LMFDB - Embedded newform 700.4.e.f.449.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [700,4,Mod(449,700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(700, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("700.449");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 700 = 2^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	700.e (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:	$41.3013370040$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	700.449
Dual form			700.4.e.f.449.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000i q^{3} +7.00000i q^{7} +11.0000 q^{9} +O(q^{10})$	$q-4.00000i q^{3} +7.00000i q^{7} +11.0000 q^{9} -12.0000 q^{11} +82.0000i q^{13} -30.0000i q^{17} -68.0000 q^{19} +28.0000 q^{21} -216.000i q^{23} -152.000i q^{27} -246.000 q^{29} -112.000 q^{31} +48.0000i q^{33} +110.000i q^{37} +328.000 q^{39} -246.000 q^{41} +172.000i q^{43} +192.000i q^{47} -49.0000 q^{49} -120.000 q^{51} -558.000i q^{53} +272.000i q^{57} -540.000 q^{59} +110.000 q^{61} +77.0000i q^{63} +140.000i q^{67} -864.000 q^{69} -840.000 q^{71} +550.000i q^{73} -84.0000i q^{77} +208.000 q^{79} -311.000 q^{81} -516.000i q^{83} +984.000i q^{87} +1398.00 q^{89} -574.000 q^{91} +448.000i q^{93} +1586.00i q^{97} -132.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 22 q^{9}+O(q^{10}) $ 2 * q + 22 * q^9	$ 2 q + 22 q^{9} - 24 q^{11} - 136 q^{19} + 56 q^{21} - 492 q^{29} - 224 q^{31} + 656 q^{39} - 492 q^{41} - 98 q^{49} - 240 q^{51} - 1080 q^{59} + 220 q^{61} - 1728 q^{69} - 1680 q^{71} + 416 q^{79} - 622 q^{81} + 2796 q^{89} - 1148 q^{91} - 264 q^{99}+O(q^{100}) $ 2 * q + 22 * q^9 - 24 * q^11 - 136 * q^19 + 56 * q^21 - 492 * q^29 - 224 * q^31 + 656 * q^39 - 492 * q^41 - 98 * q^49 - 240 * q^51 - 1080 * q^59 + 220 * q^61 - 1728 * q^69 - 1680 * q^71 + 416 * q^79 - 622 * q^81 + 2796 * q^89 - 1148 * q^91 - 264 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/700\mathbb{Z}\right)^\times$.

$n$	$101$	$351$	$477$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	− 4.00000i	− 0.769800i	−0.922958	−	0.384900i	$-0.874236\pi$
$3$	− 4.00000i	− 0.769800i	0.922958	−	0.384900i	$-0.125764\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	7.00000i	0.377964i
$7$	7.00000i	0.377964i
$8$	0	0
$9$	11.0000	0.407407
$10$	0	0
$11$	−12.0000	−0.328921	−0.164461	−	0.986384i	$-0.552588\pi$
$11$	−12.0000	−0.328921	−0.164461	+	0.986384i	$0.552588\pi$
$12$	0	0
$13$	82.0000i	1.74944i	0.484629	+	0.874720i	$0.338954\pi$
$13$	82.0000i	1.74944i	−0.484629	+	0.874720i	$0.661046\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 30.0000i	− 0.428004i	−0.976833	−	0.214002i	$-0.931350\pi$
$17$	− 30.0000i	− 0.428004i	0.976833	−	0.214002i	$-0.0686499\pi$
$18$	0	0
$19$	−68.0000	−0.821067	−0.410533	−	0.911846i	$-0.634657\pi$
$19$	−68.0000	−0.821067	−0.410533	+	0.911846i	$0.634657\pi$
$20$	0	0
$21$	28.0000	0.290957
$22$	0	0
$23$	− 216.000i	− 1.95822i	−0.203326	−	0.979111i	$-0.565175\pi$
$23$	− 216.000i	− 1.95822i	0.203326	−	0.979111i	$-0.434825\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 152.000i	− 1.08342i
$28$	0	0
$29$	−246.000	−1.57521	−0.787604	−	0.616181i	$-0.788679\pi$
$29$	−246.000	−1.57521	−0.787604	+	0.616181i	$0.788679\pi$
$30$	0	0
$31$	−112.000	−0.648897	−0.324448	−	0.945903i	$-0.605179\pi$
$31$	−112.000	−0.648897	−0.324448	+	0.945903i	$0.605179\pi$
$32$	0	0
$33$	48.0000i	0.253204i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	110.000i	0.488754i	0.969680	+	0.244377i	$0.0785834\pi$
$37$	110.000i	0.488754i	−0.969680	+	0.244377i	$0.921417\pi$
$38$	0	0
$39$	328.000	1.34672
$40$	0	0
$41$	−246.000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−246.000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	172.000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$	172.000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	192.000i	0.595874i	0.954586	+	0.297937i	$0.0962985\pi$
$47$	192.000i	0.595874i	−0.954586	+	0.297937i	$0.903701\pi$
$48$	0	0
$49$	−49.0000	−0.142857
$50$	0	0
$51$	−120.000	−0.329478
$52$	0	0
$53$	− 558.000i	− 1.44617i	−0.690757	−	0.723087i	$-0.742723\pi$
$53$	− 558.000i	− 1.44617i	0.690757	−	0.723087i	$-0.257277\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	272.000i	0.632058i
$58$	0	0
$59$	−540.000	−1.19156	−0.595780	−	0.803148i	$-0.703157\pi$
$59$	−540.000	−1.19156	−0.595780	+	0.803148i	$0.703157\pi$
$60$	0	0
$61$	110.000	0.230886	0.115443	−	0.993314i	$-0.463171\pi$
$61$	110.000	0.230886	0.115443	+	0.993314i	$0.463171\pi$
$62$	0	0
$63$	77.0000i	0.153986i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	140.000i	0.255279i	0.991821	+	0.127640i	$0.0407401\pi$
$67$	140.000i	0.255279i	−0.991821	+	0.127640i	$0.959260\pi$
$68$	0	0
$69$	−864.000	−1.50744
$70$	0	0
$71$	−840.000	−1.40408	−0.702040	−	0.712138i	$-0.747727\pi$
$71$	−840.000	−1.40408	−0.702040	+	0.712138i	$0.747727\pi$
$72$	0	0
$73$	550.000i	0.881817i	0.897552	+	0.440908i	$0.145344\pi$
$73$	550.000i	0.881817i	−0.897552	+	0.440908i	$0.854656\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 84.0000i	− 0.124321i
$78$	0	0
$79$	208.000	0.296226	0.148113	−	0.988970i	$-0.452680\pi$
$79$	208.000	0.296226	0.148113	+	0.988970i	$0.452680\pi$
$80$	0	0
$81$	−311.000	−0.426612
$82$	0	0
$83$	− 516.000i	− 0.682390i	−0.939993	−	0.341195i	$-0.889168\pi$
$83$	− 516.000i	− 0.682390i	0.939993	−	0.341195i	$-0.110832\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	984.000i	1.21260i
$88$	0	0
$89$	1398.00	1.66503	0.832515	−	0.554002i	$-0.186900\pi$
$89$	1398.00	1.66503	0.832515	+	0.554002i	$0.186900\pi$
$90$	0	0
$91$	−574.000	−0.661226
$92$	0	0
$93$	448.000i	0.499521i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1586.00i	1.66014i	0.557657	+	0.830072i	$0.311701\pi$
$97$	1586.00i	1.66014i	−0.557657	+	0.830072i	$0.688299\pi$
$98$	0	0
$99$	−132.000	−0.134005
$100$	0	0
$101$	−1242.00	−1.22360	−0.611800	−	0.791012i	$-0.709554\pi$
$101$	−1242.00	−1.22360	−0.611800	+	0.791012i	$0.709554\pi$
$102$	0	0
$103$	− 680.000i	− 0.650509i	−0.945627	−	0.325254i	$-0.894550\pi$
$103$	− 680.000i	− 0.650509i	0.945627	−	0.325254i	$-0.105450\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	996.000i	0.899878i	0.893059	+	0.449939i	$0.148554\pi$
$107$	996.000i	0.899878i	−0.893059	+	0.449939i	$0.851446\pi$
$108$	0	0
$109$	−1382.00	−1.21442	−0.607209	−	0.794542i	$-0.707711\pi$
$109$	−1382.00	−1.21442	−0.607209	+	0.794542i	$0.707711\pi$
$110$	0	0
$111$	440.000	0.376243
$112$	0	0
$113$	750.000i	0.624372i	0.950021	+	0.312186i	$0.101061\pi$
$113$	750.000i	0.624372i	−0.950021	+	0.312186i	$0.898939\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	902.000i	0.712734i
$118$	0	0
$119$	210.000	0.161770
$120$	0	0
$121$	−1187.00	−0.891811
$122$	0	0
$123$	984.000i	0.721336i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	176.000i	0.122972i	0.998108	+	0.0614861i	$0.0195840\pi$
$127$	176.000i	0.122972i	−0.998108	+	0.0614861i	$0.980416\pi$
$128$	0	0
$129$	688.000	0.469574
$130$	0	0
$131$	−1548.00	−1.03244	−0.516219	−	0.856457i	$-0.672661\pi$
$131$	−1548.00	−1.03244	−0.516219	+	0.856457i	$0.672661\pi$
$132$	0	0
$133$	− 476.000i	− 0.310334i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	378.000i	0.235728i	0.993030	+	0.117864i	$0.0376047\pi$
$137$	378.000i	0.235728i	−0.993030	+	0.117864i	$0.962395\pi$
$138$	0	0
$139$	2500.00	1.52552	0.762760	−	0.646682i	$-0.223844\pi$
$139$	2500.00	1.52552	0.762760	+	0.646682i	$0.223844\pi$
$140$	0	0
$141$	768.000	0.458704
$142$	0	0
$143$	− 984.000i	− 0.575428i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	196.000i	0.109971i
$148$	0	0
$149$	−846.000	−0.465148	−0.232574	−	0.972579i	$-0.574715\pi$
$149$	−846.000	−0.465148	−0.232574	+	0.972579i	$0.574715\pi$
$150$	0	0
$151$	−2536.00	−1.36673	−0.683367	−	0.730075i	$-0.739485\pi$
$151$	−2536.00	−1.36673	−0.683367	+	0.730075i	$0.739485\pi$
$152$	0	0
$153$	− 330.000i	− 0.174372i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 1186.00i	− 0.602886i	−0.953484	−	0.301443i	$-0.902532\pi$
$157$	− 1186.00i	− 0.602886i	0.953484	−	0.301443i	$-0.0974683\pi$
$158$	0	0
$159$	−2232.00	−1.11326
$160$	0	0
$161$	1512.00	0.740138
$162$	0	0
$163$	− 2108.00i	− 1.01295i	−0.862254	−	0.506476i	$-0.830948\pi$
$163$	− 2108.00i	− 1.01295i	0.862254	−	0.506476i	$-0.169052\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 1944.00i	− 0.900786i	−0.892830	−	0.450393i	$-0.851284\pi$
$167$	− 1944.00i	− 0.900786i	0.892830	−	0.450393i	$-0.148716\pi$
$168$	0	0
$169$	−4527.00	−2.06054
$170$	0	0
$171$	−748.000	−0.334509
$172$	0	0
$173$	1362.00i	0.598560i	0.954165	+	0.299280i	$0.0967465\pi$
$173$	1362.00i	0.598560i	−0.954165	+	0.299280i	$0.903253\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2160.00i	0.917263i
$178$	0	0
$179$	−1596.00	−0.666428	−0.333214	−	0.942851i	$-0.608133\pi$
$179$	−1596.00	−0.666428	−0.333214	+	0.942851i	$0.608133\pi$
$180$	0	0
$181$	−1690.00	−0.694015	−0.347007	−	0.937862i	$-0.612802\pi$
$181$	−1690.00	−0.694015	−0.347007	+	0.937862i	$0.612802\pi$
$182$	0	0
$183$	− 440.000i	− 0.177736i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	360.000i	0.140780i
$188$	0	0
$189$	1064.00	0.409495
$190$	0	0
$191$	3552.00	1.34562	0.672811	−	0.739815i	$-0.265087\pi$
$191$	3552.00	1.34562	0.672811	+	0.739815i	$0.265087\pi$
$192$	0	0
$193$	2686.00i	1.00177i	0.865512	+	0.500887i	$0.166993\pi$
$193$	2686.00i	1.00177i	−0.865512	+	0.500887i	$0.833007\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 1410.00i	− 0.509941i	−0.966949	−	0.254970i	$-0.917934\pi$
$197$	− 1410.00i	− 0.509941i	0.966949	−	0.254970i	$-0.0820657\pi$
$198$	0	0
$199$	2968.00	1.05727	0.528633	−	0.848850i	$-0.322705\pi$
$199$	2968.00	1.05727	0.528633	+	0.848850i	$0.322705\pi$
$200$	0	0
$201$	560.000	0.196514
$202$	0	0
$203$	− 1722.00i	− 0.595373i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 2376.00i	− 0.797794i
$208$	0	0
$209$	816.000	0.270067
$210$	0	0
$211$	−1348.00	−0.439811	−0.219906	−	0.975521i	$-0.570575\pi$
$211$	−1348.00	−0.439811	−0.219906	+	0.975521i	$0.570575\pi$
$212$	0	0
$213$	3360.00i	1.08086i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 784.000i	− 0.245260i
$218$	0	0
$219$	2200.00	0.678823
$220$	0	0
$221$	2460.00	0.748767
$222$	0	0
$223$	− 3872.00i	− 1.16273i	−0.813644	−	0.581364i	$-0.802519\pi$
$223$	− 3872.00i	− 1.16273i	0.813644	−	0.581364i	$-0.197481\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5364.00i	1.56838i	0.620524	+	0.784188i	$0.286920\pi$
$227$	5364.00i	1.56838i	−0.620524	+	0.784188i	$0.713080\pi$
$228$	0	0
$229$	874.000	0.252208	0.126104	−	0.992017i	$-0.459753\pi$
$229$	874.000	0.252208	0.126104	+	0.992017i	$0.459753\pi$
$230$	0	0
$231$	−336.000	−0.0957021
$232$	0	0
$233$	− 378.000i	− 0.106282i	−0.998587	−	0.0531408i	$-0.983077\pi$
$233$	− 378.000i	− 0.106282i	0.998587	−	0.0531408i	$-0.0169232\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 832.000i	− 0.228035i
$238$	0	0
$239$	−1920.00	−0.519642	−0.259821	−	0.965657i	$-0.583664\pi$
$239$	−1920.00	−0.519642	−0.259821	+	0.965657i	$0.583664\pi$
$240$	0	0
$241$	4322.00	1.15521	0.577603	−	0.816318i	$-0.303988\pi$
$241$	4322.00	1.15521	0.577603	+	0.816318i	$0.303988\pi$
$242$	0	0
$243$	− 2860.00i	− 0.755017i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 5576.00i	− 1.43641i
$248$	0	0
$249$	−2064.00	−0.525304
$250$	0	0
$251$	5292.00	1.33079	0.665395	−	0.746492i	$-0.268263\pi$
$251$	5292.00	1.33079	0.665395	+	0.746492i	$0.268263\pi$
$252$	0	0
$253$	2592.00i	0.644101i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 5118.00i	− 1.24223i	−0.783721	−	0.621113i	$-0.786681\pi$
$257$	− 5118.00i	− 1.24223i	0.783721	−	0.621113i	$-0.213319\pi$
$258$	0	0
$259$	−770.000	−0.184732
$260$	0	0
$261$	−2706.00	−0.641752
$262$	0	0
$263$	− 3768.00i	− 0.883440i	−0.897153	−	0.441720i	$-0.854368\pi$
$263$	− 3768.00i	− 0.883440i	0.897153	−	0.441720i	$-0.145632\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 5592.00i	− 1.28174i
$268$	0	0
$269$	−3918.00	−0.888047	−0.444024	−	0.896015i	$-0.646449\pi$
$269$	−3918.00	−0.888047	−0.444024	+	0.896015i	$0.646449\pi$
$270$	0	0
$271$	4880.00	1.09387	0.546935	−	0.837175i	$-0.315794\pi$
$271$	4880.00	1.09387	0.546935	+	0.837175i	$0.315794\pi$
$272$	0	0
$273$	2296.00i	0.509012i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 3538.00i	− 0.767429i	−0.923452	−	0.383714i	$-0.874645\pi$
$277$	− 3538.00i	− 0.767429i	0.923452	−	0.383714i	$-0.125355\pi$
$278$	0	0
$279$	−1232.00	−0.264365
$280$	0	0
$281$	−5430.00	−1.15276	−0.576382	−	0.817180i	$-0.695536\pi$
$281$	−5430.00	−1.15276	−0.576382	+	0.817180i	$0.695536\pi$
$282$	0	0
$283$	6436.00i	1.35187i	0.736959	+	0.675937i	$0.236261\pi$
$283$	6436.00i	1.35187i	−0.736959	+	0.675937i	$0.763739\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 1722.00i	− 0.354169i
$288$	0	0
$289$	4013.00	0.816813
$290$	0	0
$291$	6344.00	1.27798
$292$	0	0
$293$	− 1350.00i	− 0.269174i	−0.990902	−	0.134587i	$-0.957029\pi$
$293$	− 1350.00i	− 0.269174i	0.990902	−	0.134587i	$-0.0429707\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1824.00i	0.356361i
$298$	0	0
$299$	17712.0	3.42579
$300$	0	0
$301$	−1204.00	−0.230556
$302$	0	0
$303$	4968.00i	0.941928i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	3332.00i	0.619437i	0.950828	+	0.309719i	$0.100235\pi$
$307$	3332.00i	0.619437i	−0.950828	+	0.309719i	$0.899765\pi$
$308$	0	0
$309$	−2720.00	−0.500762
$310$	0	0
$311$	−4728.00	−0.862059	−0.431029	−	0.902338i	$-0.641849\pi$
$311$	−4728.00	−0.862059	−0.431029	+	0.902338i	$0.641849\pi$
$312$	0	0
$313$	− 5114.00i	− 0.923516i	−0.887006	−	0.461758i	$-0.847219\pi$
$313$	− 5114.00i	− 0.923516i	0.887006	−	0.461758i	$-0.152781\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7206.00i	1.27675i	0.769726	+	0.638374i	$0.220393\pi$
$317$	7206.00i	1.27675i	−0.769726	+	0.638374i	$0.779607\pi$
$318$	0	0
$319$	2952.00	0.518120
$320$	0	0
$321$	3984.00	0.692726
$322$	0	0
$323$	2040.00i	0.351420i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	5528.00i	0.934860i
$328$	0	0
$329$	−1344.00	−0.225219
$330$	0	0
$331$	6260.00	1.03952	0.519759	−	0.854313i	$-0.326022\pi$
$331$	6260.00	1.03952	0.519759	+	0.854313i	$0.326022\pi$
$332$	0	0
$333$	1210.00i	0.199122i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 5326.00i	− 0.860907i	−0.902613	−	0.430454i	$-0.858354\pi$
$337$	− 5326.00i	− 0.860907i	0.902613	−	0.430454i	$-0.141646\pi$
$338$	0	0
$339$	3000.00	0.480642
$340$	0	0
$341$	1344.00	0.213436
$342$	0	0
$343$	− 343.000i	− 0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	36.0000i	0.00556940i	0.999996	+	0.00278470i	$0.000886398\pi$
$347$	36.0000i	0.00556940i	−0.999996	+	0.00278470i	$0.999114\pi$
$348$	0	0
$349$	−3134.00	−0.480685	−0.240343	−	0.970688i	$-0.577260\pi$
$349$	−3134.00	−0.480685	−0.240343	+	0.970688i	$0.577260\pi$
$350$	0	0
$351$	12464.0	1.89538
$352$	0	0
$353$	− 1218.00i	− 0.183648i	−0.995775	−	0.0918238i	$-0.970730\pi$
$353$	− 1218.00i	− 0.183648i	0.995775	−	0.0918238i	$-0.0292697\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 840.000i	− 0.124531i
$358$	0	0
$359$	10008.0	1.47131	0.735657	−	0.677354i	$-0.236873\pi$
$359$	10008.0	1.47131	0.735657	+	0.677354i	$0.236873\pi$
$360$	0	0
$361$	−2235.00	−0.325849
$362$	0	0
$363$	4748.00i	0.686516i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 1072.00i	− 0.152474i	−0.997090	−	0.0762370i	$-0.975709\pi$
$367$	− 1072.00i	− 0.152474i	0.997090	−	0.0762370i	$-0.0242906\pi$
$368$	0	0
$369$	−2706.00	−0.381758
$370$	0	0
$371$	3906.00	0.546602
$372$	0	0
$373$	274.000i	0.0380353i	0.999819	+	0.0190177i	$0.00605388\pi$
$373$	274.000i	0.0380353i	−0.999819	+	0.0190177i	$0.993946\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 20172.0i	− 2.75573i
$378$	0	0
$379$	−7652.00	−1.03709	−0.518545	−	0.855051i	$-0.673526\pi$
$379$	−7652.00	−1.03709	−0.518545	+	0.855051i	$0.673526\pi$
$380$	0	0
$381$	704.000	0.0946641
$382$	0	0
$383$	− 2160.00i	− 0.288175i	−0.989565	−	0.144087i	$-0.953975\pi$
$383$	− 2160.00i	− 0.288175i	0.989565	−	0.144087i	$-0.0460246\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1892.00i	0.248516i
$388$	0	0
$389$	1074.00	0.139984	0.0699922	−	0.997548i	$-0.477703\pi$
$389$	1074.00	0.139984	0.0699922	+	0.997548i	$0.477703\pi$
$390$	0	0
$391$	−6480.00	−0.838127
$392$	0	0
$393$	6192.00i	0.794771i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	6926.00i	0.875582i	0.899077	+	0.437791i	$0.144239\pi$
$397$	6926.00i	0.875582i	−0.899077	+	0.437791i	$0.855761\pi$
$398$	0	0
$399$	−1904.00	−0.238895
$400$	0	0
$401$	1938.00	0.241344	0.120672	−	0.992692i	$-0.461495\pi$
$401$	1938.00	0.241344	0.120672	+	0.992692i	$0.461495\pi$
$402$	0	0
$403$	− 9184.00i	− 1.13521i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 1320.00i	− 0.160762i
$408$	0	0
$409$	9574.00	1.15747	0.578733	−	0.815517i	$-0.303547\pi$
$409$	9574.00	1.15747	0.578733	+	0.815517i	$0.303547\pi$
$410$	0	0
$411$	1512.00	0.181463
$412$	0	0
$413$	− 3780.00i	− 0.450367i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 10000.0i	− 1.17435i
$418$	0	0
$419$	5052.00	0.589037	0.294518	−	0.955646i	$-0.404841\pi$
$419$	5052.00	0.589037	0.294518	+	0.955646i	$0.404841\pi$
$420$	0	0
$421$	3422.00	0.396147	0.198074	−	0.980187i	$-0.436531\pi$
$421$	3422.00	0.396147	0.198074	+	0.980187i	$0.436531\pi$
$422$	0	0
$423$	2112.00i	0.242763i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	770.000i	0.0872668i
$428$	0	0
$429$	−3936.00	−0.442965
$430$	0	0
$431$	2208.00	0.246765	0.123382	−	0.992359i	$-0.460626\pi$
$431$	2208.00	0.246765	0.123382	+	0.992359i	$0.460626\pi$
$432$	0	0
$433$	6814.00i	0.756259i	0.925753	+	0.378129i	$0.123433\pi$
$433$	6814.00i	0.756259i	−0.925753	+	0.378129i	$0.876567\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	14688.0i	1.60783i
$438$	0	0
$439$	−12584.0	−1.36811	−0.684056	−	0.729429i	$-0.739786\pi$
$439$	−12584.0	−1.36811	−0.684056	+	0.729429i	$0.739786\pi$
$440$	0	0
$441$	−539.000	−0.0582011
$442$	0	0
$443$	− 6996.00i	− 0.750316i	−0.926961	−	0.375158i	$-0.877588\pi$
$443$	− 6996.00i	− 0.750316i	0.926961	−	0.375158i	$-0.122412\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	3384.00i	0.358071i
$448$	0	0
$449$	−9474.00	−0.995781	−0.497891	−	0.867240i	$-0.665892\pi$
$449$	−9474.00	−0.995781	−0.497891	+	0.867240i	$0.665892\pi$
$450$	0	0
$451$	2952.00	0.308213
$452$	0	0
$453$	10144.0i	1.05211i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5786.00i	0.592249i	0.955149	+	0.296124i	$0.0956943\pi$
$457$	5786.00i	0.592249i	−0.955149	+	0.296124i	$0.904306\pi$
$458$	0	0
$459$	−4560.00	−0.463709
$460$	0	0
$461$	3438.00	0.347340	0.173670	−	0.984804i	$-0.444437\pi$
$461$	3438.00	0.347340	0.173670	+	0.984804i	$0.444437\pi$
$462$	0	0
$463$	− 9392.00i	− 0.942728i	−0.881939	−	0.471364i	$-0.843762\pi$
$463$	− 9392.00i	− 0.942728i	0.881939	−	0.471364i	$-0.156238\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 4956.00i	− 0.491084i	−0.969386	−	0.245542i	$-0.921034\pi$
$467$	− 4956.00i	− 0.491084i	0.969386	−	0.245542i	$-0.0789660\pi$
$468$	0	0
$469$	−980.000	−0.0964866
$470$	0	0
$471$	−4744.00	−0.464102
$472$	0	0
$473$	− 2064.00i	− 0.200640i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 6138.00i	− 0.589182i
$478$	0	0
$479$	20592.0	1.96424	0.982122	−	0.188248i	$-0.0602807\pi$
$479$	20592.0	1.96424	0.982122	+	0.188248i	$0.0602807\pi$
$480$	0	0
$481$	−9020.00	−0.855045
$482$	0	0
$483$	− 6048.00i	− 0.569759i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 13432.0i	− 1.24982i	−0.780697	−	0.624910i	$-0.785136\pi$
$487$	− 13432.0i	− 1.24982i	0.780697	−	0.624910i	$-0.214864\pi$
$488$	0	0
$489$	−8432.00	−0.779771
$490$	0	0
$491$	−14172.0	−1.30259	−0.651297	−	0.758823i	$-0.725775\pi$
$491$	−14172.0	−1.30259	−0.651297	+	0.758823i	$0.725775\pi$
$492$	0	0
$493$	7380.00i	0.674196i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 5880.00i	− 0.530692i
$498$	0	0
$499$	5956.00	0.534323	0.267162	−	0.963652i	$-0.413914\pi$
$499$	5956.00	0.534323	0.267162	+	0.963652i	$0.413914\pi$
$500$	0	0
$501$	−7776.00	−0.693425
$502$	0	0
$503$	− 16968.0i	− 1.50411i	−0.659102	−	0.752053i	$-0.729064\pi$
$503$	− 16968.0i	− 1.50411i	0.659102	−	0.752053i	$-0.270936\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	18108.0i	1.58620i
$508$	0	0
$509$	−5214.00	−0.454040	−0.227020	−	0.973890i	$-0.572898\pi$
$509$	−5214.00	−0.454040	−0.227020	+	0.973890i	$0.572898\pi$
$510$	0	0
$511$	−3850.00	−0.333295
$512$	0	0
$513$	10336.0i	0.889562i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 2304.00i	− 0.195996i
$518$	0	0
$519$	5448.00	0.460772
$520$	0	0
$521$	−1398.00	−0.117558	−0.0587788	−	0.998271i	$-0.518721\pi$
$521$	−1398.00	−0.117558	−0.0587788	+	0.998271i	$0.518721\pi$
$522$	0	0
$523$	18580.0i	1.55344i	0.629849	+	0.776718i	$0.283117\pi$
$523$	18580.0i	1.55344i	−0.629849	+	0.776718i	$0.716883\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3360.00i	0.277730i
$528$	0	0
$529$	−34489.0	−2.83463
$530$	0	0
$531$	−5940.00	−0.485450
$532$	0	0
$533$	− 20172.0i	− 1.63930i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	6384.00i	0.513017i
$538$	0	0
$539$	588.000	0.0469888
$540$	0	0
$541$	−18970.0	−1.50755	−0.753774	−	0.657133i	$-0.771769\pi$
$541$	−18970.0	−1.50755	−0.753774	+	0.657133i	$0.771769\pi$
$542$	0	0
$543$	6760.00i	0.534253i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 16036.0i	− 1.25347i	−0.779231	−	0.626737i	$-0.784390\pi$
$547$	− 16036.0i	− 1.25347i	0.779231	−	0.626737i	$-0.215610\pi$
$548$	0	0
$549$	1210.00	0.0940647
$550$	0	0
$551$	16728.0	1.29335
$552$	0	0
$553$	1456.00i	0.111963i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	8310.00i	0.632147i	0.948735	+	0.316074i	$0.102365\pi$
$557$	8310.00i	0.632147i	−0.948735	+	0.316074i	$0.897635\pi$
$558$	0	0
$559$	−14104.0	−1.06715
$560$	0	0
$561$	1440.00	0.108372
$562$	0	0
$563$	− 7092.00i	− 0.530892i	−0.964126	−	0.265446i	$-0.914481\pi$
$563$	− 7092.00i	− 0.530892i	0.964126	−	0.265446i	$-0.0855192\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 2177.00i	− 0.161244i
$568$	0	0
$569$	7158.00	0.527380	0.263690	−	0.964608i	$-0.415060\pi$
$569$	7158.00	0.527380	0.263690	+	0.964608i	$0.415060\pi$
$570$	0	0
$571$	6500.00	0.476386	0.238193	−	0.971218i	$-0.423445\pi$
$571$	6500.00	0.476386	0.238193	+	0.971218i	$0.423445\pi$
$572$	0	0
$573$	− 14208.0i	− 1.03586i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	21794.0i	1.57244i	0.617949	+	0.786218i	$0.287964\pi$
$577$	21794.0i	1.57244i	−0.617949	+	0.786218i	$0.712036\pi$
$578$	0	0
$579$	10744.0	0.771166
$580$	0	0
$581$	3612.00	0.257919
$582$	0	0
$583$	6696.00i	0.475678i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	9756.00i	0.685985i	0.939338	+	0.342993i	$0.111441\pi$
$587$	9756.00i	0.685985i	−0.939338	+	0.342993i	$0.888559\pi$
$588$	0	0
$589$	7616.00	0.532787
$590$	0	0
$591$	−5640.00	−0.392553
$592$	0	0
$593$	− 5586.00i	− 0.386829i	−0.981117	−	0.193414i	$-0.938044\pi$
$593$	− 5586.00i	− 0.386829i	0.981117	−	0.193414i	$-0.0619562\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 11872.0i	− 0.813884i
$598$	0	0
$599$	24.0000	0.00163708	0.000818542	−	1.00000i	$-0.499739\pi$
$599$	24.0000	0.00163708	0.000818542		1.00000i	$0.499739\pi$
$600$	0	0
$601$	4298.00	0.291712	0.145856	−	0.989306i	$-0.453406\pi$
$601$	4298.00	0.291712	0.145856	+	0.989306i	$0.453406\pi$
$602$	0	0
$603$	1540.00i	0.104003i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	8480.00i	0.567039i	0.958966	+	0.283519i	$0.0915020\pi$
$607$	8480.00i	0.567039i	−0.958966	+	0.283519i	$0.908498\pi$
$608$	0	0
$609$	−6888.00	−0.458318
$610$	0	0
$611$	−15744.0	−1.04245
$612$	0	0
$613$	1906.00i	0.125583i	0.998027	+	0.0627917i	$0.0200004\pi$
$613$	1906.00i	0.125583i	−0.998027	+	0.0627917i	$0.980000\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	7482.00i	0.488191i	0.969751	+	0.244096i	$0.0784911\pi$
$617$	7482.00i	0.488191i	−0.969751	+	0.244096i	$0.921509\pi$
$618$	0	0
$619$	7348.00	0.477126	0.238563	−	0.971127i	$-0.423324\pi$
$619$	7348.00	0.477126	0.238563	+	0.971127i	$0.423324\pi$
$620$	0	0
$621$	−32832.0	−2.12158
$622$	0	0
$623$	9786.00i	0.629322i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 3264.00i	− 0.207897i
$628$	0	0
$629$	3300.00	0.209189
$630$	0	0
$631$	4520.00	0.285164	0.142582	−	0.989783i	$-0.454460\pi$
$631$	4520.00	0.285164	0.142582	+	0.989783i	$0.454460\pi$
$632$	0	0
$633$	5392.00i	0.338567i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 4018.00i	− 0.249920i
$638$	0	0
$639$	−9240.00	−0.572032
$640$	0	0
$641$	−19806.0	−1.22042	−0.610211	−	0.792239i	$-0.708915\pi$
$641$	−19806.0	−1.22042	−0.610211	+	0.792239i	$0.708915\pi$
$642$	0	0
$643$	5020.00i	0.307884i	0.988080	+	0.153942i	$0.0491969\pi$
$643$	5020.00i	0.307884i	−0.988080	+	0.153942i	$0.950803\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	28392.0i	1.72520i	0.505886	+	0.862600i	$0.331166\pi$
$647$	28392.0i	1.72520i	−0.505886	+	0.862600i	$0.668834\pi$
$648$	0	0
$649$	6480.00	0.391930
$650$	0	0
$651$	−3136.00	−0.188801
$652$	0	0
$653$	17562.0i	1.05246i	0.850343	+	0.526228i	$0.176394\pi$
$653$	17562.0i	1.05246i	−0.850343	+	0.526228i	$0.823606\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	6050.00i	0.359259i
$658$	0	0
$659$	−4716.00	−0.278770	−0.139385	−	0.990238i	$-0.544513\pi$
$659$	−4716.00	−0.278770	−0.139385	+	0.990238i	$0.544513\pi$
$660$	0	0
$661$	−22762.0	−1.33939	−0.669697	−	0.742635i	$-0.733576\pi$
$661$	−22762.0	−1.33939	−0.669697	+	0.742635i	$0.733576\pi$
$662$	0	0
$663$	− 9840.00i	− 0.576401i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	53136.0i	3.08461i
$668$	0	0
$669$	−15488.0	−0.895068
$670$	0	0
$671$	−1320.00	−0.0759434
$672$	0	0
$673$	− 4802.00i	− 0.275042i	−0.990499	−	0.137521i	$-0.956086\pi$
$673$	− 4802.00i	− 0.275042i	0.990499	−	0.137521i	$-0.0439135\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	21558.0i	1.22384i	0.790919	+	0.611921i	$0.209603\pi$
$677$	21558.0i	1.22384i	−0.790919	+	0.611921i	$0.790397\pi$
$678$	0	0
$679$	−11102.0	−0.627475
$680$	0	0
$681$	21456.0	1.20734
$682$	0	0
$683$	− 3780.00i	− 0.211768i	−0.994378	−	0.105884i	$-0.966233\pi$
$683$	− 3780.00i	− 0.211768i	0.994378	−	0.105884i	$-0.0337673\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 3496.00i	− 0.194150i
$688$	0	0
$689$	45756.0	2.52999
$690$	0	0
$691$	−5500.00	−0.302793	−0.151396	−	0.988473i	$-0.548377\pi$
$691$	−5500.00	−0.302793	−0.151396	+	0.988473i	$0.548377\pi$
$692$	0	0
$693$	− 924.000i	− 0.0506491i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	7380.00i	0.401058i
$698$	0	0
$699$	−1512.00	−0.0818156
$700$	0	0
$701$	10230.0	0.551187	0.275593	−	0.961274i	$-0.411126\pi$
$701$	10230.0	0.551187	0.275593	+	0.961274i	$0.411126\pi$
$702$	0	0
$703$	− 7480.00i	− 0.401299i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 8694.00i	− 0.462477i
$708$	0	0
$709$	−10190.0	−0.539765	−0.269883	−	0.962893i	$-0.586985\pi$
$709$	−10190.0	−0.539765	−0.269883	+	0.962893i	$0.586985\pi$
$710$	0	0
$711$	2288.00	0.120685
$712$	0	0
$713$	24192.0i	1.27068i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	7680.00i	0.400021i
$718$	0	0
$719$	−9408.00	−0.487982	−0.243991	−	0.969777i	$-0.578457\pi$
$719$	−9408.00	−0.487982	−0.243991	+	0.969777i	$0.578457\pi$
$720$	0	0
$721$	4760.00	0.245869
$722$	0	0
$723$	− 17288.0i	− 0.889278i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 33064.0i	− 1.68676i	−0.537316	−	0.843381i	$-0.680562\pi$
$727$	− 33064.0i	− 1.68676i	0.537316	−	0.843381i	$-0.319438\pi$
$728$	0	0
$729$	−19837.0	−1.00782
$730$	0	0
$731$	5160.00	0.261080
$732$	0	0
$733$	6322.00i	0.318565i	0.987233	+	0.159283i	$0.0509181\pi$
$733$	6322.00i	0.318565i	−0.987233	+	0.159283i	$0.949082\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 1680.00i	− 0.0839669i
$738$	0	0
$739$	20740.0	1.03239	0.516193	−	0.856472i	$-0.327349\pi$
$739$	20740.0	1.03239	0.516193	+	0.856472i	$0.327349\pi$
$740$	0	0
$741$	−22304.0	−1.10575
$742$	0	0
$743$	− 32040.0i	− 1.58201i	−0.611810	−	0.791005i	$-0.709558\pi$
$743$	− 32040.0i	− 1.58201i	0.611810	−	0.791005i	$-0.290442\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 5676.00i	− 0.278011i
$748$	0	0
$749$	−6972.00	−0.340122
$750$	0	0
$751$	−12832.0	−0.623497	−0.311749	−	0.950165i	$-0.600915\pi$
$751$	−12832.0	−0.623497	−0.311749	+	0.950165i	$0.600915\pi$
$752$	0	0
$753$	− 21168.0i	− 1.02444i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 19906.0i	− 0.955741i	−0.878430	−	0.477870i	$-0.841409\pi$
$757$	− 19906.0i	− 0.955741i	0.878430	−	0.477870i	$-0.158591\pi$
$758$	0	0
$759$	10368.0	0.495829
$760$	0	0
$761$	10842.0	0.516455	0.258227	−	0.966084i	$-0.416862\pi$
$761$	10842.0	0.516455	0.258227	+	0.966084i	$0.416862\pi$
$762$	0	0
$763$	− 9674.00i	− 0.459007i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 44280.0i	− 2.08456i
$768$	0	0
$769$	−28274.0	−1.32586	−0.662930	−	0.748681i	$-0.730687\pi$
$769$	−28274.0	−1.32586	−0.662930	+	0.748681i	$0.730687\pi$
$770$	0	0
$771$	−20472.0	−0.956266
$772$	0	0
$773$	32346.0i	1.50505i	0.658563	+	0.752526i	$0.271165\pi$
$773$	32346.0i	1.50505i	−0.658563	+	0.752526i	$0.728835\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	3080.00i	0.142206i
$778$	0	0
$779$	16728.0	0.769375
$780$	0	0
$781$	10080.0	0.461832
$782$	0	0
$783$	37392.0i	1.70662i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	30116.0i	1.36407i	0.731322	+	0.682033i	$0.238904\pi$
$787$	30116.0i	1.36407i	−0.731322	+	0.682033i	$0.761096\pi$
$788$	0	0
$789$	−15072.0	−0.680073
$790$	0	0
$791$	−5250.00	−0.235991
$792$	0	0
$793$	9020.00i	0.403921i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 6594.00i	− 0.293063i	−0.989206	−	0.146532i	$-0.953189\pi$
$797$	− 6594.00i	− 0.293063i	0.989206	−	0.146532i	$-0.0468110\pi$
$798$	0	0
$799$	5760.00	0.255036
$800$	0	0
$801$	15378.0	0.678346
$802$	0	0
$803$	− 6600.00i	− 0.290048i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	15672.0i	0.683619i
$808$	0	0
$809$	43014.0	1.86933	0.934667	−	0.355524i	$-0.115697\pi$
$809$	43014.0	1.86933	0.934667	+	0.355524i	$0.115697\pi$
$810$	0	0
$811$	−14164.0	−0.613274	−0.306637	−	0.951827i	$-0.599204\pi$
$811$	−14164.0	−0.613274	−0.306637	+	0.951827i	$0.599204\pi$
$812$	0	0
$813$	− 19520.0i	− 0.842062i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 11696.0i	− 0.500846i
$818$	0	0
$819$	−6314.00	−0.269388
$820$	0	0
$821$	34830.0	1.48060	0.740302	−	0.672275i	$-0.234683\pi$
$821$	34830.0	1.48060	0.740302	+	0.672275i	$0.234683\pi$
$822$	0	0
$823$	− 31016.0i	− 1.31367i	−0.754035	−	0.656835i	$-0.771895\pi$
$823$	− 31016.0i	− 1.31367i	0.754035	−	0.656835i	$-0.228105\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	9876.00i	0.415263i	0.978207	+	0.207631i	$0.0665754\pi$
$827$	9876.00i	0.415263i	−0.978207	+	0.207631i	$0.933425\pi$
$828$	0	0
$829$	3154.00	0.132139	0.0660693	−	0.997815i	$-0.478954\pi$
$829$	3154.00	0.132139	0.0660693	+	0.997815i	$0.478954\pi$
$830$	0	0
$831$	−14152.0	−0.590767
$832$	0	0
$833$	1470.00i	0.0611434i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	17024.0i	0.703029i
$838$	0	0
$839$	−36936.0	−1.51987	−0.759936	−	0.649998i	$-0.774770\pi$
$839$	−36936.0	−1.51987	−0.759936	+	0.649998i	$0.774770\pi$
$840$	0	0
$841$	36127.0	1.48128
$842$	0	0
$843$	21720.0i	0.887398i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 8309.00i	− 0.337073i
$848$	0	0
$849$	25744.0	1.04067
$850$	0	0
$851$	23760.0	0.957088
$852$	0	0
$853$	− 9638.00i	− 0.386869i	−0.981113	−	0.193434i	$-0.938037\pi$
$853$	− 9638.00i	− 0.386869i	0.981113	−	0.193434i	$-0.0619626\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	10266.0i	0.409195i	0.978846	+	0.204597i	$0.0655885\pi$
$857$	10266.0i	0.409195i	−0.978846	+	0.204597i	$0.934411\pi$
$858$	0	0
$859$	4084.00	0.162217	0.0811084	−	0.996705i	$-0.474154\pi$
$859$	4084.00	0.162217	0.0811084	+	0.996705i	$0.474154\pi$
$860$	0	0
$861$	−6888.00	−0.272639
$862$	0	0
$863$	192.000i	0.00757330i	0.999993	+	0.00378665i	$0.00120533\pi$
$863$	192.000i	0.00757330i	−0.999993	+	0.00378665i	$0.998795\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 16052.0i	− 0.628783i
$868$	0	0
$869$	−2496.00	−0.0974350
$870$	0	0
$871$	−11480.0	−0.446596
$872$	0	0
$873$	17446.0i	0.676355i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	19910.0i	0.766605i	0.923623	+	0.383303i	$0.125213\pi$
$877$	19910.0i	0.766605i	−0.923623	+	0.383303i	$0.874787\pi$
$878$	0	0
$879$	−5400.00	−0.207210
$880$	0	0
$881$	14802.0	0.566052	0.283026	−	0.959112i	$-0.408662\pi$
$881$	14802.0	0.566052	0.283026	+	0.959112i	$0.408662\pi$
$882$	0	0
$883$	32548.0i	1.24046i	0.784419	+	0.620231i	$0.212961\pi$
$883$	32548.0i	1.24046i	−0.784419	+	0.620231i	$0.787039\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1464.00i	0.0554186i	0.999616	+	0.0277093i	$0.00882128\pi$
$887$	1464.00i	0.0554186i	−0.999616	+	0.0277093i	$0.991179\pi$
$888$	0	0
$889$	−1232.00	−0.0464791
$890$	0	0
$891$	3732.00	0.140322
$892$	0	0
$893$	− 13056.0i	− 0.489252i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 70848.0i	− 2.63717i
$898$	0	0
$899$	27552.0	1.02215
$900$	0	0
$901$	−16740.0	−0.618968
$902$	0	0
$903$	4816.00i	0.177482i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 49564.0i	− 1.81449i	−0.420599	−	0.907247i	$-0.638180\pi$
$907$	− 49564.0i	− 1.81449i	0.420599	−	0.907247i	$-0.361820\pi$
$908$	0	0
$909$	−13662.0	−0.498504
$910$	0	0
$911$	8448.00	0.307239	0.153619	−	0.988130i	$-0.450907\pi$
$911$	8448.00	0.307239	0.153619	+	0.988130i	$0.450907\pi$
$912$	0	0
$913$	6192.00i	0.224453i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 10836.0i	− 0.390225i
$918$	0	0
$919$	−14600.0	−0.524058	−0.262029	−	0.965060i	$-0.584392\pi$
$919$	−14600.0	−0.524058	−0.262029	+	0.965060i	$0.584392\pi$
$920$	0	0
$921$	13328.0	0.476843
$922$	0	0
$923$	− 68880.0i	− 2.45635i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 7480.00i	− 0.265022i
$928$	0	0
$929$	21102.0	0.745247	0.372623	−	0.927983i	$-0.378458\pi$
$929$	21102.0	0.745247	0.372623	+	0.927983i	$0.378458\pi$
$930$	0	0
$931$	3332.00	0.117295
$932$	0	0
$933$	18912.0i	0.663613i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 20806.0i	− 0.725403i	−0.931905	−	0.362701i	$-0.881854\pi$
$937$	− 20806.0i	− 0.725403i	0.931905	−	0.362701i	$-0.118146\pi$
$938$	0	0
$939$	−20456.0	−0.710923
$940$	0	0
$941$	24510.0	0.849100	0.424550	−	0.905404i	$-0.360432\pi$
$941$	24510.0	0.849100	0.424550	+	0.905404i	$0.360432\pi$
$942$	0	0
$943$	53136.0i	1.83494i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 44148.0i	− 1.51491i	−0.652889	−	0.757454i	$-0.726443\pi$
$947$	− 44148.0i	− 1.51491i	0.652889	−	0.757454i	$-0.273557\pi$
$948$	0	0
$949$	−45100.0	−1.54268
$950$	0	0
$951$	28824.0	0.982841
$952$	0	0
$953$	− 27114.0i	− 0.921625i	−0.887498	−	0.460812i	$-0.847558\pi$
$953$	− 27114.0i	− 0.921625i	0.887498	−	0.460812i	$-0.152442\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 11808.0i	− 0.398849i
$958$	0	0
$959$	−2646.00	−0.0890968
$960$	0	0
$961$	−17247.0	−0.578933
$962$	0	0
$963$	10956.0i	0.366617i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 10264.0i	− 0.341332i	−0.985329	−	0.170666i	$-0.945408\pi$
$967$	− 10264.0i	− 0.341332i	0.985329	−	0.170666i	$-0.0545919\pi$
$968$	0	0
$969$	8160.00	0.270523
$970$	0	0
$971$	51468.0	1.70102	0.850508	−	0.525962i	$-0.176295\pi$
$971$	51468.0	1.70102	0.850508	+	0.525962i	$0.176295\pi$
$972$	0	0
$973$	17500.0i	0.576592i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 23790.0i	− 0.779027i	−0.921021	−	0.389514i	$-0.872643\pi$
$977$	− 23790.0i	− 0.779027i	0.921021	−	0.389514i	$-0.127357\pi$
$978$	0	0
$979$	−16776.0	−0.547664
$980$	0	0
$981$	−15202.0	−0.494763
$982$	0	0
$983$	− 26424.0i	− 0.857370i	−0.903454	−	0.428685i	$-0.858977\pi$
$983$	− 26424.0i	− 0.857370i	0.903454	−	0.428685i	$-0.141023\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	5376.00i	0.173374i
$988$	0	0
$989$	37152.0	1.19450
$990$	0	0
$991$	39488.0	1.26577	0.632885	−	0.774246i	$-0.281871\pi$
$991$	39488.0	1.26577	0.632885	+	0.774246i	$0.281871\pi$
$992$	0	0
$993$	− 25040.0i	− 0.800222i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	30854.0i	0.980096i	0.871695	+	0.490048i	$0.163021\pi$
$997$	30854.0i	0.980096i	−0.871695	+	0.490048i	$0.836979\pi$
$998$	0	0
$999$	16720.0	0.529527

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	700.4.e.f.449.1		2
5.2	odd	4		700.4.a.e.1.1		1
5.3	odd	4		28.4.a.b.1.1	✓	1
5.4	even	2	inner	700.4.e.f.449.2		2
15.8	even	4		252.4.a.c.1.1		1
20.3	even	4		112.4.a.c.1.1		1
35.3	even	12		196.4.e.d.177.1		2
35.13	even	4		196.4.a.b.1.1		1
35.18	odd	12		196.4.e.c.177.1		2
35.23	odd	12		196.4.e.c.165.1		2
35.33	even	12		196.4.e.d.165.1		2
40.3	even	4		448.4.a.m.1.1		1
40.13	odd	4		448.4.a.d.1.1		1
60.23	odd	4		1008.4.a.f.1.1		1
105.23	even	12		1764.4.k.k.361.1		2
105.38	odd	12		1764.4.k.e.1549.1		2
105.53	even	12		1764.4.k.k.1549.1		2
105.68	odd	12		1764.4.k.e.361.1		2
105.83	odd	4		1764.4.a.k.1.1		1
140.83	odd	4		784.4.a.n.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.4.a.b.1.1	✓	1	5.3	odd	4
112.4.a.c.1.1		1	20.3	even	4
196.4.a.b.1.1		1	35.13	even	4
196.4.e.c.165.1		2	35.23	odd	12
196.4.e.c.177.1		2	35.18	odd	12
196.4.e.d.165.1		2	35.33	even	12
196.4.e.d.177.1		2	35.3	even	12
252.4.a.c.1.1		1	15.8	even	4
448.4.a.d.1.1		1	40.13	odd	4
448.4.a.m.1.1		1	40.3	even	4
700.4.a.e.1.1		1	5.2	odd	4
700.4.e.f.449.1		2	1.1	even	1	trivial
700.4.e.f.449.2		2	5.4	even	2	inner
784.4.a.n.1.1		1	140.83	odd	4
1008.4.a.f.1.1		1	60.23	odd	4
1764.4.a.k.1.1		1	105.83	odd	4
1764.4.k.e.361.1		2	105.68	odd	12
1764.4.k.e.1549.1		2	105.38	odd	12
1764.4.k.k.361.1		2	105.23	even	12
1764.4.k.k.1549.1		2	105.53	even	12

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:	\( N \)	\(=\)	\( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	700.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.00000i q^{3} +7.00000i q^{7} +11.0000 q^{9} +O(q^{10})\)	\(q-4.00000i q^{3} +7.00000i q^{7} +11.0000 q^{9} -12.0000 q^{11} +82.0000i q^{13} -30.0000i q^{17} -68.0000 q^{19} +28.0000 q^{21} -216.000i q^{23} -152.000i q^{27} -246.000 q^{29} -112.000 q^{31} +48.0000i q^{33} +110.000i q^{37} +328.000 q^{39} -246.000 q^{41} +172.000i q^{43} +192.000i q^{47} -49.0000 q^{49} -120.000 q^{51} -558.000i q^{53} +272.000i q^{57} -540.000 q^{59} +110.000 q^{61} +77.0000i q^{63} +140.000i q^{67} -864.000 q^{69} -840.000 q^{71} +550.000i q^{73} -84.0000i q^{77} +208.000 q^{79} -311.000 q^{81} -516.000i q^{83} +984.000i q^{87} +1398.00 q^{89} -574.000 q^{91} +448.000i q^{93} +1586.00i q^{97} -132.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 22 q^{9}+O(q^{10}) \) 2 * q + 22 * q^9	\( 2 q + 22 q^{9} - 24 q^{11} - 136 q^{19} + 56 q^{21} - 492 q^{29} - 224 q^{31} + 656 q^{39} - 492 q^{41} - 98 q^{49} - 240 q^{51} - 1080 q^{59} + 220 q^{61} - 1728 q^{69} - 1680 q^{71} + 416 q^{79} - 622 q^{81} + 2796 q^{89} - 1148 q^{91} - 264 q^{99}+O(q^{100}) \) 2 * q + 22 * q^9 - 24 * q^11 - 136 * q^19 + 56 * q^21 - 492 * q^29 - 224 * q^31 + 656 * q^39 - 492 * q^41 - 98 * q^49 - 240 * q^51 - 1080 * q^59 + 220 * q^61 - 1728 * q^69 - 1680 * q^71 + 416 * q^79 - 622 * q^81 + 2796 * q^89 - 1148 * q^91 - 264 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

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