q^{61} -19.0789i q^{62} +376.546i q^{63} +24.6588 q^{64} +100.932 q^{66} -849.923i q^{67} +110.628i q^{68} -8.30957 q^{69} -911.836 q^{71} +454.787i q^{72} -704.352i q^{73} +81.8568 q^{74} -559.380 q^{76} -1047.37i q^{77} -40.1662i q^{78} -55.8414 q^{79} +622.266 q^{81} -47.1202i q^{82} +1235.19i q^{83} -110.536 q^{84} -307.245 q^{86} +31.4396i q^{87} -1265.00i q^{88} +72.8069 q^{89} -416.803 q^{91} +46.7155i q^{92} -18.0771i q^{93} -70.7499 q^{94} -207.124 q^{96} +972.939i q^{97} +155.964i q^{98} -1831.47 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 20 q^{4} + 72 q^{6} - 78 q^{9} - 236 q^{11} + 136 q^{14} + 484 q^{16} + 320 q^{19} - 944 q^{21} - 1608 q^{24} - 88 q^{26} - 748 q^{29} + 140 q^{31} + 204 q^{34} + 2516 q^{36} - 496 q^{39} - 708 q^{41}+ \cdots - 1540 q^{99}+O(q^{100}) $ 6 * q - 20 * q^4 + 72 * q^6 - 78 * q^9 - 236 * q^11 + 136 * q^14 + 484 * q^16 + 320 * q^19 - 944 * q^21 - 1608 * q^24 - 88 * q^26 - 748 * q^29 + 140 * q^31 + 204 * q^34 + 2516 * q^36 - 496 * q^39 - 708 * q^41 - 696 * q^44 - 8 * q^46 - 2182 * q^49 - 136 * q^51 - 3120 * q^54 - 5992 * q^56 + 608 * q^59 + 492 * q^61 - 3652 * q^64 - 304 * q^66 + 2288 * q^69 - 1204 * q^71 - 1832 * q^74 - 3776 * q^76 - 412 * q^79 + 870 * q^81 + 8240 * q^84 + 512 * q^86 + 1740 * q^89 - 2912 * q^91 - 4448 * q^94 + 8136 * q^96 - 1540 * 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 $

$2$	1.22168i	0.431928i	0.976401	+	0.215964i	$0.0692894\pi$
$2$	1.22168i	0.431928i	−0.976401	+	0.215964i	$0.930711\pi$
$3$	1.15753i	0.222766i	0.993778	+	0.111383i	$0.0355281\pi$
$3$	1.15753i	0.222766i	−0.993778	+	0.111383i	$0.964472\pi$
$4$	6.50750	0.813438
$5$	0	0
$5$	0	0
$6$	−1.41413	−0.0962191
$7$	14.6743i	0.792340i	0.918177	+	0.396170i	$0.129661\pi$
$7$	14.6743i	0.792340i	−0.918177	+	0.396170i	$0.870339\pi$
$8$	17.7235i	0.783275i
$9$	25.6601	0.950375
$10$	0	0
$11$	−71.3743	−1.95638	−0.978189	−	0.207716i	$-0.933397\pi$
$11$	−71.3743	−1.95638	−0.978189	+	0.207716i	$0.933397\pi$
$12$	7.53262i	0.181207i
$13$	28.4035i	0.605979i	0.952994	+	0.302989i	$0.0979847\pi$
$13$	28.4035i	0.605979i	−0.952994	+	0.302989i	$0.902015\pi$
$14$	−17.9273	−0.342234
$15$	0	0
$16$	30.4076	0.475119
$17$	17.0000i	0.242536i
$17$	17.0000i	0.242536i
$18$	31.3484i	0.410494i
$19$	−85.9592	−1.03792	−0.518958	−	0.854800i	$-0.673680\pi$
$19$	−85.9592	−1.03792	−0.518958	+	0.854800i	$0.673680\pi$
$20$	0	0
$21$	−16.9860	−0.176507
$22$	− 87.1964i	− 0.845015i
$23$	7.17872i	0.0650811i	0.999470	+	0.0325406i	$0.0103598\pi$
$23$	7.17872i	0.0650811i	−0.999470	+	0.0325406i	$0.989640\pi$
$24$	−20.5154	−0.174487
$25$	0	0
$26$	−34.7000	−0.261739
$27$	60.9556i	0.434478i
$28$	95.4934i	0.644520i
$29$	27.1610	0.173919	0.0869597	−	0.996212i	$-0.472285\pi$
$29$	27.1610	0.173919	0.0869597	+	0.996212i	$0.472285\pi$
$30$	0	0
$31$	−15.6170	−0.0904803	−0.0452401	−	0.998976i	$-0.514405\pi$
$31$	−15.6170	−0.0904803	−0.0452401	+	0.998976i	$0.514405\pi$
$32$	178.936i	0.988493i
$33$	− 82.6178i	− 0.435815i
$34$	−20.7685	−0.104758
$35$	0	0
$36$	166.983	0.773071
$37$	− 67.0036i	− 0.297712i	−0.988859	−	0.148856i	$-0.952441\pi$
$37$	− 67.0036i	− 0.297712i	0.988859	−	0.148856i	$-0.0475590\pi$
$38$	− 105.014i	− 0.448305i
$39$	−32.8779	−0.134992
$40$	0	0
$41$	−38.5701	−0.146918	−0.0734590	−	0.997298i	$-0.523404\pi$
$41$	−38.5701	−0.146918	−0.0734590	+	0.997298i	$0.523404\pi$
$42$	− 20.7514i	− 0.0762383i
$43$	251.495i	0.891920i	0.895053	+	0.445960i	$0.147138\pi$
$43$	251.495i	0.891920i	−0.895053	+	0.445960i	$0.852862\pi$
$44$	−464.469	−1.59139
$45$	0	0
$46$	−8.77008	−0.0281104
$47$	57.9121i	0.179731i	0.995954	+	0.0898654i	$0.0286437\pi$
$47$	57.9121i	0.179731i	−0.995954	+	0.0898654i	$0.971356\pi$
$48$	35.1977i	0.105841i
$49$	127.663	0.372197
$50$	0	0
$51$	−19.6780	−0.0540288
$52$	184.836i	0.492926i
$53$	677.807i	1.75668i	0.478039	+	0.878339i	$0.341348\pi$
$53$	677.807i	1.75668i	−0.478039	+	0.878339i	$0.658652\pi$
$54$	−74.4680	−0.187663
$55$	0	0
$56$	−260.081	−0.620620
$57$	− 99.5002i	− 0.231213i
$58$	33.1819i	0.0751207i
$59$	−598.645	−1.32097	−0.660483	−	0.750841i	$-0.729648\pi$
$59$	−598.645	−1.32097	−0.660483	+	0.750841i	$0.729648\pi$
$60$	0	0
$61$	346.063	0.726375	0.363187	−	0.931716i	$-0.381689\pi$
$61$	346.063	0.726375	0.363187	+	0.931716i	$0.381689\pi$
$62$	− 19.0789i	− 0.0390810i
$63$	376.546i	0.753021i
$64$	24.6588	0.0481617
$65$	0	0
$66$	100.932	0.188241
$67$	− 849.923i	− 1.54977i	−0.632102	−	0.774885i	$-0.717808\pi$
$67$	− 849.923i	− 1.54977i	0.632102	−	0.774885i	$-0.282192\pi$
$68$	110.628i	0.197288i
$69$	−8.30957	−0.0144979
$70$	0	0
$71$	−911.836	−1.52415	−0.762077	−	0.647486i	$-0.775820\pi$
$71$	−911.836	−1.52415	−0.762077	+	0.647486i	$0.775820\pi$
$72$	454.787i	0.744405i
$73$	− 704.352i	− 1.12929i	−0.825334	−	0.564645i	$-0.809013\pi$
$73$	− 704.352i	− 1.12929i	0.825334	−	0.564645i	$-0.190987\pi$
$74$	81.8568	0.128590
$75$	0	0
$76$	−559.380	−0.844280
$77$	− 1047.37i	− 1.55012i
$78$	− 40.1662i	− 0.0583067i
$79$	−55.8414	−0.0795272	−0.0397636	−	0.999209i	$-0.512660\pi$
$79$	−55.8414	−0.0795272	−0.0397636	+	0.999209i	$0.512660\pi$
$80$	0	0
$81$	622.266	0.853588
$82$	− 47.1202i	− 0.0634580i
$83$	1235.19i	1.63350i	0.576995	+	0.816748i	$0.304225\pi$
$83$	1235.19i	1.63350i	−0.576995	+	0.816748i	$0.695775\pi$
$84$	−110.536	−0.143577
$85$	0	0
$86$	−307.245	−0.385246
$87$	31.4396i	0.0387434i
$88$	− 1265.00i	− 1.53238i
$89$	72.8069	0.0867136	0.0433568	−	0.999060i	$-0.486195\pi$
$89$	72.8069	0.0867136	0.0433568	+	0.999060i	$0.486195\pi$
$90$	0	0
$91$	−416.803	−0.480141
$92$	46.7155i	0.0529395i
$93$	− 18.0771i	− 0.0201560i
$94$	−70.7499	−0.0776308
$95$	0	0
$96$	−207.124	−0.220203
$97$	972.939i	1.01842i	0.860642	+	0.509211i	$0.170063\pi$
$97$	972.939i	1.01842i	−0.860642	+	0.509211i	$0.829937\pi$
$98$	155.964i	0.160762i
$99$	−1831.47	−1.85929
$100$	0	0
$101$	−658.119	−0.648369	−0.324185	−	0.945994i	$-0.605090\pi$
$101$	−658.119	−0.648369	−0.324185	+	0.945994i	$0.605090\pi$
$102$	− 24.0401i	− 0.0233366i
$103$	1727.68i	1.65275i	0.563120	+	0.826375i	$0.309601\pi$
$103$	1727.68i	1.65275i	−0.563120	+	0.826375i	$0.690399\pi$
$104$	−503.410	−0.474648
$105$	0	0
$106$	−828.061	−0.758758
$107$	− 903.004i	− 0.815857i	−0.913014	−	0.407929i	$-0.866251\pi$
$107$	− 903.004i	− 0.815857i	0.913014	−	0.407929i	$-0.133749\pi$
$108$	396.669i	0.353421i
$109$	−833.557	−0.732480	−0.366240	−	0.930520i	$-0.619355\pi$
$109$	−833.557	−0.732480	−0.366240	+	0.930520i	$0.619355\pi$
$110$	0	0
$111$	77.5586	0.0663201
$112$	446.212i	0.376456i
$113$	− 1006.49i	− 0.837900i	−0.908009	−	0.418950i	$-0.862398\pi$
$113$	− 1006.49i	− 0.837900i	0.908009	−	0.418950i	$-0.137602\pi$
$114$	121.557	0.0998673
$115$	0	0
$116$	176.750	0.141473
$117$	728.838i	0.575907i
$118$	− 731.352i	− 0.570562i
$119$	−249.464	−0.192171
$120$	0	0
$121$	3763.29	2.82742
$122$	422.778i	0.313742i
$123$	− 44.6459i	− 0.0327284i
$124$	−101.627	−0.0736001
$125$	0	0
$126$	−460.017	−0.325251
$127$	1114.87i	0.778970i	0.921033	+	0.389485i	$0.127347\pi$
$127$	1114.87i	0.778970i	−0.921033	+	0.389485i	$0.872653\pi$
$128$	1461.62i	1.00929i
$129$	−291.112	−0.198690
$130$	0	0
$131$	1558.43	1.03939	0.519697	−	0.854351i	$-0.326045\pi$
$131$	1558.43	1.03939	0.519697	+	0.854351i	$0.326045\pi$
$132$	− 537.635i	− 0.354509i
$133$	− 1261.40i	− 0.822382i
$134$	1038.33	0.669389
$135$	0	0
$136$	−301.299	−0.189972
$137$	235.190i	0.146669i	0.997307	+	0.0733346i	$0.0233641\pi$
$137$	235.190i	0.146669i	−0.997307	+	0.0733346i	$0.976636\pi$
$138$	− 10.1516i	− 0.00626205i
$139$	2138.46	1.30490	0.652452	−	0.757830i	$-0.273740\pi$
$139$	2138.46	1.30490	0.652452	+	0.757830i	$0.273740\pi$
$140$	0	0
$141$	−67.0349	−0.0400380
$142$	− 1113.97i	− 0.658325i
$143$	− 2027.28i	− 1.18552i
$144$	780.264	0.451542
$145$	0	0
$146$	860.491	0.487772
$147$	147.774i	0.0829129i
$148$	− 436.026i	− 0.242170i
$149$	3317.71	1.82414	0.912072	−	0.410030i	$-0.134482\pi$
$149$	3317.71	1.82414	0.912072	+	0.410030i	$0.134482\pi$
$150$	0	0
$151$	848.486	0.457277	0.228638	−	0.973511i	$-0.426573\pi$
$151$	848.486	0.457277	0.228638	+	0.973511i	$0.426573\pi$
$152$	− 1523.50i	− 0.812973i
$153$	436.222i	0.230500i
$154$	1279.55	0.669539
$155$	0	0
$156$	−213.953	−0.109807
$157$	949.821i	0.482828i	0.970422	+	0.241414i	$0.0776111\pi$
$157$	949.821i	0.482828i	−0.970422	+	0.241414i	$0.922389\pi$
$158$	− 68.2202i	− 0.0343500i
$159$	−784.580	−0.391329
$160$	0	0
$161$	−105.343	−0.0515664
$162$	760.208i	0.368689i
$163$	− 1942.09i	− 0.933229i	−0.884461	−	0.466615i	$-0.845473\pi$
$163$	− 1942.09i	− 0.933229i	0.884461	−	0.466615i	$-0.154527\pi$
$164$	−250.995	−0.119509
$165$	0	0
$166$	−1509.01	−0.705553
$167$	− 3358.46i	− 1.55620i	−0.628141	−	0.778100i	$-0.716184\pi$
$167$	− 3358.46i	− 1.55620i	0.628141	−	0.778100i	$-0.283816\pi$
$168$	− 301.051i	− 0.138253i
$169$	1390.24	0.632790
$170$	0	0
$171$	−2205.72	−0.986409
$172$	1636.60i	0.725522i
$173$	− 216.466i	− 0.0951308i	−0.998868	−	0.0475654i	$-0.984854\pi$
$173$	− 216.466i	− 0.0951308i	0.998868	−	0.0475654i	$-0.0151463\pi$
$174$	−38.4090	−0.0167344
$175$	0	0
$176$	−2170.32	−0.929513
$177$	− 692.949i	− 0.294267i
$178$	88.9465i	0.0374541i
$179$	1142.30	0.476981	0.238490	−	0.971145i	$-0.423347\pi$
$179$	1142.30	0.476981	0.238490	+	0.971145i	$0.423347\pi$
$180$	0	0
$181$	1884.85	0.774031	0.387015	−	0.922073i	$-0.373506\pi$
$181$	1884.85	0.774031	0.387015	+	0.922073i	$0.373506\pi$
$182$	− 509.199i	− 0.207387i
$183$	400.578i	0.161812i
$184$	−127.232	−0.0509764
$185$	0	0
$186$	22.0843	0.00870593
$187$	− 1213.36i	− 0.474491i
$188$	376.863i	0.146200i
$189$	−894.483	−0.344254
$190$	0	0
$191$	2274.84	0.861790	0.430895	−	0.902402i	$-0.358198\pi$
$191$	2274.84	0.861790	0.430895	+	0.902402i	$0.358198\pi$
$192$	28.5432i	0.0107288i
$193$	2905.75i	1.08373i	0.840465	+	0.541866i	$0.182282\pi$
$193$	2905.75i	1.08373i	−0.840465	+	0.541866i	$0.817718\pi$
$194$	−1188.62	−0.439885
$195$	0	0
$196$	830.771	0.302759
$197$	4837.99i	1.74971i	0.484388	+	0.874853i	$0.339042\pi$
$197$	4837.99i	1.74971i	−0.484388	+	0.874853i	$0.660958\pi$
$198$	− 2237.47i	− 0.803081i
$199$	2365.26	0.842558	0.421279	−	0.906931i	$-0.361581\pi$
$199$	2365.26	0.842558	0.421279	+	0.906931i	$0.361581\pi$
$200$	0	0
$201$	983.809	0.345237
$202$	− 804.009i	− 0.280049i
$203$	398.569i	0.137803i
$204$	−128.055	−0.0439491
$205$	0	0
$206$	−2110.67	−0.713869
$207$	184.207i	0.0618515i
$208$	863.685i	0.287912i
$209$	6135.28	2.03056
$210$	0	0
$211$	594.164	0.193858	0.0969288	−	0.995291i	$-0.469098\pi$
$211$	594.164	0.193858	0.0969288	+	0.995291i	$0.469098\pi$
$212$	4410.83i	1.42895i
$213$	− 1055.48i	− 0.339530i
$214$	1103.18	0.352392
$215$	0	0
$216$	−1080.35	−0.340316
$217$	− 229.169i	− 0.0716912i
$218$	− 1018.34i	− 0.316379i
$219$	815.307	0.251568
$220$	0	0
$221$	−482.860	−0.146971
$222$	94.7516i	0.0286455i
$223$	− 3040.15i	− 0.912931i	−0.889741	−	0.456466i	$-0.849115\pi$
$223$	− 3040.15i	− 0.912931i	0.889741	−	0.456466i	$-0.150885\pi$
$224$	−2625.77	−0.783223
$225$	0	0
$226$	1229.61	0.361912
$227$	− 459.136i	− 0.134246i	−0.997745	−	0.0671232i	$-0.978618\pi$
$227$	− 459.136i	− 0.134246i	0.997745	−	0.0671232i	$-0.0213821\pi$
$228$	− 647.498i	− 0.188077i
$229$	−2153.49	−0.621427	−0.310713	−	0.950504i	$-0.600568\pi$
$229$	−2153.49	−0.621427	−0.310713	+	0.950504i	$0.600568\pi$
$230$	0	0
$231$	1212.36	0.345314
$232$	481.387i	0.136227i
$233$	− 5456.45i	− 1.53418i	−0.641540	−	0.767090i	$-0.721704\pi$
$233$	− 5456.45i	− 1.53418i	0.641540	−	0.767090i	$-0.278296\pi$
$234$	−890.405	−0.248750
$235$	0	0
$236$	−3895.69	−1.07452
$237$	− 64.6380i	− 0.0177160i
$238$	− 304.764i	− 0.0830040i
$239$	6073.58	1.64380	0.821898	−	0.569634i	$-0.192915\pi$
$239$	6073.58	1.64380	0.821898	+	0.569634i	$0.192915\pi$
$240$	0	0
$241$	7191.69	1.92223	0.961115	−	0.276148i	$-0.0890581\pi$
$241$	7191.69	1.92223	0.961115	+	0.276148i	$0.0890581\pi$
$242$	4597.53i	1.22124i
$243$	2366.09i	0.624629i
$244$	2252.01	0.590861
$245$	0	0
$246$	54.5429	0.0141363
$247$	− 2441.54i	− 0.628955i
$248$	− 276.787i	− 0.0708710i
$249$	−1429.77	−0.363888
$250$	0	0
$251$	−3778.49	−0.950185	−0.475092	−	0.879936i	$-0.657585\pi$
$251$	−3778.49	−0.950185	−0.475092	+	0.879936i	$0.657585\pi$
$252$	2450.37i	0.612536i
$253$	− 512.376i	− 0.127323i
$254$	−1362.02	−0.336459
$255$	0	0
$256$	−1588.35	−0.387781
$257$	− 5002.13i	− 1.21410i	−0.794663	−	0.607051i	$-0.792352\pi$
$257$	− 5002.13i	− 1.21410i	0.794663	−	0.607051i	$-0.207648\pi$
$258$	− 355.645i	− 0.0858197i
$259$	983.235	0.235889
$260$	0	0
$261$	696.954	0.165289
$262$	1903.90i	0.448943i
$263$	− 4745.06i	− 1.11252i	−0.831008	−	0.556260i	$-0.812236\pi$
$263$	− 4745.06i	− 1.11252i	0.831008	−	0.556260i	$-0.187764\pi$
$264$	1464.27	0.341363
$265$	0	0
$266$	1541.02	0.355210
$267$	84.2760i	0.0193169i
$268$	− 5530.88i	− 1.26064i
$269$	−7844.29	−1.77797	−0.888987	−	0.457932i	$-0.848590\pi$
$269$	−7844.29	−1.77797	−0.888987	+	0.457932i	$0.848590\pi$
$270$	0	0
$271$	3730.24	0.836148	0.418074	−	0.908413i	$-0.362705\pi$
$271$	3730.24	0.836148	0.418074	+	0.908413i	$0.362705\pi$
$272$	516.930i	0.115233i
$273$	− 482.462i	− 0.106959i
$274$	−287.327	−0.0633505
$275$	0	0
$276$	−54.0745	−0.0117931
$277$	− 1888.55i	− 0.409646i	−0.978799	−	0.204823i	$-0.934338\pi$
$277$	− 1888.55i	− 0.409646i	0.978799	−	0.204823i	$-0.0656619\pi$
$278$	2612.51i	0.563625i
$279$	−400.733	−0.0859902
$280$	0	0
$281$	−3340.77	−0.709231	−0.354615	−	0.935012i	$-0.615388\pi$
$281$	−3340.77	−0.709231	−0.354615	+	0.935012i	$0.615388\pi$
$282$	− 81.8950i	− 0.0172935i
$283$	7783.21i	1.63485i	0.576032	+	0.817427i	$0.304600\pi$
$283$	7783.21i	1.63485i	−0.576032	+	0.817427i	$0.695400\pi$
$284$	−5933.77	−1.23981
$285$	0	0
$286$	2476.68	0.512061
$287$	− 565.991i	− 0.116409i
$288$	4591.53i	0.939439i
$289$	−289.000	−0.0588235
$290$	0	0
$291$	−1126.20	−0.226870
$292$	− 4583.57i	− 0.918608i
$293$	8514.61i	1.69771i	0.528626	+	0.848855i	$0.322707\pi$
$293$	8514.61i	1.69771i	−0.528626	+	0.848855i	$0.677293\pi$
$294$	−180.532	−0.0358124
$295$	0	0
$296$	1187.54	0.233190
$297$	− 4350.66i	− 0.850003i
$298$	4053.17i	0.787899i
$299$	−203.901	−0.0394378
$300$	0	0
$301$	−3690.52	−0.706705
$302$	1036.58i	0.197511i
$303$	− 761.792i	− 0.144435i
$304$	−2613.82	−0.493134
$305$	0	0
$306$	−532.923	−0.0995594
$307$	− 8954.07i	− 1.66461i	−0.554317	−	0.832306i	$-0.687020\pi$
$307$	− 8954.07i	− 1.66461i	0.554317	−	0.832306i	$-0.312980\pi$
$308$	− 6815.77i	− 1.26092i
$309$	−1999.84	−0.368177
$310$	0	0
$311$	−3842.82	−0.700664	−0.350332	−	0.936626i	$-0.613931\pi$
$311$	−3842.82	−0.700664	−0.350332	+	0.936626i	$0.613931\pi$
$312$	− 582.711i	− 0.105736i
$313$	4432.84i	0.800507i	0.916404	+	0.400254i	$0.131078\pi$
$313$	4432.84i	0.800507i	−0.916404	+	0.400254i	$0.868922\pi$
$314$	−1160.37	−0.208547
$315$	0	0
$316$	−363.388	−0.0646904
$317$	7694.21i	1.36325i	0.731702	+	0.681625i	$0.238726\pi$
$317$	7694.21i	1.36325i	−0.731702	+	0.681625i	$0.761274\pi$
$318$	− 958.504i	− 0.169026i
$319$	−1938.59	−0.340252
$320$	0	0
$321$	1045.25	0.181746
$322$	− 128.695i	− 0.0222730i
$323$	− 1461.31i	− 0.251731i
$324$	4049.40	0.694341
$325$	0	0
$326$	2372.61	0.403088
$327$	− 964.866i	− 0.163172i
$328$	− 683.596i	− 0.115077i
$329$	−849.823	−0.142408
$330$	0	0
$331$	7276.70	1.20835	0.604175	−	0.796852i	$-0.293503\pi$
$331$	7276.70	1.20835	0.604175	+	0.796852i	$0.293503\pi$
$332$	8038.03i	1.32875i
$333$	− 1719.32i	− 0.282938i
$334$	4102.95	0.672166
$335$	0	0
$336$	−516.503	−0.0838618
$337$	− 3300.91i	− 0.533566i	−0.963757	−	0.266783i	$-0.914039\pi$
$337$	− 3300.91i	− 0.533566i	0.963757	−	0.266783i	$-0.0859607\pi$
$338$	1698.42i	0.273320i
$339$	1165.04	0.186656
$340$	0	0
$341$	1114.65	0.177014
$342$	− 2694.68i	− 0.426058i
$343$	6906.68i	1.08725i
$344$	−4457.36	−0.698619
$345$	0	0
$346$	264.452	0.0410897
$347$	− 7998.41i	− 1.23740i	−0.785628	−	0.618699i	$-0.787660\pi$
$347$	− 7998.41i	− 1.23740i	0.785628	−	0.618699i	$-0.212340\pi$
$348$	204.593i	0.0315154i
$349$	4378.96	0.671634	0.335817	−	0.941927i	$-0.390988\pi$
$349$	4378.96	0.671634	0.335817	+	0.941927i	$0.390988\pi$
$350$	0	0
$351$	−1731.35	−0.263284
$352$	− 12771.5i	− 1.93387i
$353$	− 1508.37i	− 0.227430i	−0.993513	−	0.113715i	$-0.963725\pi$
$353$	− 1508.37i	− 0.227430i	0.993513	−	0.113715i	$-0.0362750\pi$
$354$	846.560	0.127102
$355$	0	0
$356$	473.791	0.0705362
$357$	− 288.761i	− 0.0428092i
$358$	1395.52i	0.206021i
$359$	−3436.61	−0.505229	−0.252614	−	0.967567i	$-0.581290\pi$
$359$	−3436.61	−0.505229	−0.252614	+	0.967567i	$0.581290\pi$
$360$	0	0
$361$	529.984	0.0772684
$362$	2302.67i	0.334326i
$363$	4356.11i	0.629853i
$364$	−2712.35	−0.390565
$365$	0	0
$366$	−489.377	−0.0698911
$367$	10656.6i	1.51573i	0.652413	+	0.757863i	$0.273757\pi$
$367$	10656.6i	1.51573i	−0.652413	+	0.757863i	$0.726243\pi$
$368$	218.288i	0.0309213i
$369$	−989.713	−0.139627
$370$	0	0
$371$	−9946.37	−1.39189
$372$	− 117.637i	− 0.0163956i
$373$	9062.32i	1.25799i	0.777410	+	0.628994i	$0.216533\pi$
$373$	9062.32i	1.25799i	−0.777410	+	0.628994i	$0.783467\pi$
$374$	1482.34	0.204946
$375$	0	0
$376$	−1026.40	−0.140779
$377$	771.467i	0.105391i
$378$	− 1092.77i	− 0.148693i
$379$	−1080.44	−0.146434	−0.0732169	−	0.997316i	$-0.523327\pi$
$379$	−1080.44	−0.146434	−0.0732169	+	0.997316i	$0.523327\pi$
$380$	0	0
$381$	−1290.50	−0.173528
$382$	2779.13i	0.372232i
$383$	− 6920.65i	− 0.923313i	−0.887059	−	0.461656i	$-0.847255\pi$
$383$	− 6920.65i	− 0.923313i	0.887059	−	0.461656i	$-0.152745\pi$
$384$	−1691.86	−0.224837
$385$	0	0
$386$	−3549.88	−0.468094
$387$	6453.39i	0.847659i
$388$	6331.40i	0.828423i
$389$	−9664.52	−1.25967	−0.629833	−	0.776730i	$-0.716877\pi$
$389$	−9664.52	−1.25967	−0.629833	+	0.776730i	$0.716877\pi$
$390$	0	0
$391$	−122.038	−0.0157845
$392$	2262.64i	0.291532i
$393$	1803.92i	0.231542i
$394$	−5910.46	−0.755748
$395$	0	0
$396$	−11918.3	−1.51242
$397$	− 7464.80i	− 0.943697i	−0.881680	−	0.471848i	$-0.843587\pi$
$397$	− 7464.80i	− 0.943697i	0.881680	−	0.471848i	$-0.156413\pi$
$398$	2889.59i	0.363924i
$399$	1460.10	0.183199
$400$	0	0
$401$	2948.70	0.367209	0.183605	−	0.983000i	$-0.441223\pi$
$401$	2948.70	0.367209	0.183605	+	0.983000i	$0.441223\pi$
$402$	1201.90i	0.149117i
$403$	− 443.577i	− 0.0548291i
$404$	−4282.71	−0.527408
$405$	0	0
$406$	−486.923	−0.0595212
$407$	4782.34i	0.582436i
$408$	− 348.762i	− 0.0423194i
$409$	−5546.04	−0.670499	−0.335250	−	0.942129i	$-0.608821\pi$
$409$	−5546.04	−0.670499	−0.335250	+	0.942129i	$0.608821\pi$
$410$	0	0
$411$	−272.239	−0.0326730
$412$	11242.9i	1.34441i
$413$	− 8784.73i	− 1.04665i
$414$	−225.041	−0.0267154
$415$	0	0
$416$	−5082.42	−0.599005
$417$	2475.33i	0.290689i
$418$	7495.33i	0.877054i
$419$	−8245.75	−0.961411	−0.480706	−	0.876882i	$-0.659619\pi$
$419$	−8245.75	−0.961411	−0.480706	+	0.876882i	$0.659619\pi$
$420$	0	0
$421$	−9972.70	−1.15449	−0.577244	−	0.816571i	$-0.695872\pi$
$421$	−9972.70	−1.15449	−0.577244	+	0.816571i	$0.695872\pi$
$422$	725.877i	0.0837325i
$423$	1486.03i	0.170812i
$424$	−12013.1	−1.37596
$425$	0	0
$426$	1289.45	0.146653
$427$	5078.25i	0.575536i
$428$	− 5876.30i	− 0.663649i
$429$	2346.64	0.264095
$430$	0	0
$431$	10506.0	1.17415	0.587073	−	0.809534i	$-0.300280\pi$
$431$	10506.0	1.17415	0.587073	+	0.809534i	$0.300280\pi$
$432$	1853.52i	0.206429i
$433$	− 13866.9i	− 1.53903i	−0.638631	−	0.769514i	$-0.720499\pi$
$433$	− 13866.9i	− 1.53903i	0.638631	−	0.769514i	$-0.279501\pi$
$434$	279.970	0.0309654
$435$	0	0
$436$	−5424.38	−0.595827
$437$	− 617.077i	− 0.0675487i
$438$	996.043i	0.108659i
$439$	−270.717	−0.0294319	−0.0147160	−	0.999892i	$-0.504684\pi$
$439$	−270.717	−0.0294319	−0.0147160	+	0.999892i	$0.504684\pi$
$440$	0	0
$441$	3275.86	0.353726
$442$	− 589.899i	− 0.0634811i
$443$	6809.29i	0.730291i	0.930950	+	0.365146i	$0.118981\pi$
$443$	6809.29i	0.730291i	−0.930950	+	0.365146i	$0.881019\pi$
$444$	504.713	0.0539473
$445$	0	0
$446$	3714.09	0.394321
$447$	3840.34i	0.406358i
$448$	361.852i	0.0381605i
$449$	9130.09	0.959633	0.479817	−	0.877369i	$-0.340703\pi$
$449$	9130.09	0.959633	0.479817	+	0.877369i	$0.340703\pi$
$450$	0	0
$451$	2752.91	0.287427
$452$	− 6549.74i	− 0.681579i
$453$	982.146i	0.101866i
$454$	560.916	0.0579848
$455$	0	0
$456$	1763.49	0.181103
$457$	− 2524.80i	− 0.258436i	−0.991616	−	0.129218i	$-0.958753\pi$
$457$	− 2524.80i	− 0.258436i	0.991616	−	0.129218i	$-0.0412467\pi$
$458$	− 2630.87i	− 0.268412i
$459$	−1036.24	−0.105376
$460$	0	0
$461$	11102.1	1.12164	0.560819	−	0.827938i	$-0.310486\pi$
$461$	11102.1	1.12164	0.560819	+	0.827938i	$0.310486\pi$
$462$	1481.11i	0.149151i
$463$	3045.92i	0.305736i	0.988247	+	0.152868i	$0.0488510\pi$
$463$	3045.92i	0.305736i	−0.988247	+	0.152868i	$0.951149\pi$
$464$	825.901	0.0826325
$465$	0	0
$466$	6666.02	0.662655
$467$	− 14515.5i	− 1.43832i	−0.694844	−	0.719161i	$-0.744527\pi$
$467$	− 14515.5i	− 1.43832i	0.694844	−	0.719161i	$-0.255473\pi$
$468$	4742.92i	0.468465i
$469$	12472.1	1.22795
$470$	0	0
$471$	−1099.44	−0.107558
$472$	− 10610.1i	− 1.03468i
$473$	− 17950.3i	− 1.74493i
$474$	78.9668	0.00765203
$475$	0	0
$476$	−1623.39	−0.156319
$477$	17392.6i	1.66950i
$478$	7419.96i	0.710002i
$479$	5511.93	0.525775	0.262888	−	0.964826i	$-0.415325\pi$
$479$	5511.93	0.525775	0.262888	+	0.964826i	$0.415325\pi$
$480$	0	0
$481$	1903.14	0.180407
$482$	8785.92i	0.830265i
$483$	− 121.937i	− 0.0114873i
$484$	24489.6	2.29993
$485$	0	0
$486$	−2890.60	−0.269795
$487$	− 10132.2i	− 0.942777i	−0.881926	−	0.471388i	$-0.843753\pi$
$487$	− 10132.2i	− 0.942777i	0.881926	−	0.471388i	$-0.156247\pi$
$488$	6133.45i	0.568951i
$489$	2248.03	0.207892
$490$	0	0
$491$	−5965.90	−0.548345	−0.274172	−	0.961681i	$-0.588404\pi$
$491$	−5965.90	−0.548345	−0.274172	+	0.961681i	$0.588404\pi$
$492$	− 290.534i	− 0.0266225i
$493$	461.736i	0.0421817i
$494$	2982.78	0.271663
$495$	0	0
$496$	−474.875	−0.0429890
$497$	− 13380.6i	− 1.20765i
$498$	− 1746.72i	− 0.157173i
$499$	−19690.2	−1.76644	−0.883220	−	0.468959i	$-0.844629\pi$
$499$	−19690.2	−1.76644	−0.883220	+	0.468959i	$0.844629\pi$
$500$	0	0
$501$	3887.51	0.346669
$502$	− 4616.10i	− 0.410412i
$503$	18780.3i	1.66476i	0.554209	+	0.832378i	$0.313021\pi$
$503$	18780.3i	1.66476i	−0.554209	+	0.832378i	$0.686979\pi$
$504$	−6673.70	−0.589822
$505$	0	0
$506$	625.958	0.0549945
$507$	1609.24i	0.140964i
$508$	7255.05i	0.633644i
$509$	4447.70	0.387310	0.193655	−	0.981070i	$-0.437966\pi$
$509$	4447.70	0.387310	0.193655	+	0.981070i	$0.437966\pi$
$510$	0	0
$511$	10335.9	0.894782
$512$	9752.47i	0.841801i
$513$	− 5239.69i	− 0.450951i
$514$	6110.99	0.524405
$515$	0	0
$516$	−1894.41	−0.161622
$517$	− 4133.44i	− 0.351622i
$518$	1201.20i	0.101887i
$519$	250.566	0.0211920
$520$	0	0
$521$	−9380.60	−0.788813	−0.394407	−	0.918936i	$-0.629050\pi$
$521$	−9380.60	−0.788813	−0.394407	+	0.918936i	$0.629050\pi$
$522$	851.453i	0.0713929i
$523$	2039.00i	0.170477i	0.996361	+	0.0852384i	$0.0271652\pi$
$523$	2039.00i	0.170477i	−0.996361	+	0.0852384i	$0.972835\pi$
$524$	10141.5	0.845482
$525$	0	0
$526$	5796.93	0.480529
$527$	− 265.488i	− 0.0219447i
$528$	− 2512.21i	− 0.207064i
$529$	12115.5	0.995764
$530$	0	0
$531$	−15361.3	−1.25541
$532$	− 8208.54i	− 0.668957i
$533$	− 1095.53i	− 0.0890291i
$534$	−102.958	−0.00834351
$535$	0	0
$536$	15063.6	1.21390
$537$	1322.24i	0.106255i
$538$	− 9583.20i	− 0.767957i
$539$	−9111.89	−0.728158
$540$	0	0
$541$	−10362.5	−0.823510	−0.411755	−	0.911295i	$-0.635084\pi$
$541$	−10362.5	−0.823510	−0.411755	+	0.911295i	$0.635084\pi$
$542$	4557.15i	0.361156i
$543$	2181.76i	0.172428i
$544$	−3041.92	−0.239745
$545$	0	0
$546$	589.412	0.0461988
$547$	1466.22i	0.114609i	0.998357	+	0.0573044i	$0.0182506\pi$
$547$	1466.22i	0.114609i	−0.998357	+	0.0573044i	$0.981749\pi$
$548$	1530.50i	0.119306i
$549$	8880.03	0.690328
$550$	0	0
$551$	−2334.73	−0.180514
$552$	− 147.275i	− 0.0113558i
$553$	− 819.436i	− 0.0630126i
$554$	2307.20	0.176938
$555$	0	0
$556$	13916.0	1.06146
$557$	2400.26i	0.182590i	0.995824	+	0.0912949i	$0.0291006\pi$
$557$	2400.26i	0.182590i	−0.995824	+	0.0912949i	$0.970899\pi$
$558$	− 489.567i	− 0.0371416i
$559$	−7143.34	−0.540485
$560$	0	0
$561$	1404.50	0.105701
$562$	− 4081.35i	− 0.306337i
$563$	14902.0i	1.11553i	0.829998	+	0.557766i	$0.188341\pi$
$563$	14902.0i	1.11553i	−0.829998	+	0.557766i	$0.811659\pi$
$564$	−436.230	−0.0325684
$565$	0	0
$566$	−9508.57	−0.706140
$567$	9131.34i	0.676332i
$568$	− 16160.9i	− 1.19383i
$569$	9568.31	0.704964	0.352482	−	0.935819i	$-0.385338\pi$
$569$	9568.31	0.704964	0.352482	+	0.935819i	$0.385338\pi$
$570$	0	0
$571$	10645.8	0.780235	0.390117	−	0.920765i	$-0.372434\pi$
$571$	10645.8	0.780235	0.390117	+	0.920765i	$0.372434\pi$
$572$	− 13192.5i	− 0.964350i
$573$	2633.20i	0.191978i
$574$	691.458	0.0502803
$575$	0	0
$576$	632.748	0.0457717
$577$	335.528i	0.0242084i	0.999927	+	0.0121042i	$0.00385298\pi$
$577$	335.528i	0.0242084i	−0.999927	+	0.0121042i	$0.996147\pi$
$578$	− 353.065i	− 0.0254075i
$579$	−3363.48	−0.241419
$580$	0	0
$581$	−18125.7	−1.29428
$582$	− 1375.86i	− 0.0979916i
$583$	− 48378.0i	− 3.43673i
$584$	12483.6	0.884545
$585$	0	0
$586$	−10402.1	−0.733289
$587$	12512.7i	0.879817i	0.898043	+	0.439908i	$0.144989\pi$
$587$	12512.7i	0.879817i	−0.898043	+	0.439908i	$0.855011\pi$
$588$	961.640i	0.0674445i
$589$	1342.42	0.0939109
$590$	0	0
$591$	−5600.10	−0.389776
$592$	− 2037.42i	− 0.141449i
$593$	11061.3i	0.765994i	0.923749	+	0.382997i	$0.125108\pi$
$593$	11061.3i	0.765994i	−0.923749	+	0.382997i	$0.874892\pi$
$594$	5315.10	0.367140
$595$	0	0
$596$	21590.0	1.48383
$597$	2737.86i	0.187694i
$598$	− 249.101i	− 0.0170343i
$599$	−4672.09	−0.318692	−0.159346	−	0.987223i	$-0.550938\pi$
$599$	−4672.09	−0.318692	−0.159346	+	0.987223i	$0.550938\pi$
$600$	0	0
$601$	26076.4	1.76985	0.884924	−	0.465736i	$-0.154210\pi$
$601$	26076.4	1.76985	0.884924	+	0.465736i	$0.154210\pi$
$602$	− 4508.63i	− 0.305246i
$603$	− 21809.1i	− 1.47286i
$604$	5521.53	0.371966
$605$	0	0
$606$	930.663	0.0623855
$607$	11437.6i	0.764810i	0.923995	+	0.382405i	$0.124904\pi$
$607$	11437.6i	0.764810i	−0.923995	+	0.382405i	$0.875096\pi$
$608$	− 15381.2i	− 1.02597i
$609$	−461.355	−0.0306980
$610$	0	0
$611$	−1644.91	−0.108913
$612$	2838.72i	0.187497i
$613$	− 28634.7i	− 1.88670i	−0.331806	−	0.943348i	$-0.607658\pi$
$613$	− 28634.7i	− 1.88670i	0.331806	−	0.943348i	$-0.392342\pi$
$614$	10939.0	0.718993
$615$	0	0
$616$	18563.1	1.21417
$617$	− 3394.42i	− 0.221482i	−0.993849	−	0.110741i	$-0.964678\pi$
$617$	− 3394.42i	− 0.221482i	0.993849	−	0.110741i	$-0.0353224\pi$
$618$	− 2443.15i	− 0.159026i
$619$	−10197.3	−0.662139	−0.331069	−	0.943606i	$-0.607409\pi$
$619$	−10197.3	−0.662139	−0.331069	+	0.943606i	$0.607409\pi$
$620$	0	0
$621$	−437.583	−0.0282763
$622$	− 4694.69i	− 0.302637i
$623$	1068.39i	0.0687067i
$624$	−999.739	−0.0641372
$625$	0	0
$626$	−5415.50	−0.345762
$627$	7101.76i	0.452339i
$628$	6180.96i	0.392751i
$629$	1139.06	0.0722057
$630$	0	0
$631$	13195.9	0.832522	0.416261	−	0.909245i	$-0.363340\pi$
$631$	13195.9	0.832522	0.416261	+	0.909245i	$0.363340\pi$
$632$	− 989.705i	− 0.0622917i
$633$	687.761i	0.0431849i
$634$	−9399.85	−0.588826
$635$	0	0
$636$	−5105.66	−0.318322
$637$	3626.09i	0.225543i
$638$	− 2368.34i	− 0.146965i
$639$	−23397.8	−1.44852
$640$	0	0
$641$	−10280.5	−0.633469	−0.316734	−	0.948514i	$-0.602586\pi$
$641$	−10280.5	−0.633469	−0.316734	+	0.948514i	$0.602586\pi$
$642$	1276.96i	0.0785010i
$643$	16401.3i	1.00592i	0.864311	+	0.502959i	$0.167755\pi$
$643$	16401.3i	1.00592i	−0.864311	+	0.502959i	$0.832245\pi$
$644$	−685.520	−0.0419461
$645$	0	0
$646$	1785.24	0.108730
$647$	18960.3i	1.15210i	0.817416	+	0.576048i	$0.195406\pi$
$647$	18960.3i	1.15210i	−0.817416	+	0.576048i	$0.804594\pi$
$648$	11028.7i	0.668594i
$649$	42727.9	2.58431
$650$	0	0
$651$	265.269	0.0159704
$652$	− 12638.2i	− 0.759124i
$653$	− 1539.55i	− 0.0922623i	−0.998935	−	0.0461311i	$-0.985311\pi$
$653$	− 1539.55i	− 0.0922623i	0.998935	−	0.0461311i	$-0.0146892\pi$
$654$	1178.76	0.0704785
$655$	0	0
$656$	−1172.83	−0.0698036
$657$	− 18073.8i	− 1.07325i
$658$	− 1038.21i	− 0.0615100i
$659$	9245.33	0.546505	0.273253	−	0.961942i	$-0.411901\pi$
$659$	9245.33	0.546505	0.273253	+	0.961942i	$0.411901\pi$
$660$	0	0
$661$	28073.9	1.65197	0.825983	−	0.563696i	$-0.190621\pi$
$661$	28073.9	1.65197	0.825983	+	0.563696i	$0.190621\pi$
$662$	8889.78i	0.521920i
$663$	− 558.924i	− 0.0327403i
$664$	−21891.9	−1.27948
$665$	0	0
$666$	2100.46	0.122209
$667$	194.981i	0.0113189i
$668$	− 21855.2i	− 1.26587i
$669$	3519.06	0.203370
$670$	0	0
$671$	−24700.0	−1.42106
$672$	− 3039.41i	− 0.174476i
$673$	19690.2i	1.12779i	0.825848	+	0.563893i	$0.190697\pi$
$673$	19690.2i	1.12779i	−0.825848	+	0.563893i	$0.809303\pi$
$674$	4032.64	0.230462
$675$	0	0
$676$	9046.99	0.514735
$677$	− 12023.4i	− 0.682566i	−0.939961	−	0.341283i	$-0.889139\pi$
$677$	− 12023.4i	− 0.682566i	0.939961	−	0.341283i	$-0.110861\pi$
$678$	1423.30i	0.0806219i
$679$	−14277.2	−0.806937
$680$	0	0
$681$	531.463	0.0299056
$682$	1361.74i	0.0764572i
$683$	6517.76i	0.365147i	0.983192	+	0.182573i	$0.0584427\pi$
$683$	6517.76i	0.365147i	−0.983192	+	0.182573i	$0.941557\pi$
$684$	−14353.8	−0.802383
$685$	0	0
$686$	−8437.74	−0.469613
$687$	− 2492.73i	− 0.138433i
$688$	7647.36i	0.423769i
$689$	−19252.1	−1.06451
$690$	0	0
$691$	−25926.3	−1.42733	−0.713664	−	0.700488i	$-0.752966\pi$
$691$	−25926.3	−1.42733	−0.713664	+	0.700488i	$0.752966\pi$
$692$	− 1408.66i	− 0.0773830i
$693$	− 26875.7i	− 1.47319i
$694$	9771.47	0.534467
$695$	0	0
$696$	−557.219	−0.0303467
$697$	− 655.691i	− 0.0356328i
$698$	5349.68i	0.290098i
$699$	6315.99	0.341764
$700$	0	0
$701$	1635.68	0.0881294	0.0440647	−	0.999029i	$-0.485969\pi$
$701$	1635.68	0.0881294	0.0440647	+	0.999029i	$0.485969\pi$
$702$	− 2115.16i	− 0.113720i
$703$	5759.58i	0.308999i
$704$	−1760.00	−0.0942225
$705$	0	0
$706$	1842.75	0.0982333
$707$	− 9657.47i	− 0.513729i
$708$	− 4509.37i	− 0.239368i
$709$	26057.0	1.38024	0.690122	−	0.723693i	$-0.257557\pi$
$709$	26057.0	1.38024	0.690122	+	0.723693i	$0.257557\pi$
$710$	0	0
$711$	−1432.90	−0.0755807
$712$	1290.39i	0.0679206i
$713$	− 112.110i	− 0.00588856i
$714$	352.773	0.0184905
$715$	0	0
$716$	7433.53	0.387994
$717$	7030.34i	0.366183i
$718$	− 4198.42i	− 0.218223i
$719$	−106.968	−0.00554828	−0.00277414	−	0.999996i	$-0.500883\pi$
$719$	−106.968	−0.00554828	−0.00277414	+	0.999996i	$0.500883\pi$
$720$	0	0
$721$	−25352.6	−1.30954
$722$	647.469i	0.0333744i
$723$	8324.58i	0.428208i
$724$	12265.6	0.629626
$725$	0	0
$726$	−5321.77	−0.272051
$727$	− 17019.9i	− 0.868273i	−0.900847	−	0.434137i	$-0.857053\pi$
$727$	− 17019.9i	− 0.868273i	0.900847	−	0.434137i	$-0.142947\pi$
$728$	− 7387.21i	− 0.376083i
$729$	14062.4	0.714442
$730$	0	0
$731$	−4275.41	−0.216322
$732$	2606.76i	0.131624i
$733$	27064.0i	1.36375i	0.731467	+	0.681877i	$0.238836\pi$
$733$	27064.0i	1.36375i	−0.731467	+	0.681877i	$0.761164\pi$
$734$	−13019.0	−0.654685
$735$	0	0
$736$	−1284.53	−0.0643322
$737$	60662.6i	3.03194i
$738$	− 1209.11i	− 0.0603089i
$739$	13177.0	0.655919	0.327960	−	0.944692i	$-0.393639\pi$
$739$	13177.0	0.655919	0.327960	+	0.944692i	$0.393639\pi$
$740$	0	0
$741$	2826.16	0.140110
$742$	− 12151.3i	− 0.601195i
$743$	− 1164.23i	− 0.0574854i	−0.999587	−	0.0287427i	$-0.990850\pi$
$743$	− 1164.23i	− 0.0574854i	0.999587	−	0.0287427i	$-0.00915034\pi$
$744$	320.389	0.0157877
$745$	0	0
$746$	−11071.2	−0.543360
$747$	31695.2i	1.55243i
$748$	− 7895.97i	− 0.385969i
$749$	13251.0	0.646437
$750$	0	0
$751$	5632.54	0.273681	0.136840	−	0.990593i	$-0.456305\pi$
$751$	5632.54	0.273681	0.136840	+	0.990593i	$0.456305\pi$
$752$	1760.97i	0.0853936i
$753$	− 4373.71i	− 0.211669i
$754$	−942.484	−0.0455215
$755$	0	0
$756$	−5820.85	−0.280030
$757$	− 40260.8i	− 1.93303i	−0.256611	−	0.966515i	$-0.582606\pi$
$757$	− 40260.8i	− 1.93303i	0.256611	−	0.966515i	$-0.417394\pi$
$758$	− 1319.95i	− 0.0632489i
$759$	593.089	0.0283634
$760$	0	0
$761$	−22060.7	−1.05085	−0.525427	−	0.850839i	$-0.676094\pi$
$761$	−22060.7	−1.05085	−0.525427	+	0.850839i	$0.676094\pi$
$762$	− 1576.57i	− 0.0749517i
$763$	− 12231.9i	− 0.580374i
$764$	14803.6	0.701013
$765$	0	0
$766$	8454.80	0.398805
$767$	− 17003.6i	− 0.800477i
$768$	− 1838.56i	− 0.0863846i
$769$	−8923.07	−0.418432	−0.209216	−	0.977869i	$-0.567091\pi$
$769$	−8923.07	−0.418432	−0.209216	+	0.977869i	$0.567091\pi$
$770$	0	0
$771$	5790.10	0.270461
$772$	18909.2i	0.881548i
$773$	− 3076.31i	− 0.143140i	−0.997436	−	0.0715699i	$-0.977199\pi$
$773$	− 3076.31i	− 0.143140i	0.997436	−	0.0715699i	$-0.0228009\pi$
$774$	−7883.96	−0.366128
$775$	0	0
$776$	−17243.9	−0.797705
$777$	1138.12i	0.0525481i
$778$	− 11806.9i	− 0.544086i
$779$	3315.45	0.152488
$780$	0	0
$781$	65081.6	2.98182
$782$	− 149.091i	− 0.00681777i
$783$	1655.61i	0.0755642i
$784$	3881.95	0.176838
$785$	0	0
$786$	−2203.81	−0.100009
$787$	− 20627.3i	− 0.934286i	−0.884182	−	0.467143i	$-0.845283\pi$
$787$	− 20627.3i	− 0.934286i	0.884182	−	0.467143i	$-0.154717\pi$
$788$	31483.2i	1.42328i
$789$	5492.54	0.247832
$790$	0	0
$791$	14769.6	0.663902
$792$	− 32460.1i	− 1.45634i
$793$	9829.42i	0.440167i
$794$	9119.58	0.407609
$795$	0	0
$796$	15392.0	0.685369
$797$	13976.5i	0.621172i	0.950545	+	0.310586i	$0.100525\pi$
$797$	13976.5i	0.621172i	−0.950545	+	0.310586i	$0.899475\pi$
$798$	1783.77i	0.0791289i
$799$	−984.506	−0.0435911
$800$	0	0
$801$	1868.23	0.0824105
$802$	3602.36i	0.158608i
$803$	50272.6i	2.20932i
$804$	6402.14	0.280829
$805$	0	0
$806$	541.908	0.0236822
$807$	− 9079.99i	− 0.396073i
$808$	− 11664.2i	− 0.507852i
$809$	−5425.25	−0.235774	−0.117887	−	0.993027i	$-0.537612\pi$
$809$	−5425.25	−0.235774	−0.117887	+	0.993027i	$0.537612\pi$
$810$	0	0
$811$	−36180.8	−1.56656	−0.783279	−	0.621670i	$-0.786455\pi$
$811$	−36180.8	−1.56656	−0.783279	+	0.621670i	$0.786455\pi$
$812$	2593.69i	0.112095i
$813$	4317.86i	0.186266i
$814$	−5842.47	−0.251571
$815$	0	0
$816$	−598.361	−0.0256701
$817$	− 21618.3i	− 0.925738i
$818$	− 6775.48i	− 0.289608i
$819$	−10695.2	−0.456314
$820$	0	0
$821$	−17712.7	−0.752955	−0.376478	−	0.926426i	$-0.622865\pi$
$821$	−17712.7	−0.752955	−0.376478	+	0.926426i	$0.622865\pi$
$822$	− 332.589i	− 0.0141124i
$823$	15244.9i	0.645691i	0.946452	+	0.322845i	$0.104639\pi$
$823$	15244.9i	0.645691i	−0.946452	+	0.322845i	$0.895361\pi$
$824$	−30620.5	−1.29456
$825$	0	0
$826$	10732.1	0.452080
$827$	4079.89i	0.171550i	0.996315	+	0.0857750i	$0.0273366\pi$
$827$	4079.89i	0.171550i	−0.996315	+	0.0857750i	$0.972663\pi$
$828$	1198.73i	0.0503124i
$829$	−13959.9	−0.584857	−0.292429	−	0.956287i	$-0.594463\pi$
$829$	−13959.9	−0.584857	−0.292429	+	0.956287i	$0.594463\pi$
$830$	0	0
$831$	2186.05	0.0912554
$832$	700.397i	0.0291850i
$833$	2170.28i	0.0902710i
$834$	−3024.05	−0.125557
$835$	0	0
$836$	39925.3	1.65173
$837$	− 951.941i	− 0.0393117i
$838$	− 10073.6i	− 0.415261i
$839$	−45111.6	−1.85629	−0.928144	−	0.372221i	$-0.878596\pi$
$839$	−45111.6	−1.85629	−0.928144	+	0.372221i	$0.878596\pi$
$840$	0	0
$841$	−23651.3	−0.969752
$842$	− 12183.4i	− 0.498656i
$843$	− 3867.04i	− 0.157993i
$844$	3866.52	0.157691
$845$	0	0
$846$	−1815.45	−0.0737784
$847$	55223.8i	2.24028i
$848$	20610.5i	0.834632i
$849$	−9009.28	−0.364191
$850$	0	0
$851$	481.000	0.0193754
$852$	− 6868.51i	− 0.276187i
$853$	37653.2i	1.51140i	0.654920	+	0.755698i	$0.272702\pi$
$853$	37653.2i	1.51140i	−0.654920	+	0.755698i	$0.727298\pi$
$854$	−6203.99	−0.248590
$855$	0	0
$856$	16004.4	0.639041
$857$	7916.64i	0.315551i	0.987475	+	0.157776i	$0.0504322\pi$
$857$	7916.64i	0.315551i	−0.987475	+	0.157776i	$0.949568\pi$
$858$	2866.83i	0.114070i
$859$	−4012.33	−0.159370	−0.0796850	−	0.996820i	$-0.525391\pi$
$859$	−4012.33	−0.159370	−0.0796850	+	0.996820i	$0.525391\pi$
$860$	0	0
$861$	655.150	0.0259320
$862$	12835.0i	0.507147i
$863$	− 22547.6i	− 0.889374i	−0.895686	−	0.444687i	$-0.853315\pi$
$863$	− 22547.6i	− 0.889374i	0.895686	−	0.444687i	$-0.146685\pi$
$864$	−10907.2	−0.429478
$865$	0	0
$866$	16940.8	0.664749
$867$	− 334.526i	− 0.0131039i
$868$	− 1491.32i	− 0.0583163i
$869$	3985.64	0.155585
$870$	0	0
$871$	24140.8	0.939127
$872$	− 14773.5i	− 0.573733i
$873$	24965.7i	0.967883i
$874$	753.869	0.0291762
$875$	0	0
$876$	5305.62	0.204635
$877$	16964.9i	0.653210i	0.945161	+	0.326605i	$0.105905\pi$
$877$	16964.9i	0.653210i	−0.945161	+	0.326605i	$0.894095\pi$
$878$	− 330.729i	− 0.0127125i
$879$	−9855.90	−0.378193
$880$	0	0
$881$	−28213.2	−1.07892	−0.539459	−	0.842012i	$-0.681371\pi$
$881$	−28213.2	−1.07892	−0.539459	+	0.842012i	$0.681371\pi$
$882$	4002.05i	0.152784i
$883$	44019.1i	1.67764i	0.544405	+	0.838822i	$0.316755\pi$
$883$	44019.1i	1.67764i	−0.544405	+	0.838822i	$0.683245\pi$
$884$	−3142.21	−0.119552
$885$	0	0
$886$	−8318.75	−0.315433
$887$	14078.8i	0.532941i	0.963843	+	0.266471i	$0.0858576\pi$
$887$	14078.8i	0.532941i	−0.963843	+	0.266471i	$0.914142\pi$
$888$	1374.61i	0.0519469i
$889$	−16360.1	−0.617209
$890$	0	0
$891$	−44413.8	−1.66994
$892$	− 19783.8i	− 0.742613i
$893$	− 4978.08i	− 0.186545i
$894$	−4691.66	−0.175517
$895$	0	0
$896$	−21448.3	−0.799705
$897$	− 236.021i	− 0.00878541i
$898$	11154.0i	0.414493i
$899$	−424.172	−0.0157363
$900$	0	0
$901$	−11522.7	−0.426057
$902$	3363.17i	0.124148i
$903$	− 4271.88i	− 0.157430i
$904$	17838.5	0.656306
$905$	0	0
$906$	−1199.87	−0.0439988
$907$	− 12670.6i	− 0.463858i	−0.972733	−	0.231929i	$-0.925496\pi$
$907$	− 12670.6i	− 0.463858i	0.972733	−	0.231929i	$-0.0745037\pi$
$908$	− 2987.83i	− 0.109201i
$909$	−16887.4	−0.616194
$910$	0	0
$911$	15336.4	0.557760	0.278880	−	0.960326i	$-0.410037\pi$
$911$	15336.4	0.557760	0.278880	+	0.960326i	$0.410037\pi$
$912$	− 3025.57i	− 0.109854i
$913$	− 88161.1i	− 3.19574i
$914$	3084.49	0.111626
$915$	0	0
$916$	−14013.9	−0.505492
$917$	22868.9i	0.823553i
$918$	− 1265.96i	− 0.0455150i
$919$	−43929.1	−1.57681	−0.788404	−	0.615158i	$-0.789092\pi$
$919$	−43929.1	−1.57681	−0.788404	+	0.615158i	$0.789092\pi$
$920$	0	0
$921$	10364.6	0.370819
$922$	13563.2i	0.484467i
$923$	− 25899.3i	− 0.923605i
$924$	7889.45	0.280892
$925$	0	0
$926$	−3721.13	−0.132056
$927$	44332.5i	1.57073i
$928$	4860.08i	0.171918i
$929$	−28646.0	−1.01167	−0.505836	−	0.862629i	$-0.668816\pi$
$929$	−28646.0	−1.01167	−0.505836	+	0.862629i	$0.668816\pi$
$930$	0	0
$931$	−10973.8	−0.386309
$932$	− 35507.9i	− 1.24796i
$933$	− 4448.17i	− 0.156084i
$934$	17733.2	0.621252
$935$	0	0
$936$	−12917.6	−0.451094
$937$	25892.8i	0.902753i	0.892334	+	0.451376i	$0.149067\pi$
$937$	25892.8i	0.902753i	−0.892334	+	0.451376i	$0.850933\pi$
$938$	15236.8i	0.530384i
$939$	−5131.13	−0.178326
$940$	0	0
$941$	−12650.8	−0.438263	−0.219131	−	0.975695i	$-0.570322\pi$
$941$	−12650.8	−0.438263	−0.219131	+	0.975695i	$0.570322\pi$
$942$	− 1343.17i	− 0.0464572i
$943$	− 276.884i	− 0.00956158i
$944$	−18203.4	−0.627617
$945$	0	0
$946$	21929.4	0.753686
$947$	− 19766.7i	− 0.678280i	−0.940736	−	0.339140i	$-0.889864\pi$
$947$	− 19766.7i	− 0.678280i	0.940736	−	0.339140i	$-0.110136\pi$
$948$	− 420.632i	− 0.0144109i
$949$	20006.1	0.684326
$950$	0	0
$951$	−8906.27	−0.303686
$952$	− 4421.37i	− 0.150523i
$953$	6066.08i	0.206190i	0.994671	+	0.103095i	$0.0328746\pi$
$953$	6066.08i	0.206190i	−0.994671	+	0.103095i	$0.967125\pi$
$954$	−21248.2	−0.721105
$955$	0	0
$956$	39523.9	1.33713
$957$	− 2243.98i	− 0.0757968i
$958$	6733.80i	0.227097i
$959$	−3451.27	−0.116212
$960$	0	0
$961$	−29547.1	−0.991813
$962$	2325.02i	0.0779228i
$963$	− 23171.2i	− 0.775370i
$964$	46799.9	1.56361
$965$	0	0
$966$	148.968	0.00496167
$967$	− 31669.2i	− 1.05317i	−0.850123	−	0.526584i	$-0.823473\pi$
$967$	− 31669.2i	− 1.05317i	0.850123	−	0.526584i	$-0.176527\pi$
$968$	66698.7i	2.21464i
$969$	1691.50	0.0560773
$970$	0	0
$971$	−24982.5	−0.825671	−0.412835	−	0.910806i	$-0.635462\pi$
$971$	−24982.5	−0.825671	−0.412835	+	0.910806i	$0.635462\pi$
$972$	15397.3i	0.508097i
$973$	31380.5i	1.03393i
$974$	12378.2	0.407212
$975$	0	0
$976$	10523.0	0.345115
$977$	26574.1i	0.870194i	0.900384	+	0.435097i	$0.143286\pi$
$977$	26574.1i	0.870194i	−0.900384	+	0.435097i	$0.856714\pi$
$978$	2746.36i	0.0897945i
$979$	−5196.54	−0.169645
$980$	0	0
$981$	−21389.2	−0.696131
$982$	− 7288.41i	− 0.236846i
$983$	19222.4i	0.623703i	0.950131	+	0.311852i	$0.100949\pi$
$983$	19222.4i	0.623703i	−0.950131	+	0.311852i	$0.899051\pi$
$984$	791.282	0.0256353
$985$	0	0
$986$	−564.093	−0.0182195
$987$	− 983.694i	− 0.0317237i
$988$	− 15888.4i	− 0.511616i
$989$	−1805.41	−0.0580472
$990$	0	0
$991$	−22396.5	−0.717909	−0.358954	−	0.933355i	$-0.616867\pi$
$991$	−22396.5	−0.717909	−0.358954	+	0.933355i	$0.616867\pi$
$992$	− 2794.44i	− 0.0894391i
$993$	8422.99i	0.269180i
$994$	16346.8	0.521618
$995$	0	0
$996$	−9304.24	−0.296000
$997$	26548.7i	0.843337i	0.906750	+	0.421669i	$0.138555\pi$
$997$	26548.7i	0.843337i	−0.906750	+	0.421669i	$0.861445\pi$
$998$	− 24055.0i	− 0.762975i
$999$	4084.24	0.129349

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	425.4.b.g.324.4		6
5.2	odd	4		425.4.a.h.1.1		3
5.3	odd	4		85.4.a.e.1.3	✓	3
5.4	even	2	inner	425.4.b.g.324.3		6
15.8	even	4		765.4.a.l.1.1		3
20.3	even	4		1360.4.a.s.1.2		3
85.33	odd	4		1445.4.a.j.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.4.a.e.1.3	✓	3	5.3	odd	4
425.4.a.h.1.1		3	5.2	odd	4
425.4.b.g.324.3		6	5.4	even	2	inner
425.4.b.g.324.4		6	1.1	even	1	trivial
765.4.a.l.1.1		3	15.8	even	4
1360.4.a.s.1.2		3	20.3	even	4
1445.4.a.j.1.3		3	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)

\(f(q)\)	\(=\)	\(q+1.22168i q^{2} +1.15753i q^{3} +6.50750 q^{4} -1.41413 q^{6} +14.6743i q^{7} +17.7235i q^{8} +25.6601 q^{9} -71.3743 q^{11} +7.53262i q^{12} +28.4035i q^{13} -17.9273 q^{14} +30.4076 q^{16} +17.0000i q^{17} +31.3484i q^{18} -85.9592 q^{19} -16.9860 q^{21} -87.1964i q^{22} +7.17872i q^{23} -20.5154 q^{24} -34.7000 q^{26} +60.9556i q^{27} +95.4934i q^{28} +27.1610 q^{29} -15.6170 q^{31} +178.936i q^{32} -82.6178i q^{33} -20.7685 q^{34} +166.983 q^{36} -67.0036i q^{37} -105.014i q^{38} -32.8779 q^{39} -38.5701 q^{41} -20.7514i q^{42} +251.495i q^{43} -464.469 q^{44} -8.77008 q^{46} +57.9121i q^{47} +35.1977i q^{48} +127.663 q^{49} -19.6780 q^{51} +184.836i q^{52} +677.807i q^{53} -74.4680 q^{54} -260.081 q^{56} -99.5002i q^{57} +33.1819i q^{58} -598.645 q^{59} +346.063 q^{61} -19.0789i q^{62} +376.546i q^{63} +24.6588 q^{64} +100.932 q^{66} -849.923i q^{67} +110.628i q^{68} -8.30957 q^{69} -911.836 q^{71} +454.787i q^{72} -704.352i q^{73} +81.8568 q^{74} -559.380 q^{76} -1047.37i q^{77} -40.1662i q^{78} -55.8414 q^{79} +622.266 q^{81} -47.1202i q^{82} +1235.19i q^{83} -110.536 q^{84} -307.245 q^{86} +31.4396i q^{87} -1265.00i q^{88} +72.8069 q^{89} -416.803 q^{91} +46.7155i q^{92} -18.0771i q^{93} -70.7499 q^{94} -207.124 q^{96} +972.939i q^{97} +155.964i q^{98} -1831.47 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 20 q^{4} + 72 q^{6} - 78 q^{9} - 236 q^{11} + 136 q^{14} + 484 q^{16} + 320 q^{19} - 944 q^{21} - 1608 q^{24} - 88 q^{26} - 748 q^{29} + 140 q^{31} + 204 q^{34} + 2516 q^{36} - 496 q^{39} - 708 q^{41}+ \cdots - 1540 q^{99}+O(q^{100}) \) 6 * q - 20 * q^4 + 72 * q^6 - 78 * q^9 - 236 * q^11 + 136 * q^14 + 484 * q^16 + 320 * q^19 - 944 * q^21 - 1608 * q^24 - 88 * q^26 - 748 * q^29 + 140 * q^31 + 204 * q^34 + 2516 * q^36 - 496 * q^39 - 708 * q^41 - 696 * q^44 - 8 * q^46 - 2182 * q^49 - 136 * q^51 - 3120 * q^54 - 5992 * q^56 + 608 * q^59 + 492 * q^61 - 3652 * q^64 - 304 * q^66 + 2288 * q^69 - 1204 * q^71 - 1832 * q^74 - 3776 * q^76 - 412 * q^79 + 870 * q^81 + 8240 * q^84 + 512 * q^86 + 1740 * q^89 - 2912 * q^91 - 4448 * q^94 + 8136 * q^96 - 1540 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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