LMFDB - Embedded newform 400.4.c.c.49.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,4,Mod(49,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("400.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	400.c (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:	$23.6007640023$
Analytic rank:	$0$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 10)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	400.49
Dual form			400.4.c.c.49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.00000i q^{3} -4.00000i q^{7} -37.0000 q^{9} +O(q^{10})$	$q+8.00000i q^{3} -4.00000i q^{7} -37.0000 q^{9} -12.0000 q^{11} -58.0000i q^{13} -66.0000i q^{17} -100.000 q^{19} +32.0000 q^{21} -132.000i q^{23} -80.0000i q^{27} +90.0000 q^{29} -152.000 q^{31} -96.0000i q^{33} +34.0000i q^{37} +464.000 q^{39} -438.000 q^{41} -32.0000i q^{43} -204.000i q^{47} +327.000 q^{49} +528.000 q^{51} +222.000i q^{53} -800.000i q^{57} +420.000 q^{59} +902.000 q^{61} +148.000i q^{63} -1024.00i q^{67} +1056.00 q^{69} -432.000 q^{71} +362.000i q^{73} +48.0000i q^{77} -160.000 q^{79} -359.000 q^{81} -72.0000i q^{83} +720.000i q^{87} -810.000 q^{89} -232.000 q^{91} -1216.00i q^{93} -1106.00i q^{97} +444.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 74 q^{9}+O(q^{10}) $ 2 * q - 74 * q^9	$ 2 q - 74 q^{9} - 24 q^{11} - 200 q^{19} + 64 q^{21} + 180 q^{29} - 304 q^{31} + 928 q^{39} - 876 q^{41} + 654 q^{49} + 1056 q^{51} + 840 q^{59} + 1804 q^{61} + 2112 q^{69} - 864 q^{71} - 320 q^{79} - 718 q^{81} - 1620 q^{89} - 464 q^{91} + 888 q^{99}+O(q^{100}) $ 2 * q - 74 * q^9 - 24 * q^11 - 200 * q^19 + 64 * q^21 + 180 * q^29 - 304 * q^31 + 928 * q^39 - 876 * q^41 + 654 * q^49 + 1056 * q^51 + 840 * q^59 + 1804 * q^61 + 2112 * q^69 - 864 * q^71 - 320 * q^79 - 718 * q^81 - 1620 * q^89 - 464 * q^91 + 888 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	8.00000i	1.53960i	0.638285	+	0.769800i	$0.279644\pi$
$3$	8.00000i	1.53960i	−0.638285	+	0.769800i	$0.720356\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 4.00000i	− 0.215980i	−0.994152	−	0.107990i	$-0.965559\pi$
$7$	− 4.00000i	− 0.215980i	0.994152	−	0.107990i	$-0.0344414\pi$
$8$	0	0
$9$	−37.0000	−1.37037
$10$	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
$13$	− 58.0000i	− 1.23741i	−0.785624	−	0.618704i	$-0.787658\pi$
$13$	− 58.0000i	− 1.23741i	0.785624	−	0.618704i	$-0.212342\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 66.0000i	− 0.941609i	−0.882238	−	0.470804i	$-0.843964\pi$
$17$	− 66.0000i	− 0.941609i	0.882238	−	0.470804i	$-0.156036\pi$
$18$	0	0
$19$	−100.000	−1.20745	−0.603726	−	0.797192i	$-0.706318\pi$
$19$	−100.000	−1.20745	−0.603726	+	0.797192i	$0.706318\pi$
$20$	0	0
$21$	32.0000	0.332522
$22$	0	0
$23$	− 132.000i	− 1.19669i	−0.801238	−	0.598346i	$-0.795825\pi$
$23$	− 132.000i	− 1.19669i	0.801238	−	0.598346i	$-0.204175\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 80.0000i	− 0.570222i
$28$	0	0
$29$	90.0000	0.576296	0.288148	−	0.957586i	$-0.406961\pi$
$29$	90.0000	0.576296	0.288148	+	0.957586i	$0.406961\pi$
$30$	0	0
$31$	−152.000	−0.880645	−0.440323	−	0.897840i	$-0.645136\pi$
$31$	−152.000	−0.880645	−0.440323	+	0.897840i	$0.645136\pi$
$32$	0	0
$33$	− 96.0000i	− 0.506408i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	34.0000i	0.151069i	0.997143	+	0.0755347i	$0.0240664\pi$
$37$	34.0000i	0.151069i	−0.997143	+	0.0755347i	$0.975934\pi$
$38$	0	0
$39$	464.000	1.90511
$40$	0	0
$41$	−438.000	−1.66839	−0.834196	−	0.551467i	$-0.814068\pi$
$41$	−438.000	−1.66839	−0.834196	+	0.551467i	$0.814068\pi$
$42$	0	0
$43$	− 32.0000i	− 0.113487i	−0.998389	−	0.0567437i	$-0.981928\pi$
$43$	− 32.0000i	− 0.113487i	0.998389	−	0.0567437i	$-0.0180718\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 204.000i	− 0.633116i	−0.948573	−	0.316558i	$-0.897473\pi$
$47$	− 204.000i	− 0.633116i	0.948573	−	0.316558i	$-0.102527\pi$
$48$	0	0
$49$	327.000	0.953353
$50$	0	0
$51$	528.000	1.44970
$52$	0	0
$53$	222.000i	0.575359i	0.957727	+	0.287680i	$0.0928838\pi$
$53$	222.000i	0.575359i	−0.957727	+	0.287680i	$0.907116\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 800.000i	− 1.85899i
$58$	0	0
$59$	420.000	0.926769	0.463384	−	0.886157i	$-0.346635\pi$
$59$	420.000	0.926769	0.463384	+	0.886157i	$0.346635\pi$
$60$	0	0
$61$	902.000	1.89327	0.946633	−	0.322312i	$-0.104460\pi$
$61$	902.000	1.89327	0.946633	+	0.322312i	$0.104460\pi$
$62$	0	0
$63$	148.000i	0.295972i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 1024.00i	− 1.86719i	−0.358334	−	0.933593i	$-0.616655\pi$
$67$	− 1024.00i	− 1.86719i	0.358334	−	0.933593i	$-0.383345\pi$
$68$	0	0
$69$	1056.00	1.84243
$70$	0	0
$71$	−432.000	−0.722098	−0.361049	−	0.932547i	$-0.617581\pi$
$71$	−432.000	−0.722098	−0.361049	+	0.932547i	$0.617581\pi$
$72$	0	0
$73$	362.000i	0.580396i	0.956967	+	0.290198i	$0.0937211\pi$
$73$	362.000i	0.580396i	−0.956967	+	0.290198i	$0.906279\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	48.0000i	0.0710404i
$78$	0	0
$79$	−160.000	−0.227866	−0.113933	−	0.993488i	$-0.536345\pi$
$79$	−160.000	−0.227866	−0.113933	+	0.993488i	$0.536345\pi$
$80$	0	0
$81$	−359.000	−0.492455
$82$	0	0
$83$	− 72.0000i	− 0.0952172i	−0.998866	−	0.0476086i	$-0.984840\pi$
$83$	− 72.0000i	− 0.0952172i	0.998866	−	0.0476086i	$-0.0151600\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	720.000i	0.887266i
$88$	0	0
$89$	−810.000	−0.964717	−0.482359	−	0.875974i	$-0.660220\pi$
$89$	−810.000	−0.964717	−0.482359	+	0.875974i	$0.660220\pi$
$90$	0	0
$91$	−232.000	−0.267255
$92$	0	0
$93$	− 1216.00i	− 1.35584i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 1106.00i	− 1.15770i	−0.815433	−	0.578852i	$-0.803501\pi$
$97$	− 1106.00i	− 1.15770i	0.815433	−	0.578852i	$-0.196499\pi$
$98$	0	0
$99$	444.000	0.450744
$100$	0	0
$101$	−258.000	−0.254178	−0.127089	−	0.991891i	$-0.540563\pi$
$101$	−258.000	−0.254178	−0.127089	+	0.991891i	$0.540563\pi$
$102$	0	0
$103$	988.000i	0.945151i	0.881290	+	0.472575i	$0.156676\pi$
$103$	988.000i	0.945151i	−0.881290	+	0.472575i	$0.843324\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 24.0000i	− 0.0216838i	−0.999941	−	0.0108419i	$-0.996549\pi$
$107$	− 24.0000i	− 0.0216838i	0.999941	−	0.0108419i	$-0.00345115\pi$
$108$	0	0
$109$	−950.000	−0.834803	−0.417401	−	0.908722i	$-0.637059\pi$
$109$	−950.000	−0.834803	−0.417401	+	0.908722i	$0.637059\pi$
$110$	0	0
$111$	−272.000	−0.232586
$112$	0	0
$113$	− 1038.00i	− 0.864131i	−0.901842	−	0.432066i	$-0.857785\pi$
$113$	− 1038.00i	− 0.864131i	0.901842	−	0.432066i	$-0.142215\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2146.00i	1.69571i
$118$	0	0
$119$	−264.000	−0.203368
$120$	0	0
$121$	−1187.00	−0.891811
$122$	0	0
$123$	− 3504.00i	− 2.56866i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 124.000i	− 0.0866395i	−0.999061	−	0.0433198i	$-0.986207\pi$
$127$	− 124.000i	− 0.0866395i	0.999061	−	0.0433198i	$-0.0137934\pi$
$128$	0	0
$129$	256.000	0.174725
$130$	0	0
$131$	−132.000	−0.0880374	−0.0440187	−	0.999031i	$-0.514016\pi$
$131$	−132.000	−0.0880374	−0.0440187	+	0.999031i	$0.514016\pi$
$132$	0	0
$133$	400.000i	0.260785i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1254.00i	0.782018i	0.920387	+	0.391009i	$0.127874\pi$
$137$	1254.00i	0.782018i	−0.920387	+	0.391009i	$0.872126\pi$
$138$	0	0
$139$	−2860.00	−1.74519	−0.872597	−	0.488440i	$-0.837566\pi$
$139$	−2860.00	−1.74519	−0.872597	+	0.488440i	$0.837566\pi$
$140$	0	0
$141$	1632.00	0.974746
$142$	0	0
$143$	696.000i	0.407010i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	2616.00i	1.46778i
$148$	0	0
$149$	−750.000	−0.412365	−0.206183	−	0.978514i	$-0.566104\pi$
$149$	−750.000	−0.412365	−0.206183	+	0.978514i	$0.566104\pi$
$150$	0	0
$151$	448.000	0.241442	0.120721	−	0.992686i	$-0.461479\pi$
$151$	448.000	0.241442	0.120721	+	0.992686i	$0.461479\pi$
$152$	0	0
$153$	2442.00i	1.29035i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 2246.00i	− 1.14172i	−0.821047	−	0.570861i	$-0.806610\pi$
$157$	− 2246.00i	− 1.14172i	0.821047	−	0.570861i	$-0.193390\pi$
$158$	0	0
$159$	−1776.00	−0.885824
$160$	0	0
$161$	−528.000	−0.258461
$162$	0	0
$163$	568.000i	0.272940i	0.990644	+	0.136470i	$0.0435757\pi$
$163$	568.000i	0.272940i	−0.990644	+	0.136470i	$0.956424\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 1524.00i	− 0.706172i	−0.935591	−	0.353086i	$-0.885132\pi$
$167$	− 1524.00i	− 0.706172i	0.935591	−	0.353086i	$-0.114868\pi$
$168$	0	0
$169$	−1167.00	−0.531179
$170$	0	0
$171$	3700.00	1.65466
$172$	0	0
$173$	3702.00i	1.62692i	0.581618	+	0.813462i	$0.302420\pi$
$173$	3702.00i	1.62692i	−0.581618	+	0.813462i	$0.697580\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	3360.00i	1.42685i
$178$	0	0
$179$	3180.00	1.32785	0.663923	−	0.747801i	$-0.268890\pi$
$179$	3180.00	1.32785	0.663923	+	0.747801i	$0.268890\pi$
$180$	0	0
$181$	−2098.00	−0.861564	−0.430782	−	0.902456i	$-0.641762\pi$
$181$	−2098.00	−0.861564	−0.430782	+	0.902456i	$0.641762\pi$
$182$	0	0
$183$	7216.00i	2.91487i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	792.000i	0.309715i
$188$	0	0
$189$	−320.000	−0.123156
$190$	0	0
$191$	−4392.00	−1.66384	−0.831921	−	0.554894i	$-0.812759\pi$
$191$	−4392.00	−1.66384	−0.831921	+	0.554894i	$0.812759\pi$
$192$	0	0
$193$	− 2158.00i	− 0.804851i	−0.915453	−	0.402425i	$-0.868167\pi$
$193$	− 2158.00i	− 0.804851i	0.915453	−	0.402425i	$-0.131833\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1074.00i	0.388423i	0.980960	+	0.194212i	$0.0622148\pi$
$197$	1074.00i	0.388423i	−0.980960	+	0.194212i	$0.937785\pi$
$198$	0	0
$199$	2840.00	1.01167	0.505835	−	0.862630i	$-0.331185\pi$
$199$	2840.00	1.01167	0.505835	+	0.862630i	$0.331185\pi$
$200$	0	0
$201$	8192.00	2.87472
$202$	0	0
$203$	− 360.000i	− 0.124468i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	4884.00i	1.63991i
$208$	0	0
$209$	1200.00	0.397157
$210$	0	0
$211$	2668.00	0.870487	0.435243	−	0.900313i	$-0.356662\pi$
$211$	2668.00	0.870487	0.435243	+	0.900313i	$0.356662\pi$
$212$	0	0
$213$	− 3456.00i	− 1.11174i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	608.000i	0.190202i
$218$	0	0
$219$	−2896.00	−0.893578
$220$	0	0
$221$	−3828.00	−1.16515
$222$	0	0
$223$	− 1772.00i	− 0.532116i	−0.963957	−	0.266058i	$-0.914279\pi$
$223$	− 1772.00i	− 0.532116i	0.963957	−	0.266058i	$-0.0857213\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 2784.00i	− 0.814011i	−0.913426	−	0.407006i	$-0.866573\pi$
$227$	− 2784.00i	− 0.814011i	0.913426	−	0.407006i	$-0.133427\pi$
$228$	0	0
$229$	−350.000	−0.100998	−0.0504992	−	0.998724i	$-0.516081\pi$
$229$	−350.000	−0.100998	−0.0504992	+	0.998724i	$0.516081\pi$
$230$	0	0
$231$	−384.000	−0.109374
$232$	0	0
$233$	1962.00i	0.551652i	0.961208	+	0.275826i	$0.0889513\pi$
$233$	1962.00i	0.551652i	−0.961208	+	0.275826i	$0.911049\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 1280.00i	− 0.350823i
$238$	0	0
$239$	−4320.00	−1.16919	−0.584597	−	0.811324i	$-0.698748\pi$
$239$	−4320.00	−1.16919	−0.584597	+	0.811324i	$0.698748\pi$
$240$	0	0
$241$	−478.000	−0.127762	−0.0638811	−	0.997958i	$-0.520348\pi$
$241$	−478.000	−0.127762	−0.0638811	+	0.997958i	$0.520348\pi$
$242$	0	0
$243$	− 5032.00i	− 1.32841i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5800.00i	1.49411i
$248$	0	0
$249$	576.000	0.146596
$250$	0	0
$251$	−2652.00	−0.666903	−0.333452	−	0.942767i	$-0.608213\pi$
$251$	−2652.00	−0.666903	−0.333452	+	0.942767i	$0.608213\pi$
$252$	0	0
$253$	1584.00i	0.393617i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2334.00i	0.566502i	0.959046	+	0.283251i	$0.0914129\pi$
$257$	2334.00i	0.566502i	−0.959046	+	0.283251i	$0.908587\pi$
$258$	0	0
$259$	136.000	0.0326279
$260$	0	0
$261$	−3330.00	−0.789739
$262$	0	0
$263$	3948.00i	0.925643i	0.886451	+	0.462822i	$0.153163\pi$
$263$	3948.00i	0.925643i	−0.886451	+	0.462822i	$0.846837\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 6480.00i	− 1.48528i
$268$	0	0
$269$	−1590.00	−0.360387	−0.180193	−	0.983631i	$-0.557672\pi$
$269$	−1590.00	−0.360387	−0.180193	+	0.983631i	$0.557672\pi$
$270$	0	0
$271$	−4952.00	−1.11001	−0.555005	−	0.831847i	$-0.687284\pi$
$271$	−4952.00	−1.11001	−0.555005	+	0.831847i	$0.687284\pi$
$272$	0	0
$273$	− 1856.00i	− 0.411466i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 1646.00i	− 0.357034i	−0.983937	−	0.178517i	$-0.942870\pi$
$277$	− 1646.00i	− 0.357034i	0.983937	−	0.178517i	$-0.0571300\pi$
$278$	0	0
$279$	5624.00	1.20681
$280$	0	0
$281$	−1158.00	−0.245838	−0.122919	−	0.992417i	$-0.539226\pi$
$281$	−1158.00	−0.245838	−0.122919	+	0.992417i	$0.539226\pi$
$282$	0	0
$283$	− 6992.00i	− 1.46866i	−0.678792	−	0.734331i	$-0.737496\pi$
$283$	− 6992.00i	− 1.46866i	0.678792	−	0.734331i	$-0.262504\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1752.00i	0.360339i
$288$	0	0
$289$	557.000	0.113373
$290$	0	0
$291$	8848.00	1.78240
$292$	0	0
$293$	− 258.000i	− 0.0514421i	−0.999669	−	0.0257210i	$-0.991812\pi$
$293$	− 258.000i	− 0.0514421i	0.999669	−	0.0257210i	$-0.00818816\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	960.000i	0.187558i
$298$	0	0
$299$	−7656.00	−1.48080
$300$	0	0
$301$	−128.000	−0.0245110
$302$	0	0
$303$	− 2064.00i	− 0.391332i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 8944.00i	− 1.66274i	−0.555720	−	0.831370i	$-0.687557\pi$
$307$	− 8944.00i	− 1.66274i	0.555720	−	0.831370i	$-0.312443\pi$
$308$	0	0
$309$	−7904.00	−1.45515
$310$	0	0
$311$	−1392.00	−0.253804	−0.126902	−	0.991915i	$-0.540503\pi$
$311$	−1392.00	−0.253804	−0.126902	+	0.991915i	$0.540503\pi$
$312$	0	0
$313$	− 5878.00i	− 1.06148i	−0.847534	−	0.530742i	$-0.821913\pi$
$313$	− 5878.00i	− 1.06148i	0.847534	−	0.530742i	$-0.178087\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 10326.0i	− 1.82955i	−0.403969	−	0.914773i	$-0.632370\pi$
$317$	− 10326.0i	− 1.82955i	0.403969	−	0.914773i	$-0.367630\pi$
$318$	0	0
$319$	−1080.00	−0.189556
$320$	0	0
$321$	192.000	0.0333844
$322$	0	0
$323$	6600.00i	1.13695i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 7600.00i	− 1.28526i
$328$	0	0
$329$	−816.000	−0.136740
$330$	0	0
$331$	4228.00	0.702090	0.351045	−	0.936359i	$-0.385826\pi$
$331$	4228.00	0.702090	0.351045	+	0.936359i	$0.385826\pi$
$332$	0	0
$333$	− 1258.00i	− 0.207021i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 1106.00i	− 0.178776i	−0.995997	−	0.0893882i	$-0.971509\pi$
$337$	− 1106.00i	− 0.178776i	0.995997	−	0.0893882i	$-0.0284912\pi$
$338$	0	0
$339$	8304.00	1.33042
$340$	0	0
$341$	1824.00	0.289663
$342$	0	0
$343$	− 2680.00i	− 0.421885i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	9336.00i	1.44433i	0.691720	+	0.722165i	$0.256853\pi$
$347$	9336.00i	1.44433i	−0.691720	+	0.722165i	$0.743147\pi$
$348$	0	0
$349$	11770.0	1.80525	0.902627	−	0.430424i	$-0.141636\pi$
$349$	11770.0	1.80525	0.902627	+	0.430424i	$0.141636\pi$
$350$	0	0
$351$	−4640.00	−0.705598
$352$	0	0
$353$	8322.00i	1.25477i	0.778707	+	0.627387i	$0.215876\pi$
$353$	8322.00i	1.25477i	−0.778707	+	0.627387i	$0.784124\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 2112.00i	− 0.313106i
$358$	0	0
$359$	10680.0	1.57011	0.785054	−	0.619427i	$-0.212635\pi$
$359$	10680.0	1.57011	0.785054	+	0.619427i	$0.212635\pi$
$360$	0	0
$361$	3141.00	0.457938
$362$	0	0
$363$	− 9496.00i	− 1.37303i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 5884.00i	− 0.836900i	−0.908240	−	0.418450i	$-0.862574\pi$
$367$	− 5884.00i	− 0.836900i	0.908240	−	0.418450i	$-0.137426\pi$
$368$	0	0
$369$	16206.0	2.28632
$370$	0	0
$371$	888.000	0.124266
$372$	0	0
$373$	− 2098.00i	− 0.291234i	−0.989341	−	0.145617i	$-0.953483\pi$
$373$	− 2098.00i	− 0.291234i	0.989341	−	0.145617i	$-0.0465167\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 5220.00i	− 0.713113i
$378$	0	0
$379$	3860.00	0.523153	0.261576	−	0.965183i	$-0.415758\pi$
$379$	3860.00	0.523153	0.261576	+	0.965183i	$0.415758\pi$
$380$	0	0
$381$	992.000	0.133390
$382$	0	0
$383$	9588.00i	1.27917i	0.768718	+	0.639587i	$0.220895\pi$
$383$	9588.00i	1.27917i	−0.768718	+	0.639587i	$0.779105\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1184.00i	0.155520i
$388$	0	0
$389$	13410.0	1.74785	0.873925	−	0.486060i	$-0.161566\pi$
$389$	13410.0	1.74785	0.873925	+	0.486060i	$0.161566\pi$
$390$	0	0
$391$	−8712.00	−1.12682
$392$	0	0
$393$	− 1056.00i	− 0.135542i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	13114.0i	1.65787i	0.559348	+	0.828933i	$0.311052\pi$
$397$	13114.0i	1.65787i	−0.559348	+	0.828933i	$0.688948\pi$
$398$	0	0
$399$	−3200.00	−0.401505
$400$	0	0
$401$	−5838.00	−0.727022	−0.363511	−	0.931590i	$-0.618422\pi$
$401$	−5838.00	−0.727022	−0.363511	+	0.931590i	$0.618422\pi$
$402$	0	0
$403$	8816.00i	1.08972i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 408.000i	− 0.0496899i
$408$	0	0
$409$	−9530.00	−1.15215	−0.576074	−	0.817398i	$-0.695416\pi$
$409$	−9530.00	−1.15215	−0.576074	+	0.817398i	$0.695416\pi$
$410$	0	0
$411$	−10032.0	−1.20400
$412$	0	0
$413$	− 1680.00i	− 0.200163i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 22880.0i	− 2.68690i
$418$	0	0
$419$	7260.00	0.846478	0.423239	−	0.906018i	$-0.360893\pi$
$419$	7260.00	0.846478	0.423239	+	0.906018i	$0.360893\pi$
$420$	0	0
$421$	12062.0	1.39636	0.698178	−	0.715924i	$-0.253994\pi$
$421$	12062.0	1.39636	0.698178	+	0.715924i	$0.253994\pi$
$422$	0	0
$423$	7548.00i	0.867604i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 3608.00i	− 0.408907i
$428$	0	0
$429$	−5568.00	−0.626633
$430$	0	0
$431$	13608.0	1.52082	0.760411	−	0.649442i	$-0.224998\pi$
$431$	13608.0	1.52082	0.760411	+	0.649442i	$0.224998\pi$
$432$	0	0
$433$	− 3838.00i	− 0.425964i	−0.977056	−	0.212982i	$-0.931682\pi$
$433$	− 3838.00i	− 0.425964i	0.977056	−	0.212982i	$-0.0683176\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	13200.0i	1.44495i
$438$	0	0
$439$	7400.00	0.804516	0.402258	−	0.915526i	$-0.368225\pi$
$439$	7400.00	0.804516	0.402258	+	0.915526i	$0.368225\pi$
$440$	0	0
$441$	−12099.0	−1.30645
$442$	0	0
$443$	− 8352.00i	− 0.895746i	−0.894097	−	0.447873i	$-0.852182\pi$
$443$	− 8352.00i	− 0.895746i	0.894097	−	0.447873i	$-0.147818\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 6000.00i	− 0.634878i
$448$	0	0
$449$	−10770.0	−1.13200	−0.566000	−	0.824405i	$-0.691510\pi$
$449$	−10770.0	−1.13200	−0.566000	+	0.824405i	$0.691510\pi$
$450$	0	0
$451$	5256.00	0.548770
$452$	0	0
$453$	3584.00i	0.371724i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6694.00i	0.685191i	0.939483	+	0.342595i	$0.111306\pi$
$457$	6694.00i	0.685191i	−0.939483	+	0.342595i	$0.888694\pi$
$458$	0	0
$459$	−5280.00	−0.536927
$460$	0	0
$461$	−3018.00	−0.304907	−0.152454	−	0.988311i	$-0.548717\pi$
$461$	−3018.00	−0.304907	−0.152454	+	0.988311i	$0.548717\pi$
$462$	0	0
$463$	− 14492.0i	− 1.45464i	−0.686296	−	0.727322i	$-0.740765\pi$
$463$	− 14492.0i	− 1.45464i	0.686296	−	0.727322i	$-0.259235\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	7776.00i	0.770515i	0.922809	+	0.385257i	$0.125887\pi$
$467$	7776.00i	0.770515i	−0.922809	+	0.385257i	$0.874113\pi$
$468$	0	0
$469$	−4096.00	−0.403274
$470$	0	0
$471$	17968.0	1.75780
$472$	0	0
$473$	384.000i	0.0373284i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 8214.00i	− 0.788455i
$478$	0	0
$479$	−13680.0	−1.30492	−0.652458	−	0.757825i	$-0.726262\pi$
$479$	−13680.0	−1.30492	−0.652458	+	0.757825i	$0.726262\pi$
$480$	0	0
$481$	1972.00	0.186934
$482$	0	0
$483$	− 4224.00i	− 0.397927i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	7916.00i	0.736567i	0.929714	+	0.368284i	$0.120054\pi$
$487$	7916.00i	0.736567i	−0.929714	+	0.368284i	$0.879946\pi$
$488$	0	0
$489$	−4544.00	−0.420218
$490$	0	0
$491$	−13932.0	−1.28053	−0.640267	−	0.768152i	$-0.721176\pi$
$491$	−13932.0	−1.28053	−0.640267	+	0.768152i	$0.721176\pi$
$492$	0	0
$493$	− 5940.00i	− 0.542645i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1728.00i	0.155959i
$498$	0	0
$499$	−8260.00	−0.741019	−0.370509	−	0.928829i	$-0.620817\pi$
$499$	−8260.00	−0.741019	−0.370509	+	0.928829i	$0.620817\pi$
$500$	0	0
$501$	12192.0	1.08722
$502$	0	0
$503$	11148.0i	0.988200i	0.869405	+	0.494100i	$0.164502\pi$
$503$	11148.0i	0.988200i	−0.869405	+	0.494100i	$0.835498\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 9336.00i	− 0.817803i
$508$	0	0
$509$	9690.00	0.843815	0.421907	−	0.906639i	$-0.361361\pi$
$509$	9690.00	0.843815	0.421907	+	0.906639i	$0.361361\pi$
$510$	0	0
$511$	1448.00	0.125354
$512$	0	0
$513$	8000.00i	0.688516i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	2448.00i	0.208245i
$518$	0	0
$519$	−29616.0	−2.50481
$520$	0	0
$521$	−16038.0	−1.34863	−0.674316	−	0.738443i	$-0.735562\pi$
$521$	−16038.0	−1.34863	−0.674316	+	0.738443i	$0.735562\pi$
$522$	0	0
$523$	− 992.000i	− 0.0829391i	−0.999140	−	0.0414695i	$-0.986796\pi$
$523$	− 992.000i	− 0.0829391i	0.999140	−	0.0414695i	$-0.0132039\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	10032.0i	0.829223i
$528$	0	0
$529$	−5257.00	−0.432070
$530$	0	0
$531$	−15540.0	−1.27002
$532$	0	0
$533$	25404.0i	2.06448i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	25440.0i	2.04435i
$538$	0	0
$539$	−3924.00	−0.313578
$540$	0	0
$541$	7142.00	0.567576	0.283788	−	0.958887i	$-0.408409\pi$
$541$	7142.00	0.567576	0.283788	+	0.958887i	$0.408409\pi$
$542$	0	0
$543$	− 16784.0i	− 1.32646i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	7616.00i	0.595314i	0.954673	+	0.297657i	$0.0962051\pi$
$547$	7616.00i	0.595314i	−0.954673	+	0.297657i	$0.903795\pi$
$548$	0	0
$549$	−33374.0	−2.59448
$550$	0	0
$551$	−9000.00	−0.695849
$552$	0	0
$553$	640.000i	0.0492144i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	10314.0i	0.784593i	0.919839	+	0.392296i	$0.128319\pi$
$557$	10314.0i	0.784593i	−0.919839	+	0.392296i	$0.871681\pi$
$558$	0	0
$559$	−1856.00	−0.140430
$560$	0	0
$561$	−6336.00	−0.476838
$562$	0	0
$563$	7128.00i	0.533587i	0.963754	+	0.266793i	$0.0859641\pi$
$563$	7128.00i	0.533587i	−0.963754	+	0.266793i	$0.914036\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1436.00i	0.106360i
$568$	0	0
$569$	−2010.00	−0.148091	−0.0740453	−	0.997255i	$-0.523591\pi$
$569$	−2010.00	−0.148091	−0.0740453	+	0.997255i	$0.523591\pi$
$570$	0	0
$571$	23188.0	1.69945	0.849726	−	0.527224i	$-0.176767\pi$
$571$	23188.0	1.69945	0.849726	+	0.527224i	$0.176767\pi$
$572$	0	0
$573$	− 35136.0i	− 2.56165i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 22466.0i	− 1.62092i	−0.585793	−	0.810461i	$-0.699217\pi$
$577$	− 22466.0i	− 1.62092i	0.585793	−	0.810461i	$-0.300783\pi$
$578$	0	0
$579$	17264.0	1.23915
$580$	0	0
$581$	−288.000	−0.0205650
$582$	0	0
$583$	− 2664.00i	− 0.189248i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	22776.0i	1.60148i	0.599015	+	0.800738i	$0.295559\pi$
$587$	22776.0i	1.60148i	−0.599015	+	0.800738i	$0.704441\pi$
$588$	0	0
$589$	15200.0	1.06334
$590$	0	0
$591$	−8592.00	−0.598016
$592$	0	0
$593$	− 21198.0i	− 1.46796i	−0.679174	−	0.733978i	$-0.737662\pi$
$593$	− 21198.0i	− 1.46796i	0.679174	−	0.733978i	$-0.262338\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	22720.0i	1.55757i
$598$	0	0
$599$	15960.0	1.08866	0.544330	−	0.838871i	$-0.316784\pi$
$599$	15960.0	1.08866	0.544330	+	0.838871i	$0.316784\pi$
$600$	0	0
$601$	5882.00	0.399221	0.199610	−	0.979875i	$-0.436032\pi$
$601$	5882.00	0.399221	0.199610	+	0.979875i	$0.436032\pi$
$602$	0	0
$603$	37888.0i	2.55874i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	8516.00i	0.569446i	0.958610	+	0.284723i	$0.0919016\pi$
$607$	8516.00i	0.569446i	−0.958610	+	0.284723i	$0.908098\pi$
$608$	0	0
$609$	2880.00	0.191631
$610$	0	0
$611$	−11832.0	−0.783423
$612$	0	0
$613$	8462.00i	0.557548i	0.960357	+	0.278774i	$0.0899280\pi$
$613$	8462.00i	0.557548i	−0.960357	+	0.278774i	$0.910072\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	11094.0i	0.723870i	0.932203	+	0.361935i	$0.117884\pi$
$617$	11094.0i	0.723870i	−0.932203	+	0.361935i	$0.882116\pi$
$618$	0	0
$619$	2180.00	0.141553	0.0707767	−	0.997492i	$-0.477452\pi$
$619$	2180.00	0.141553	0.0707767	+	0.997492i	$0.477452\pi$
$620$	0	0
$621$	−10560.0	−0.682380
$622$	0	0
$623$	3240.00i	0.208359i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	9600.00i	0.611463i
$628$	0	0
$629$	2244.00	0.142248
$630$	0	0
$631$	26848.0	1.69382	0.846911	−	0.531734i	$-0.178459\pi$
$631$	26848.0	1.69382	0.846911	+	0.531734i	$0.178459\pi$
$632$	0	0
$633$	21344.0i	1.34020i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 18966.0i	− 1.17969i
$638$	0	0
$639$	15984.0	0.989542
$640$	0	0
$641$	26322.0	1.62193	0.810965	−	0.585095i	$-0.198943\pi$
$641$	26322.0	1.62193	0.810965	+	0.585095i	$0.198943\pi$
$642$	0	0
$643$	10168.0i	0.623619i	0.950145	+	0.311809i	$0.100935\pi$
$643$	10168.0i	0.623619i	−0.950145	+	0.311809i	$0.899065\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 23604.0i	− 1.43426i	−0.696937	−	0.717132i	$-0.745454\pi$
$647$	− 23604.0i	− 1.43426i	0.696937	−	0.717132i	$-0.254546\pi$
$648$	0	0
$649$	−5040.00	−0.304834
$650$	0	0
$651$	−4864.00	−0.292834
$652$	0	0
$653$	16422.0i	0.984139i	0.870556	+	0.492069i	$0.163759\pi$
$653$	16422.0i	0.984139i	−0.870556	+	0.492069i	$0.836241\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 13394.0i	− 0.795357i
$658$	0	0
$659$	−26100.0	−1.54281	−0.771405	−	0.636345i	$-0.780446\pi$
$659$	−26100.0	−1.54281	−0.771405	+	0.636345i	$0.780446\pi$
$660$	0	0
$661$	−3058.00	−0.179943	−0.0899716	−	0.995944i	$-0.528678\pi$
$661$	−3058.00	−0.179943	−0.0899716	+	0.995944i	$0.528678\pi$
$662$	0	0
$663$	− 30624.0i	− 1.79387i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 11880.0i	− 0.689648i
$668$	0	0
$669$	14176.0	0.819246
$670$	0	0
$671$	−10824.0	−0.622736
$672$	0	0
$673$	10802.0i	0.618702i	0.950948	+	0.309351i	$0.100112\pi$
$673$	10802.0i	0.618702i	−0.950948	+	0.309351i	$0.899888\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	10674.0i	0.605960i	0.952997	+	0.302980i	$0.0979816\pi$
$677$	10674.0i	0.605960i	−0.952997	+	0.302980i	$0.902018\pi$
$678$	0	0
$679$	−4424.00	−0.250041
$680$	0	0
$681$	22272.0	1.25325
$682$	0	0
$683$	28608.0i	1.60272i	0.598185	+	0.801358i	$0.295889\pi$
$683$	28608.0i	1.60272i	−0.598185	+	0.801358i	$0.704111\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 2800.00i	− 0.155497i
$688$	0	0
$689$	12876.0	0.711954
$690$	0	0
$691$	2428.00	0.133669	0.0668346	−	0.997764i	$-0.478710\pi$
$691$	2428.00	0.133669	0.0668346	+	0.997764i	$0.478710\pi$
$692$	0	0
$693$	− 1776.00i	− 0.0973516i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	28908.0i	1.57097i
$698$	0	0
$699$	−15696.0	−0.849324
$700$	0	0
$701$	−6618.00	−0.356574	−0.178287	−	0.983979i	$-0.557056\pi$
$701$	−6618.00	−0.356574	−0.178287	+	0.983979i	$0.557056\pi$
$702$	0	0
$703$	− 3400.00i	− 0.182409i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1032.00i	0.0548972i
$708$	0	0
$709$	−20510.0	−1.08642	−0.543208	−	0.839598i	$-0.682791\pi$
$709$	−20510.0	−1.08642	−0.543208	+	0.839598i	$0.682791\pi$
$710$	0	0
$711$	5920.00	0.312261
$712$	0	0
$713$	20064.0i	1.05386i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 34560.0i	− 1.80009i
$718$	0	0
$719$	31680.0	1.64321	0.821603	−	0.570061i	$-0.193080\pi$
$719$	31680.0	1.64321	0.821603	+	0.570061i	$0.193080\pi$
$720$	0	0
$721$	3952.00	0.204133
$722$	0	0
$723$	− 3824.00i	− 0.196703i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	13196.0i	0.673195i	0.941649	+	0.336597i	$0.109276\pi$
$727$	13196.0i	0.673195i	−0.941649	+	0.336597i	$0.890724\pi$
$728$	0	0
$729$	30563.0	1.55276
$730$	0	0
$731$	−2112.00	−0.106861
$732$	0	0
$733$	8102.00i	0.408259i	0.978944	+	0.204130i	$0.0654364\pi$
$733$	8102.00i	0.408259i	−0.978944	+	0.204130i	$0.934564\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	12288.0i	0.614158i
$738$	0	0
$739$	−12580.0	−0.626201	−0.313101	−	0.949720i	$-0.601368\pi$
$739$	−12580.0	−0.626201	−0.313101	+	0.949720i	$0.601368\pi$
$740$	0	0
$741$	−46400.0	−2.30033
$742$	0	0
$743$	− 29892.0i	− 1.47595i	−0.674828	−	0.737975i	$-0.735782\pi$
$743$	− 29892.0i	− 1.47595i	0.674828	−	0.737975i	$-0.264218\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	2664.00i	0.130483i
$748$	0	0
$749$	−96.0000	−0.00468326
$750$	0	0
$751$	40408.0	1.96339	0.981697	−	0.190450i	$-0.0609946\pi$
$751$	40408.0	1.96339	0.981697	+	0.190450i	$0.0609946\pi$
$752$	0	0
$753$	− 21216.0i	− 1.02676i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 32366.0i	− 1.55398i	−0.629513	−	0.776990i	$-0.716746\pi$
$757$	− 32366.0i	− 1.55398i	0.629513	−	0.776990i	$-0.283254\pi$
$758$	0	0
$759$	−12672.0	−0.606014
$760$	0	0
$761$	−17238.0	−0.821126	−0.410563	−	0.911832i	$-0.634668\pi$
$761$	−17238.0	−0.821126	−0.410563	+	0.911832i	$0.634668\pi$
$762$	0	0
$763$	3800.00i	0.180300i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 24360.0i	− 1.14679i
$768$	0	0
$769$	−10850.0	−0.508792	−0.254396	−	0.967100i	$-0.581877\pi$
$769$	−10850.0	−0.508792	−0.254396	+	0.967100i	$0.581877\pi$
$770$	0	0
$771$	−18672.0	−0.872186
$772$	0	0
$773$	9102.00i	0.423514i	0.977322	+	0.211757i	$0.0679185\pi$
$773$	9102.00i	0.423514i	−0.977322	+	0.211757i	$0.932081\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1088.00i	0.0502340i
$778$	0	0
$779$	43800.0	2.01450
$780$	0	0
$781$	5184.00	0.237514
$782$	0	0
$783$	− 7200.00i	− 0.328617i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 25504.0i	− 1.15517i	−0.816330	−	0.577585i	$-0.803995\pi$
$787$	− 25504.0i	− 1.15517i	0.816330	−	0.577585i	$-0.196005\pi$
$788$	0	0
$789$	−31584.0	−1.42512
$790$	0	0
$791$	−4152.00	−0.186635
$792$	0	0
$793$	− 52316.0i	− 2.34274i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 14166.0i	− 0.629593i	−0.949159	−	0.314796i	$-0.898064\pi$
$797$	− 14166.0i	− 0.629593i	0.949159	−	0.314796i	$-0.101936\pi$
$798$	0	0
$799$	−13464.0	−0.596148
$800$	0	0
$801$	29970.0	1.32202
$802$	0	0
$803$	− 4344.00i	− 0.190905i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 12720.0i	− 0.554852i
$808$	0	0
$809$	−33210.0	−1.44327	−0.721633	−	0.692276i	$-0.756608\pi$
$809$	−33210.0	−1.44327	−0.721633	+	0.692276i	$0.756608\pi$
$810$	0	0
$811$	−39212.0	−1.69780	−0.848902	−	0.528550i	$-0.822736\pi$
$811$	−39212.0	−1.69780	−0.848902	+	0.528550i	$0.822736\pi$
$812$	0	0
$813$	− 39616.0i	− 1.70897i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	3200.00i	0.137030i
$818$	0	0
$819$	8584.00	0.366238
$820$	0	0
$821$	6222.00	0.264494	0.132247	−	0.991217i	$-0.457781\pi$
$821$	6222.00	0.264494	0.132247	+	0.991217i	$0.457781\pi$
$822$	0	0
$823$	− 31172.0i	− 1.32028i	−0.751144	−	0.660138i	$-0.770498\pi$
$823$	− 31172.0i	− 1.32028i	0.751144	−	0.660138i	$-0.229502\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 264.000i	− 0.0111006i	−0.999985	−	0.00555029i	$-0.998233\pi$
$827$	− 264.000i	− 0.0111006i	0.999985	−	0.00555029i	$-0.00176672\pi$
$828$	0	0
$829$	29050.0	1.21707	0.608533	−	0.793528i	$-0.291758\pi$
$829$	29050.0	1.21707	0.608533	+	0.793528i	$0.291758\pi$
$830$	0	0
$831$	13168.0	0.549691
$832$	0	0
$833$	− 21582.0i	− 0.897685i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	12160.0i	0.502164i
$838$	0	0
$839$	−21720.0	−0.893752	−0.446876	−	0.894596i	$-0.647463\pi$
$839$	−21720.0	−0.893752	−0.446876	+	0.894596i	$0.647463\pi$
$840$	0	0
$841$	−16289.0	−0.667883
$842$	0	0
$843$	− 9264.00i	− 0.378492i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	4748.00i	0.192613i
$848$	0	0
$849$	55936.0	2.26115
$850$	0	0
$851$	4488.00	0.180783
$852$	0	0
$853$	− 6658.00i	− 0.267252i	−0.991032	−	0.133626i	$-0.957338\pi$
$853$	− 6658.00i	− 0.267252i	0.991032	−	0.133626i	$-0.0426620\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	13974.0i	0.556993i	0.960437	+	0.278496i	$0.0898360\pi$
$857$	13974.0i	0.556993i	−0.960437	+	0.278496i	$0.910164\pi$
$858$	0	0
$859$	23780.0	0.944544	0.472272	−	0.881453i	$-0.343434\pi$
$859$	23780.0	0.944544	0.472272	+	0.881453i	$0.343434\pi$
$860$	0	0
$861$	−14016.0	−0.554778
$862$	0	0
$863$	12228.0i	0.482324i	0.970485	+	0.241162i	$0.0775286\pi$
$863$	12228.0i	0.482324i	−0.970485	+	0.241162i	$0.922471\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	4456.00i	0.174549i
$868$	0	0
$869$	1920.00	0.0749500
$870$	0	0
$871$	−59392.0	−2.31047
$872$	0	0
$873$	40922.0i	1.58648i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 11606.0i	− 0.446872i	−0.974719	−	0.223436i	$-0.928273\pi$
$877$	− 11606.0i	− 0.446872i	0.974719	−	0.223436i	$-0.0717274\pi$
$878$	0	0
$879$	2064.00	0.0792002
$880$	0	0
$881$	−32958.0	−1.26037	−0.630183	−	0.776446i	$-0.717020\pi$
$881$	−32958.0	−1.26037	−0.630183	+	0.776446i	$0.717020\pi$
$882$	0	0
$883$	− 8072.00i	− 0.307638i	−0.988099	−	0.153819i	$-0.950843\pi$
$883$	− 8072.00i	− 0.307638i	0.988099	−	0.153819i	$-0.0491573\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	15756.0i	0.596431i	0.954498	+	0.298216i	$0.0963915\pi$
$887$	15756.0i	0.596431i	−0.954498	+	0.298216i	$0.903609\pi$
$888$	0	0
$889$	−496.000	−0.0187124
$890$	0	0
$891$	4308.00	0.161979
$892$	0	0
$893$	20400.0i	0.764457i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 61248.0i	− 2.27983i
$898$	0	0
$899$	−13680.0	−0.507512
$900$	0	0
$901$	14652.0	0.541763
$902$	0	0
$903$	− 1024.00i	− 0.0377371i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	18776.0i	0.687372i	0.939085	+	0.343686i	$0.111676\pi$
$907$	18776.0i	0.687372i	−0.939085	+	0.343686i	$0.888324\pi$
$908$	0	0
$909$	9546.00	0.348318
$910$	0	0
$911$	20568.0	0.748022	0.374011	−	0.927424i	$-0.377982\pi$
$911$	20568.0	0.748022	0.374011	+	0.927424i	$0.377982\pi$
$912$	0	0
$913$	864.000i	0.0313190i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	528.000i	0.0190143i
$918$	0	0
$919$	−6280.00	−0.225417	−0.112708	−	0.993628i	$-0.535953\pi$
$919$	−6280.00	−0.225417	−0.112708	+	0.993628i	$0.535953\pi$
$920$	0	0
$921$	71552.0	2.55996
$922$	0	0
$923$	25056.0i	0.893530i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 36556.0i	− 1.29521i
$928$	0	0
$929$	20430.0	0.721514	0.360757	−	0.932660i	$-0.382518\pi$
$929$	20430.0	0.721514	0.360757	+	0.932660i	$0.382518\pi$
$930$	0	0
$931$	−32700.0	−1.15113
$932$	0	0
$933$	− 11136.0i	− 0.390757i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 8906.00i	− 0.310508i	−0.987875	−	0.155254i	$-0.950380\pi$
$937$	− 8906.00i	− 0.310508i	0.987875	−	0.155254i	$-0.0496197\pi$
$938$	0	0
$939$	47024.0	1.63426
$940$	0	0
$941$	−17418.0	−0.603412	−0.301706	−	0.953401i	$-0.597556\pi$
$941$	−17418.0	−0.603412	−0.301706	+	0.953401i	$0.597556\pi$
$942$	0	0
$943$	57816.0i	1.99655i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 2544.00i	− 0.0872956i	−0.999047	−	0.0436478i	$-0.986102\pi$
$947$	− 2544.00i	− 0.0872956i	0.999047	−	0.0436478i	$-0.0138979\pi$
$948$	0	0
$949$	20996.0	0.718187
$950$	0	0
$951$	82608.0	2.81677
$952$	0	0
$953$	15402.0i	0.523525i	0.965132	+	0.261763i	$0.0843038\pi$
$953$	15402.0i	0.523525i	−0.965132	+	0.261763i	$0.915696\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 8640.00i	− 0.291841i
$958$	0	0
$959$	5016.00	0.168900
$960$	0	0
$961$	−6687.00	−0.224464
$962$	0	0
$963$	888.000i	0.0297148i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 49444.0i	− 1.64427i	−0.569291	−	0.822136i	$-0.692782\pi$
$967$	− 49444.0i	− 1.64427i	0.569291	−	0.822136i	$-0.307218\pi$
$968$	0	0
$969$	−52800.0	−1.75044
$970$	0	0
$971$	25188.0	0.832463	0.416231	−	0.909259i	$-0.363351\pi$
$971$	25188.0	0.832463	0.416231	+	0.909259i	$0.363351\pi$
$972$	0	0
$973$	11440.0i	0.376927i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 2946.00i	− 0.0964697i	−0.998836	−	0.0482348i	$-0.984640\pi$
$977$	− 2946.00i	− 0.0964697i	0.998836	−	0.0482348i	$-0.0153596\pi$
$978$	0	0
$979$	9720.00	0.317316
$980$	0	0
$981$	35150.0	1.14399
$982$	0	0
$983$	− 15012.0i	− 0.487089i	−0.969890	−	0.243544i	$-0.921690\pi$
$983$	− 15012.0i	− 0.487089i	0.969890	−	0.243544i	$-0.0783102\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 6528.00i	− 0.210525i
$988$	0	0
$989$	−4224.00	−0.135809
$990$	0	0
$991$	5128.00	0.164376	0.0821878	−	0.996617i	$-0.473809\pi$
$991$	5128.00	0.164376	0.0821878	+	0.996617i	$0.473809\pi$
$992$	0	0
$993$	33824.0i	1.08094i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	49714.0i	1.57920i	0.613625	+	0.789598i	$0.289711\pi$
$997$	49714.0i	1.57920i	−0.613625	+	0.789598i	$0.710289\pi$
$998$	0	0
$999$	2720.00	0.0861431

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.4.c.c.49.2		2
4.3	odd	2		50.4.b.a.49.1		2
5.2	odd	4		80.4.a.f.1.1		1
5.3	odd	4		400.4.a.b.1.1		1
5.4	even	2	inner	400.4.c.c.49.1		2
12.11	even	2		450.4.c.d.199.2		2
15.2	even	4		720.4.a.j.1.1		1
20.3	even	4		50.4.a.c.1.1		1
20.7	even	4		10.4.a.a.1.1	✓	1
20.19	odd	2		50.4.b.a.49.2		2
40.3	even	4		1600.4.a.d.1.1		1
40.13	odd	4		1600.4.a.bx.1.1		1
40.27	even	4		320.4.a.m.1.1		1
40.37	odd	4		320.4.a.b.1.1		1
60.23	odd	4		450.4.a.q.1.1		1
60.47	odd	4		90.4.a.a.1.1		1
60.59	even	2		450.4.c.d.199.1		2
80.27	even	4		1280.4.d.j.641.2		2
80.37	odd	4		1280.4.d.g.641.1		2
80.67	even	4		1280.4.d.j.641.1		2
80.77	odd	4		1280.4.d.g.641.2		2
140.27	odd	4		490.4.a.o.1.1		1
140.47	odd	12		490.4.e.a.361.1		2
140.67	even	12		490.4.e.i.471.1		2
140.83	odd	4		2450.4.a.b.1.1		1
140.87	odd	12		490.4.e.a.471.1		2
140.107	even	12		490.4.e.i.361.1		2
180.7	even	12		810.4.e.c.271.1		2
180.47	odd	12		810.4.e.w.271.1		2
180.67	even	12		810.4.e.c.541.1		2
180.167	odd	12		810.4.e.w.541.1		2
220.87	odd	4		1210.4.a.b.1.1		1
260.207	even	4		1690.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.4.a.a.1.1	✓	1	20.7	even	4
50.4.a.c.1.1		1	20.3	even	4
50.4.b.a.49.1		2	4.3	odd	2
50.4.b.a.49.2		2	20.19	odd	2
80.4.a.f.1.1		1	5.2	odd	4
90.4.a.a.1.1		1	60.47	odd	4
320.4.a.b.1.1		1	40.37	odd	4
320.4.a.m.1.1		1	40.27	even	4
400.4.a.b.1.1		1	5.3	odd	4
400.4.c.c.49.1		2	5.4	even	2	inner
400.4.c.c.49.2		2	1.1	even	1	trivial
450.4.a.q.1.1		1	60.23	odd	4
450.4.c.d.199.1		2	60.59	even	2
450.4.c.d.199.2		2	12.11	even	2
490.4.a.o.1.1		1	140.27	odd	4
490.4.e.a.361.1		2	140.47	odd	12
490.4.e.a.471.1		2	140.87	odd	12
490.4.e.i.361.1		2	140.107	even	12
490.4.e.i.471.1		2	140.67	even	12
720.4.a.j.1.1		1	15.2	even	4
810.4.e.c.271.1		2	180.7	even	12
810.4.e.c.541.1		2	180.67	even	12
810.4.e.w.271.1		2	180.47	odd	12
810.4.e.w.541.1		2	180.167	odd	12
1210.4.a.b.1.1		1	220.87	odd	4
1280.4.d.g.641.1		2	80.37	odd	4
1280.4.d.g.641.2		2	80.77	odd	4
1280.4.d.j.641.1		2	80.67	even	4
1280.4.d.j.641.2		2	80.27	even	4
1600.4.a.d.1.1		1	40.3	even	4
1600.4.a.bx.1.1		1	40.13	odd	4
1690.4.a.a.1.1		1	260.207	even	4
2450.4.a.b.1.1		1	140.83	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	400.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+8.00000i q^{3} -4.00000i q^{7} -37.0000 q^{9} +O(q^{10})\)	\(q+8.00000i q^{3} -4.00000i q^{7} -37.0000 q^{9} -12.0000 q^{11} -58.0000i q^{13} -66.0000i q^{17} -100.000 q^{19} +32.0000 q^{21} -132.000i q^{23} -80.0000i q^{27} +90.0000 q^{29} -152.000 q^{31} -96.0000i q^{33} +34.0000i q^{37} +464.000 q^{39} -438.000 q^{41} -32.0000i q^{43} -204.000i q^{47} +327.000 q^{49} +528.000 q^{51} +222.000i q^{53} -800.000i q^{57} +420.000 q^{59} +902.000 q^{61} +148.000i q^{63} -1024.00i q^{67} +1056.00 q^{69} -432.000 q^{71} +362.000i q^{73} +48.0000i q^{77} -160.000 q^{79} -359.000 q^{81} -72.0000i q^{83} +720.000i q^{87} -810.000 q^{89} -232.000 q^{91} -1216.00i q^{93} -1106.00i q^{97} +444.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 74 q^{9}+O(q^{10}) \) 2 * q - 74 * q^9	\( 2 q - 74 q^{9} - 24 q^{11} - 200 q^{19} + 64 q^{21} + 180 q^{29} - 304 q^{31} + 928 q^{39} - 876 q^{41} + 654 q^{49} + 1056 q^{51} + 840 q^{59} + 1804 q^{61} + 2112 q^{69} - 864 q^{71} - 320 q^{79} - 718 q^{81} - 1620 q^{89} - 464 q^{91} + 888 q^{99}+O(q^{100}) \) 2 * q - 74 * q^9 - 24 * q^11 - 200 * q^19 + 64 * q^21 + 180 * q^29 - 304 * q^31 + 928 * q^39 - 876 * q^41 + 654 * q^49 + 1056 * q^51 + 840 * q^59 + 1804 * q^61 + 2112 * q^69 - 864 * q^71 - 320 * q^79 - 718 * q^81 - 1620 * q^89 - 464 * q^91 + 888 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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