LMFDB - Embedded newform 1100.4.b.j.749.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1100,4,Mod(749,1100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1100, 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("1100.749");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1100 = 2^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1100.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:	$64.9021010063$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 187x^{8} + 12537x^{6} + 358849x^{4} + 3893659x^{2} + 7371225 $ x^10 + 187x^8 + 12537x^6 + 358849x^4 + 3893659x^2 + 7371225
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2}\cdot 5^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			749.10
Root			$8.44158i$ of defining polynomial
Character	$\chi$	$=$	1100.749
Dual form			1100.4.b.j.749.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.44158i q^{3} -27.8042i q^{7} -62.1434 q^{9} +11.0000 q^{11} +62.6157i q^{13} +101.549i q^{17} +113.764 q^{19} +262.516 q^{21} +94.7892i q^{23} -331.809i q^{27} -184.566 q^{29} +242.367 q^{31} +103.857i q^{33} +325.657i q^{37} -591.191 q^{39} -42.3024 q^{41} -241.298i q^{43} -294.119i q^{47} -430.075 q^{49} -958.781 q^{51} -330.830i q^{53} +1074.12i q^{57} -365.656 q^{59} -368.332 q^{61} +1727.85i q^{63} +996.600i q^{67} -894.959 q^{69} -831.332 q^{71} +349.642i q^{73} -305.847i q^{77} -473.373 q^{79} +1454.93 q^{81} -1059.45i q^{83} -1742.59i q^{87} +740.318 q^{89} +1740.98 q^{91} +2288.33i q^{93} -552.874i q^{97} -683.577 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 118 q^{9} + 110 q^{11} - 6 q^{19} + 316 q^{21} - 314 q^{29} + 114 q^{31} - 1036 q^{39} + 148 q^{41} - 1054 q^{49} - 2292 q^{51} - 1384 q^{59} - 1996 q^{61} - 2492 q^{69} - 3290 q^{71} - 2640 q^{79}+ \cdots - 1298 q^{99}+O(q^{100}) $ 10 * q - 118 * q^9 + 110 * q^11 - 6 * q^19 + 316 * q^21 - 314 * q^29 + 114 * q^31 - 1036 * q^39 + 148 * q^41 - 1054 * q^49 - 2292 * q^51 - 1384 * q^59 - 1996 * q^61 - 2492 * q^69 - 3290 * q^71 - 2640 * q^79 - 1550 * q^81 - 2482 * q^89 - 2432 * q^91 - 1298 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1100\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$551$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	9.44158i	1.81703i	0.417850	+	0.908516i	$0.362784\pi$
$3$	9.44158i	1.81703i	−0.417850	+	0.908516i	$0.637216\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 27.8042i	− 1.50129i	−0.660707	−	0.750644i	$-0.729743\pi$
$7$	− 27.8042i	− 1.50129i	0.660707	−	0.750644i	$-0.270257\pi$
$8$	0	0
$9$	−62.1434	−2.30161
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	62.6157i	1.33588i	0.744214	+	0.667941i	$0.232824\pi$
$13$	62.6157i	1.33588i	−0.744214	+	0.667941i	$0.767176\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	101.549i	1.44878i	0.689392	+	0.724389i	$0.257878\pi$
$17$	101.549i	1.44878i	−0.689392	+	0.724389i	$0.742122\pi$
$18$	0	0
$19$	113.764	1.37365	0.686825	−	0.726823i	$-0.259004\pi$
$19$	113.764	1.37365	0.686825	+	0.726823i	$0.259004\pi$
$20$	0	0
$21$	262.516	2.72789
$22$	0	0
$23$	94.7892i	0.859344i	0.902985	+	0.429672i	$0.141371\pi$
$23$	94.7892i	0.859344i	−0.902985	+	0.429672i	$0.858629\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 331.809i	− 2.36506i
$28$	0	0
$29$	−184.566	−1.18183	−0.590914	−	0.806734i	$-0.701233\pi$
$29$	−184.566	−1.18183	−0.590914	+	0.806734i	$0.701233\pi$
$30$	0	0
$31$	242.367	1.40421	0.702103	−	0.712075i	$-0.252244\pi$
$31$	242.367	1.40421	0.702103	+	0.712075i	$0.252244\pi$
$32$	0	0
$33$	103.857i	0.547856i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	325.657i	1.44696i	0.690344	+	0.723482i	$0.257459\pi$
$37$	325.657i	1.44696i	−0.690344	+	0.723482i	$0.742541\pi$
$38$	0	0
$39$	−591.191	−2.42734
$40$	0	0
$41$	−42.3024	−0.161135	−0.0805673	−	0.996749i	$-0.525673\pi$
$41$	−42.3024	−0.161135	−0.0805673	+	0.996749i	$0.525673\pi$
$42$	0	0
$43$	− 241.298i	− 0.855760i	−0.903836	−	0.427880i	$-0.859261\pi$
$43$	− 241.298i	− 0.855760i	0.903836	−	0.427880i	$-0.140739\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 294.119i	− 0.912800i	−0.889775	−	0.456400i	$-0.849139\pi$
$47$	− 294.119i	− 0.912800i	0.889775	−	0.456400i	$-0.150861\pi$
$48$	0	0
$49$	−430.075	−1.25386
$50$	0	0
$51$	−958.781	−2.63248
$52$	0	0
$53$	− 330.830i	− 0.857415i	−0.903443	−	0.428708i	$-0.858969\pi$
$53$	− 330.830i	− 0.857415i	0.903443	−	0.428708i	$-0.141031\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1074.12i	2.49597i
$58$	0	0
$59$	−365.656	−0.806854	−0.403427	−	0.915012i	$-0.632181\pi$
$59$	−365.656	−0.806854	−0.403427	+	0.915012i	$0.632181\pi$
$60$	0	0
$61$	−368.332	−0.773116	−0.386558	−	0.922265i	$-0.626336\pi$
$61$	−368.332	−0.773116	−0.386558	+	0.922265i	$0.626336\pi$
$62$	0	0
$63$	1727.85i	3.45537i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	996.600i	1.81722i	0.417641	+	0.908612i	$0.362857\pi$
$67$	996.600i	1.81722i	−0.417641	+	0.908612i	$0.637143\pi$
$68$	0	0
$69$	−894.959	−1.56146
$70$	0	0
$71$	−831.332	−1.38959	−0.694795	−	0.719207i	$-0.744505\pi$
$71$	−831.332	−1.38959	−0.694795	+	0.719207i	$0.744505\pi$
$72$	0	0
$73$	349.642i	0.560582i	0.959915	+	0.280291i	$0.0904310\pi$
$73$	349.642i	0.560582i	−0.959915	+	0.280291i	$0.909569\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 305.847i	− 0.452655i
$78$	0	0
$79$	−473.373	−0.674160	−0.337080	−	0.941476i	$-0.609439\pi$
$79$	−473.373	−0.674160	−0.337080	+	0.941476i	$0.609439\pi$
$80$	0	0
$81$	1454.93	1.99579
$82$	0	0
$83$	− 1059.45i	− 1.40109i	−0.713610	−	0.700543i	$-0.752941\pi$
$83$	− 1059.45i	− 1.40109i	0.713610	−	0.700543i	$-0.247059\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 1742.59i	− 2.14742i
$88$	0	0
$89$	740.318	0.881725	0.440863	−	0.897575i	$-0.354673\pi$
$89$	740.318	0.881725	0.440863	+	0.897575i	$0.354673\pi$
$90$	0	0
$91$	1740.98	2.00554
$92$	0	0
$93$	2288.33i	2.55149i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 552.874i	− 0.578720i	−0.957220	−	0.289360i	$-0.906557\pi$
$97$	− 552.874i	− 0.578720i	0.957220	−	0.289360i	$-0.0934425\pi$
$98$	0	0
$99$	−683.577	−0.693961
$100$	0	0
$101$	−828.889	−0.816610	−0.408305	−	0.912846i	$-0.633880\pi$
$101$	−828.889	−0.816610	−0.408305	+	0.912846i	$0.633880\pi$
$102$	0	0
$103$	398.022i	0.380760i	0.981710	+	0.190380i	$0.0609720\pi$
$103$	398.022i	0.380760i	−0.981710	+	0.190380i	$0.939028\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 930.751i	− 0.840926i	−0.907310	−	0.420463i	$-0.861868\pi$
$107$	− 930.751i	− 0.840926i	0.907310	−	0.420463i	$-0.138132\pi$
$108$	0	0
$109$	−1684.66	−1.48037	−0.740187	−	0.672401i	$-0.765263\pi$
$109$	−1684.66	−1.48037	−0.740187	+	0.672401i	$0.765263\pi$
$110$	0	0
$111$	−3074.71	−2.62918
$112$	0	0
$113$	1128.26i	0.939269i	0.882861	+	0.469634i	$0.155614\pi$
$113$	1128.26i	0.939269i	−0.882861	+	0.469634i	$0.844386\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 3891.15i	− 3.07467i
$118$	0	0
$119$	2823.49	2.17503
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	− 399.401i	− 0.292787i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	739.246i	0.516515i	0.966076	+	0.258258i	$0.0831483\pi$
$127$	739.246i	0.516515i	−0.966076	+	0.258258i	$0.916852\pi$
$128$	0	0
$129$	2278.24	1.55494
$130$	0	0
$131$	−2516.75	−1.67855	−0.839273	−	0.543710i	$-0.817019\pi$
$131$	−2516.75	−1.67855	−0.839273	+	0.543710i	$0.817019\pi$
$132$	0	0
$133$	− 3163.13i	− 2.06224i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1044.87i	− 0.651599i	−0.945439	−	0.325799i	$-0.894367\pi$
$137$	− 1044.87i	− 0.651599i	0.945439	−	0.325799i	$-0.105633\pi$
$138$	0	0
$139$	−1373.48	−0.838110	−0.419055	−	0.907961i	$-0.637639\pi$
$139$	−1373.48	−0.838110	−0.419055	+	0.907961i	$0.637639\pi$
$140$	0	0
$141$	2776.94	1.65859
$142$	0	0
$143$	688.772i	0.402783i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 4060.59i	− 2.27831i
$148$	0	0
$149$	−1927.76	−1.05992	−0.529960	−	0.848022i	$-0.677793\pi$
$149$	−1927.76	−1.05992	−0.529960	+	0.848022i	$0.677793\pi$
$150$	0	0
$151$	11.4208	0.00615505	0.00307753	−	0.999995i	$-0.499020\pi$
$151$	11.4208	0.00615505	0.00307753	+	0.999995i	$0.499020\pi$
$152$	0	0
$153$	− 6310.59i	− 3.33452i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 1368.98i	− 0.695902i	−0.937513	−	0.347951i	$-0.886878\pi$
$157$	− 1368.98i	− 0.695902i	0.937513	−	0.347951i	$-0.113122\pi$
$158$	0	0
$159$	3123.56	1.55795
$160$	0	0
$161$	2635.54	1.29012
$162$	0	0
$163$	3789.78i	1.82110i	0.413402	+	0.910549i	$0.364340\pi$
$163$	3789.78i	1.82110i	−0.413402	+	0.910549i	$0.635660\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1399.89i	0.648664i	0.945943	+	0.324332i	$0.105140\pi$
$167$	1399.89i	0.648664i	−0.945943	+	0.324332i	$0.894860\pi$
$168$	0	0
$169$	−1723.72	−0.784579
$170$	0	0
$171$	−7069.71	−3.16160
$172$	0	0
$173$	− 1267.69i	− 0.557114i	−0.960420	−	0.278557i	$-0.910144\pi$
$173$	− 1267.69i	− 0.557114i	0.960420	−	0.278557i	$-0.0898562\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 3452.37i	− 1.46608i
$178$	0	0
$179$	4692.84	1.95955	0.979774	−	0.200108i	$-0.0641294\pi$
$179$	4692.84	1.95955	0.979774	+	0.200108i	$0.0641294\pi$
$180$	0	0
$181$	−1268.84	−0.521060	−0.260530	−	0.965466i	$-0.583897\pi$
$181$	−1268.84	−0.521060	−0.260530	+	0.965466i	$0.583897\pi$
$182$	0	0
$183$	− 3477.63i	− 1.40478i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1117.04i	0.436823i
$188$	0	0
$189$	−9225.70	−3.55064
$190$	0	0
$191$	2321.50	0.879466	0.439733	−	0.898129i	$-0.355073\pi$
$191$	2321.50	0.879466	0.439733	+	0.898129i	$0.355073\pi$
$192$	0	0
$193$	1594.22i	0.594584i	0.954787	+	0.297292i	$0.0960835\pi$
$193$	1594.22i	0.594584i	−0.954787	+	0.297292i	$0.903916\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	624.197i	0.225747i	0.993609	+	0.112874i	$0.0360055\pi$
$197$	624.197i	0.225747i	−0.993609	+	0.112874i	$0.963994\pi$
$198$	0	0
$199$	−588.116	−0.209500	−0.104750	−	0.994499i	$-0.533404\pi$
$199$	−588.116	−0.209500	−0.104750	+	0.994499i	$0.533404\pi$
$200$	0	0
$201$	−9409.47	−3.30196
$202$	0	0
$203$	5131.71i	1.77426i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 5890.52i	− 1.97787i
$208$	0	0
$209$	1251.41	0.414171
$210$	0	0
$211$	−2037.99	−0.664935	−0.332467	−	0.943115i	$-0.607881\pi$
$211$	−2037.99	−0.664935	−0.332467	+	0.943115i	$0.607881\pi$
$212$	0	0
$213$	− 7849.09i	− 2.52493i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 6738.83i	− 2.10812i
$218$	0	0
$219$	−3301.17	−1.01860
$220$	0	0
$221$	−6358.55	−1.93539
$222$	0	0
$223$	6589.78i	1.97885i	0.145035	+	0.989427i	$0.453671\pi$
$223$	6589.78i	1.97885i	−0.145035	+	0.989427i	$0.546329\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3910.85i	1.14349i	0.820432	+	0.571744i	$0.193733\pi$
$227$	3910.85i	1.14349i	−0.820432	+	0.571744i	$0.806267\pi$
$228$	0	0
$229$	2419.21	0.698105	0.349053	−	0.937103i	$-0.386503\pi$
$229$	2419.21	0.698105	0.349053	+	0.937103i	$0.386503\pi$
$230$	0	0
$231$	2887.67	0.822489
$232$	0	0
$233$	1783.41i	0.501439i	0.968060	+	0.250720i	$0.0806672\pi$
$233$	1783.41i	0.501439i	−0.968060	+	0.250720i	$0.919333\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 4469.39i	− 1.22497i
$238$	0	0
$239$	5127.37	1.38771	0.693854	−	0.720116i	$-0.255911\pi$
$239$	5127.37	1.38771	0.693854	+	0.720116i	$0.255911\pi$
$240$	0	0
$241$	1919.76	0.513123	0.256561	−	0.966528i	$-0.417410\pi$
$241$	1919.76	0.513123	0.256561	+	0.966528i	$0.417410\pi$
$242$	0	0
$243$	4777.98i	1.26135i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	7123.43i	1.83503i
$248$	0	0
$249$	10002.9	2.54582
$250$	0	0
$251$	−2206.32	−0.554828	−0.277414	−	0.960750i	$-0.589477\pi$
$251$	−2206.32	−0.554828	−0.277414	+	0.960750i	$0.589477\pi$
$252$	0	0
$253$	1042.68i	0.259102i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 1265.61i	− 0.307185i	−0.988134	−	0.153592i	$-0.950916\pi$
$257$	− 1265.61i	− 0.307185i	0.988134	−	0.153592i	$-0.0490842\pi$
$258$	0	0
$259$	9054.64	2.17231
$260$	0	0
$261$	11469.6	2.72011
$262$	0	0
$263$	− 5409.05i	− 1.26820i	−0.773252	−	0.634099i	$-0.781371\pi$
$263$	− 5409.05i	− 1.26820i	0.773252	−	0.634099i	$-0.218629\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	6989.77i	1.60212i
$268$	0	0
$269$	−7112.21	−1.61204	−0.806020	−	0.591888i	$-0.798383\pi$
$269$	−7112.21	−1.61204	−0.806020	+	0.591888i	$0.798383\pi$
$270$	0	0
$271$	375.510	0.0841720	0.0420860	−	0.999114i	$-0.486600\pi$
$271$	375.510	0.0841720	0.0420860	+	0.999114i	$0.486600\pi$
$272$	0	0
$273$	16437.6i	3.64414i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 137.344i	− 0.0297914i	−0.999889	−	0.0148957i	$-0.995258\pi$
$277$	− 137.344i	− 0.0297914i	0.999889	−	0.0148957i	$-0.00474163\pi$
$278$	0	0
$279$	−15061.5	−3.23193
$280$	0	0
$281$	2229.48	0.473309	0.236655	−	0.971594i	$-0.423949\pi$
$281$	2229.48	0.473309	0.236655	+	0.971594i	$0.423949\pi$
$282$	0	0
$283$	3888.38i	0.816750i	0.912814	+	0.408375i	$0.133904\pi$
$283$	3888.38i	0.816750i	−0.912814	+	0.408375i	$0.866096\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1176.19i	0.241909i
$288$	0	0
$289$	−5399.17	−1.09896
$290$	0	0
$291$	5220.00	1.05155
$292$	0	0
$293$	− 2407.39i	− 0.480005i	−0.970772	−	0.240002i	$-0.922852\pi$
$293$	− 2407.39i	− 0.480005i	0.970772	−	0.240002i	$-0.0771482\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 3649.90i	− 0.713093i
$298$	0	0
$299$	−5935.29	−1.14798
$300$	0	0
$301$	−6709.12	−1.28474
$302$	0	0
$303$	− 7826.02i	− 1.48381i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	3946.04i	0.733591i	0.930302	+	0.366795i	$0.119545\pi$
$307$	3946.04i	0.733591i	−0.930302	+	0.366795i	$0.880455\pi$
$308$	0	0
$309$	−3757.96	−0.691853
$310$	0	0
$311$	−356.572	−0.0650140	−0.0325070	−	0.999472i	$-0.510349\pi$
$311$	−356.572	−0.0650140	−0.0325070	+	0.999472i	$0.510349\pi$
$312$	0	0
$313$	5834.56i	1.05364i	0.849978	+	0.526819i	$0.176615\pi$
$313$	5834.56i	1.05364i	−0.849978	+	0.526819i	$0.823385\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	554.834i	0.0983046i	0.998791	+	0.0491523i	$0.0156520\pi$
$317$	554.834i	0.0983046i	−0.998791	+	0.0491523i	$0.984348\pi$
$318$	0	0
$319$	−2030.23	−0.356335
$320$	0	0
$321$	8787.76	1.52799
$322$	0	0
$323$	11552.6i	1.99011i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 15905.8i	− 2.68989i
$328$	0	0
$329$	−8177.74	−1.37038
$330$	0	0
$331$	−2793.68	−0.463911	−0.231955	−	0.972726i	$-0.574512\pi$
$331$	−2793.68	−0.463911	−0.231955	+	0.972726i	$0.574512\pi$
$332$	0	0
$333$	− 20237.4i	− 3.33034i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	7499.19i	1.21219i	0.795393	+	0.606094i	$0.207264\pi$
$337$	7499.19i	1.21219i	−0.795393	+	0.606094i	$0.792736\pi$
$338$	0	0
$339$	−10652.5	−1.70668
$340$	0	0
$341$	2666.04	0.423384
$342$	0	0
$343$	2421.07i	0.381123i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5337.35i	0.825718i	0.910795	+	0.412859i	$0.135470\pi$
$347$	5337.35i	0.825718i	−0.910795	+	0.412859i	$0.864530\pi$
$348$	0	0
$349$	−2031.76	−0.311627	−0.155813	−	0.987786i	$-0.549800\pi$
$349$	−2031.76	−0.311627	−0.155813	+	0.987786i	$0.549800\pi$
$350$	0	0
$351$	20776.4	3.15944
$352$	0	0
$353$	− 6016.64i	− 0.907177i	−0.891211	−	0.453588i	$-0.850144\pi$
$353$	− 6016.64i	− 0.907177i	0.891211	−	0.453588i	$-0.149856\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	26658.2i	3.95210i
$358$	0	0
$359$	−2510.10	−0.369019	−0.184510	−	0.982831i	$-0.559070\pi$
$359$	−2510.10	−0.369019	−0.184510	+	0.982831i	$0.559070\pi$
$360$	0	0
$361$	6083.34	0.886914
$362$	0	0
$363$	1142.43i	0.165185i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	298.580i	0.0424680i	0.999775	+	0.0212340i	$0.00675950\pi$
$367$	298.580i	0.0424680i	−0.999775	+	0.0212340i	$0.993241\pi$
$368$	0	0
$369$	2628.81	0.370869
$370$	0	0
$371$	−9198.48	−1.28723
$372$	0	0
$373$	− 7300.60i	− 1.01343i	−0.862112	−	0.506717i	$-0.830859\pi$
$373$	− 7300.60i	− 1.01343i	0.862112	−	0.506717i	$-0.169141\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 11556.7i	− 1.57878i
$378$	0	0
$379$	−13348.9	−1.80920	−0.904600	−	0.426262i	$-0.859830\pi$
$379$	−13348.9	−1.80920	−0.904600	+	0.426262i	$0.859830\pi$
$380$	0	0
$381$	−6979.64	−0.938525
$382$	0	0
$383$	− 9083.43i	− 1.21186i	−0.795519	−	0.605929i	$-0.792802\pi$
$383$	− 9083.43i	− 1.21186i	0.795519	−	0.605929i	$-0.207198\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	14995.1i	1.96962i
$388$	0	0
$389$	8218.11	1.07114	0.535572	−	0.844490i	$-0.320096\pi$
$389$	8218.11	1.07114	0.535572	+	0.844490i	$0.320096\pi$
$390$	0	0
$391$	−9625.73	−1.24500
$392$	0	0
$393$	− 23762.1i	− 3.04997i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	13600.7i	1.71939i	0.510807	+	0.859695i	$0.329347\pi$
$397$	13600.7i	1.71939i	−0.510807	+	0.859695i	$0.670653\pi$
$398$	0	0
$399$	29865.0	3.74716
$400$	0	0
$401$	−8796.36	−1.09543	−0.547717	−	0.836664i	$-0.684503\pi$
$401$	−8796.36	−1.09543	−0.547717	+	0.836664i	$0.684503\pi$
$402$	0	0
$403$	15176.0i	1.87585i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3582.22i	0.436276i
$408$	0	0
$409$	8218.82	0.993629	0.496815	−	0.867857i	$-0.334503\pi$
$409$	8218.82	0.993629	0.496815	+	0.867857i	$0.334503\pi$
$410$	0	0
$411$	9865.20	1.18398
$412$	0	0
$413$	10166.8i	1.21132i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 12967.8i	− 1.52287i
$418$	0	0
$419$	−1726.86	−0.201342	−0.100671	−	0.994920i	$-0.532099\pi$
$419$	−1726.86	−0.201342	−0.100671	+	0.994920i	$0.532099\pi$
$420$	0	0
$421$	−7523.60	−0.870969	−0.435485	−	0.900196i	$-0.643423\pi$
$421$	−7523.60	−0.870969	−0.435485	+	0.900196i	$0.643423\pi$
$422$	0	0
$423$	18277.5i	2.10091i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	10241.2i	1.16067i
$428$	0	0
$429$	−6503.10	−0.731871
$430$	0	0
$431$	−4872.00	−0.544491	−0.272246	−	0.962228i	$-0.587766\pi$
$431$	−4872.00	−0.544491	−0.272246	+	0.962228i	$0.587766\pi$
$432$	0	0
$433$	12745.1i	1.41453i	0.706947	+	0.707266i	$0.250072\pi$
$433$	12745.1i	1.41453i	−0.706947	+	0.707266i	$0.749928\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	10783.6i	1.18044i
$438$	0	0
$439$	5723.64	0.622266	0.311133	−	0.950366i	$-0.399292\pi$
$439$	5723.64	0.622266	0.311133	+	0.950366i	$0.399292\pi$
$440$	0	0
$441$	26726.3	2.88590
$442$	0	0
$443$	− 3344.46i	− 0.358691i	−0.983786	−	0.179345i	$-0.942602\pi$
$443$	− 3344.46i	− 0.358691i	0.983786	−	0.179345i	$-0.0573979\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 18201.1i	− 1.92591i
$448$	0	0
$449$	5620.38	0.590740	0.295370	−	0.955383i	$-0.404557\pi$
$449$	5620.38	0.590740	0.295370	+	0.955383i	$0.404557\pi$
$450$	0	0
$451$	−465.326	−0.0485839
$452$	0	0
$453$	107.831i	0.0111839i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 9759.56i	− 0.998978i	−0.866320	−	0.499489i	$-0.833521\pi$
$457$	− 9759.56i	− 0.998978i	0.866320	−	0.499489i	$-0.166479\pi$
$458$	0	0
$459$	33694.8	3.42645
$460$	0	0
$461$	1587.33	0.160367	0.0801837	−	0.996780i	$-0.474449\pi$
$461$	1587.33	0.160367	0.0801837	+	0.996780i	$0.474449\pi$
$462$	0	0
$463$	− 377.411i	− 0.0378828i	−0.999821	−	0.0189414i	$-0.993970\pi$
$463$	− 377.411i	− 0.0378828i	0.999821	−	0.0189414i	$-0.00602960\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 16477.8i	− 1.63277i	−0.577509	−	0.816385i	$-0.695975\pi$
$467$	− 16477.8i	− 1.63277i	0.577509	−	0.816385i	$-0.304025\pi$
$468$	0	0
$469$	27709.7	2.72818
$470$	0	0
$471$	12925.3	1.26448
$472$	0	0
$473$	− 2654.28i	− 0.258021i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	20558.9i	1.97343i
$478$	0	0
$479$	12862.3	1.22691	0.613456	−	0.789729i	$-0.289779\pi$
$479$	12862.3	1.22691	0.613456	+	0.789729i	$0.289779\pi$
$480$	0	0
$481$	−20391.2	−1.93297
$482$	0	0
$483$	24883.7i	2.34419i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 12294.0i	− 1.14394i	−0.820276	−	0.571968i	$-0.806180\pi$
$487$	− 12294.0i	− 1.14394i	0.820276	−	0.571968i	$-0.193820\pi$
$488$	0	0
$489$	−35781.5	−3.30899
$490$	0	0
$491$	−1565.70	−0.143909	−0.0719544	−	0.997408i	$-0.522924\pi$
$491$	−1565.70	−0.143909	−0.0719544	+	0.997408i	$0.522924\pi$
$492$	0	0
$493$	− 18742.5i	− 1.71221i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	23114.5i	2.08618i
$498$	0	0
$499$	1615.89	0.144965	0.0724823	−	0.997370i	$-0.476908\pi$
$499$	1615.89	0.144965	0.0724823	+	0.997370i	$0.476908\pi$
$500$	0	0
$501$	−13217.2	−1.17864
$502$	0	0
$503$	− 18796.6i	− 1.66620i	−0.553122	−	0.833100i	$-0.686564\pi$
$503$	− 18796.6i	− 1.66620i	0.553122	−	0.833100i	$-0.313436\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 16274.6i	− 1.42561i
$508$	0	0
$509$	9656.86	0.840929	0.420464	−	0.907309i	$-0.361867\pi$
$509$	9656.86	0.840929	0.420464	+	0.907309i	$0.361867\pi$
$510$	0	0
$511$	9721.53	0.841595
$512$	0	0
$513$	− 37748.1i	− 3.24877i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 3235.30i	− 0.275220i
$518$	0	0
$519$	11969.0	1.01230
$520$	0	0
$521$	15729.2	1.32267	0.661333	−	0.750093i	$-0.269991\pi$
$521$	15729.2	1.32267	0.661333	+	0.750093i	$0.269991\pi$
$522$	0	0
$523$	− 15546.2i	− 1.29979i	−0.760026	−	0.649893i	$-0.774814\pi$
$523$	− 15546.2i	− 1.29979i	0.760026	−	0.649893i	$-0.225186\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	24612.1i	2.03438i
$528$	0	0
$529$	3182.01	0.261528
$530$	0	0
$531$	22723.1	1.85706
$532$	0	0
$533$	− 2648.79i	− 0.215257i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	44307.8i	3.56056i
$538$	0	0
$539$	−4730.83	−0.378054
$540$	0	0
$541$	19520.2	1.55128	0.775638	−	0.631178i	$-0.217428\pi$
$541$	19520.2	1.55128	0.775638	+	0.631178i	$0.217428\pi$
$542$	0	0
$543$	− 11979.8i	− 0.946782i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	9362.67i	0.731844i	0.930645	+	0.365922i	$0.119246\pi$
$547$	9362.67i	0.731844i	−0.930645	+	0.365922i	$0.880754\pi$
$548$	0	0
$549$	22889.4	1.77941
$550$	0	0
$551$	−20997.0	−1.62342
$552$	0	0
$553$	13161.8i	1.01211i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 8841.92i	− 0.672611i	−0.941753	−	0.336305i	$-0.890823\pi$
$557$	− 8841.92i	− 0.672611i	0.941753	−	0.336305i	$-0.109177\pi$
$558$	0	0
$559$	15109.1	1.14319
$560$	0	0
$561$	−10546.6	−0.793721
$562$	0	0
$563$	11712.5i	0.876771i	0.898787	+	0.438385i	$0.144449\pi$
$563$	11712.5i	0.876771i	−0.898787	+	0.438385i	$0.855551\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 40453.2i	− 2.99625i
$568$	0	0
$569$	20873.9	1.53793	0.768964	−	0.639292i	$-0.220773\pi$
$569$	20873.9	1.53793	0.768964	+	0.639292i	$0.220773\pi$
$570$	0	0
$571$	−5649.29	−0.414038	−0.207019	−	0.978337i	$-0.566376\pi$
$571$	−5649.29	−0.414038	−0.207019	+	0.978337i	$0.566376\pi$
$572$	0	0
$573$	21918.6i	1.59802i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	18478.2i	1.33320i	0.745414	+	0.666602i	$0.232252\pi$
$577$	18478.2i	1.33320i	−0.745414	+	0.666602i	$0.767748\pi$
$578$	0	0
$579$	−15052.0	−1.08038
$580$	0	0
$581$	−29457.3	−2.10343
$582$	0	0
$583$	− 3639.13i	− 0.258520i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 11812.2i	− 0.830565i	−0.909693	−	0.415282i	$-0.863683\pi$
$587$	− 11812.2i	− 0.830565i	0.909693	−	0.415282i	$-0.136317\pi$
$588$	0	0
$589$	27572.7	1.92889
$590$	0	0
$591$	−5893.41	−0.410190
$592$	0	0
$593$	13644.4i	0.944870i	0.881365	+	0.472435i	$0.156625\pi$
$593$	13644.4i	0.944870i	−0.881365	+	0.472435i	$0.843375\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 5552.75i	− 0.380668i
$598$	0	0
$599$	−23767.5	−1.62123	−0.810613	−	0.585582i	$-0.800866\pi$
$599$	−23767.5	−1.62123	−0.810613	+	0.585582i	$0.800866\pi$
$600$	0	0
$601$	−11513.5	−0.781442	−0.390721	−	0.920509i	$-0.627774\pi$
$601$	−11513.5	−0.781442	−0.390721	+	0.920509i	$0.627774\pi$
$602$	0	0
$603$	− 61932.1i	− 4.18254i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 10407.6i	− 0.695931i	−0.937507	−	0.347965i	$-0.886873\pi$
$607$	− 10407.6i	− 0.695931i	0.937507	−	0.347965i	$-0.113127\pi$
$608$	0	0
$609$	−48451.5	−3.22390
$610$	0	0
$611$	18416.4	1.21939
$612$	0	0
$613$	− 23276.4i	− 1.53365i	−0.641857	−	0.766824i	$-0.721836\pi$
$613$	− 23276.4i	− 1.53365i	0.641857	−	0.766824i	$-0.278164\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1308.83i	0.0853998i	0.999088	+	0.0426999i	$0.0135959\pi$
$617$	1308.83i	0.0853998i	−0.999088	+	0.0426999i	$0.986404\pi$
$618$	0	0
$619$	20644.5	1.34051	0.670254	−	0.742132i	$-0.266185\pi$
$619$	20644.5	1.34051	0.670254	+	0.742132i	$0.266185\pi$
$620$	0	0
$621$	31451.9	2.03240
$622$	0	0
$623$	− 20584.0i	− 1.32372i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	11815.3i	0.752562i
$628$	0	0
$629$	−33070.1	−2.09633
$630$	0	0
$631$	6099.94	0.384841	0.192421	−	0.981313i	$-0.438366\pi$
$631$	6099.94	0.384841	0.192421	+	0.981313i	$0.438366\pi$
$632$	0	0
$633$	− 19241.9i	− 1.20821i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 26929.5i	− 1.67501i
$638$	0	0
$639$	51661.8	3.19829
$640$	0	0
$641$	−29756.9	−1.83359	−0.916793	−	0.399363i	$-0.869231\pi$
$641$	−29756.9	−1.83359	−0.916793	+	0.399363i	$0.869231\pi$
$642$	0	0
$643$	18954.0i	1.16247i	0.813734	+	0.581237i	$0.197431\pi$
$643$	18954.0i	1.16247i	−0.813734	+	0.581237i	$0.802569\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	22637.0i	1.37550i	0.725946	+	0.687752i	$0.241402\pi$
$647$	22637.0i	1.37550i	−0.725946	+	0.687752i	$0.758598\pi$
$648$	0	0
$649$	−4022.22	−0.243276
$650$	0	0
$651$	63625.2	3.83052
$652$	0	0
$653$	10412.2i	0.623985i	0.950085	+	0.311992i	$0.100996\pi$
$653$	10412.2i	0.623985i	−0.950085	+	0.311992i	$0.899004\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 21727.9i	− 1.29024i
$658$	0	0
$659$	17751.6	1.04932	0.524661	−	0.851311i	$-0.324192\pi$
$659$	17751.6	1.04932	0.524661	+	0.851311i	$0.324192\pi$
$660$	0	0
$661$	8799.38	0.517785	0.258893	−	0.965906i	$-0.416642\pi$
$661$	8799.38	0.517785	0.258893	+	0.965906i	$0.416642\pi$
$662$	0	0
$663$	− 60034.7i	− 3.51668i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 17494.9i	− 1.01560i
$668$	0	0
$669$	−62217.9	−3.59564
$670$	0	0
$671$	−4051.65	−0.233103
$672$	0	0
$673$	30990.3i	1.77502i	0.460791	+	0.887508i	$0.347566\pi$
$673$	30990.3i	1.77502i	−0.460791	+	0.887508i	$0.652434\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 3150.39i	− 0.178847i	−0.995994	−	0.0894235i	$-0.971498\pi$
$677$	− 3150.39i	− 0.178847i	0.995994	−	0.0894235i	$-0.0285025\pi$
$678$	0	0
$679$	−15372.2	−0.868825
$680$	0	0
$681$	−36924.6	−2.07776
$682$	0	0
$683$	25295.7i	1.41715i	0.705637	+	0.708574i	$0.250661\pi$
$683$	25295.7i	1.41715i	−0.705637	+	0.708574i	$0.749339\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	22841.2i	1.26848i
$688$	0	0
$689$	20715.1	1.14540
$690$	0	0
$691$	19354.1	1.06550	0.532752	−	0.846271i	$-0.321158\pi$
$691$	19354.1	1.06550	0.532752	+	0.846271i	$0.321158\pi$
$692$	0	0
$693$	19006.3i	1.04183i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 4295.76i	− 0.233448i
$698$	0	0
$699$	−16838.2	−0.911131
$700$	0	0
$701$	−28622.8	−1.54218	−0.771089	−	0.636727i	$-0.780288\pi$
$701$	−28622.8	−1.54218	−0.771089	+	0.636727i	$0.780288\pi$
$702$	0	0
$703$	37048.2i	1.98762i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	23046.6i	1.22597i
$708$	0	0
$709$	10917.6	0.578304	0.289152	−	0.957283i	$-0.406627\pi$
$709$	10917.6	0.578304	0.289152	+	0.957283i	$0.406627\pi$
$710$	0	0
$711$	29417.0	1.55165
$712$	0	0
$713$	22973.8i	1.20670i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	48410.5i	2.52151i
$718$	0	0
$719$	11840.5	0.614152	0.307076	−	0.951685i	$-0.400649\pi$
$719$	11840.5	0.614152	0.307076	+	0.951685i	$0.400649\pi$
$720$	0	0
$721$	11066.7	0.571630
$722$	0	0
$723$	18125.6i	0.932361i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 6556.62i	− 0.334486i	−0.985916	−	0.167243i	$-0.946514\pi$
$727$	− 6556.62i	− 0.334486i	0.985916	−	0.167243i	$-0.0534865\pi$
$728$	0	0
$729$	−5828.61	−0.296124
$730$	0	0
$731$	24503.6	1.23981
$732$	0	0
$733$	817.889i	0.0412134i	0.999788	+	0.0206067i	$0.00655978\pi$
$733$	817.889i	0.0412134i	−0.999788	+	0.0206067i	$0.993440\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	10962.6i	0.547914i
$738$	0	0
$739$	−25050.6	−1.24696	−0.623480	−	0.781840i	$-0.714282\pi$
$739$	−25050.6	−1.24696	−0.623480	+	0.781840i	$0.714282\pi$
$740$	0	0
$741$	−67256.5	−3.33432
$742$	0	0
$743$	36248.2i	1.78980i	0.446271	+	0.894898i	$0.352752\pi$
$743$	36248.2i	1.78980i	−0.446271	+	0.894898i	$0.647248\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	65838.0i	3.22475i
$748$	0	0
$749$	−25878.8	−1.26247
$750$	0	0
$751$	24797.2	1.20488	0.602439	−	0.798165i	$-0.294196\pi$
$751$	24797.2	1.20488	0.602439	+	0.798165i	$0.294196\pi$
$752$	0	0
$753$	− 20831.2i	− 1.00814i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 33165.4i	− 1.59236i	−0.605061	−	0.796179i	$-0.706851\pi$
$757$	− 33165.4i	− 1.59236i	0.605061	−	0.796179i	$-0.293149\pi$
$758$	0	0
$759$	−9844.55	−0.470797
$760$	0	0
$761$	955.723	0.0455255	0.0227628	−	0.999741i	$-0.492754\pi$
$761$	955.723	0.0455255	0.0227628	+	0.999741i	$0.492754\pi$
$762$	0	0
$763$	46840.6i	2.22247i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 22895.8i	− 1.07786i
$768$	0	0
$769$	10451.0	0.490080	0.245040	−	0.969513i	$-0.421199\pi$
$769$	10451.0	0.490080	0.245040	+	0.969513i	$0.421199\pi$
$770$	0	0
$771$	11949.3	0.558164
$772$	0	0
$773$	33591.7i	1.56301i	0.623896	+	0.781507i	$0.285549\pi$
$773$	33591.7i	1.56301i	−0.623896	+	0.781507i	$0.714451\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	85490.1i	3.94715i
$778$	0	0
$779$	−4812.51	−0.221343
$780$	0	0
$781$	−9144.65	−0.418977
$782$	0	0
$783$	61240.6i	2.79510i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 18251.1i	− 0.826659i	−0.910582	−	0.413329i	$-0.864366\pi$
$787$	− 18251.1i	− 0.826659i	0.910582	−	0.413329i	$-0.135634\pi$
$788$	0	0
$789$	51069.9	2.30436
$790$	0	0
$791$	31370.3	1.41011
$792$	0	0
$793$	− 23063.3i	− 1.03279i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 11233.1i	− 0.499241i	−0.968344	−	0.249621i	$-0.919694\pi$
$797$	− 11233.1i	− 0.499241i	0.968344	−	0.249621i	$-0.0803059\pi$
$798$	0	0
$799$	29867.4	1.32244
$800$	0	0
$801$	−46005.9	−2.02939
$802$	0	0
$803$	3846.06i	0.169022i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 67150.5i	− 2.92913i
$808$	0	0
$809$	−22178.4	−0.963847	−0.481923	−	0.876213i	$-0.660062\pi$
$809$	−22178.4	−0.963847	−0.481923	+	0.876213i	$0.660062\pi$
$810$	0	0
$811$	−25481.4	−1.10329	−0.551647	−	0.834078i	$-0.686001\pi$
$811$	−25481.4	−1.10329	−0.551647	+	0.834078i	$0.686001\pi$
$812$	0	0
$813$	3545.41i	0.152943i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 27451.2i	− 1.17551i
$818$	0	0
$819$	−108190.	−4.61597
$820$	0	0
$821$	33950.1	1.44320	0.721599	−	0.692311i	$-0.243407\pi$
$821$	33950.1	1.44320	0.721599	+	0.692311i	$0.243407\pi$
$822$	0	0
$823$	34061.8i	1.44267i	0.692585	+	0.721336i	$0.256472\pi$
$823$	34061.8i	1.44267i	−0.692585	+	0.721336i	$0.743528\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	41614.8i	1.74980i	0.484300	+	0.874902i	$0.339074\pi$
$827$	41614.8i	1.74980i	−0.484300	+	0.874902i	$0.660926\pi$
$828$	0	0
$829$	−18217.3	−0.763222	−0.381611	−	0.924323i	$-0.624631\pi$
$829$	−18217.3	−0.763222	−0.381611	+	0.924323i	$0.624631\pi$
$830$	0	0
$831$	1296.75	0.0541320
$832$	0	0
$833$	− 43673.7i	− 1.81657i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 80419.6i	− 3.32104i
$838$	0	0
$839$	−39668.3	−1.63230	−0.816150	−	0.577840i	$-0.803896\pi$
$839$	−39668.3	−1.63230	−0.816150	+	0.577840i	$0.803896\pi$
$840$	0	0
$841$	9675.59	0.396719
$842$	0	0
$843$	21049.8i	0.860018i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 3364.31i	− 0.136481i
$848$	0	0
$849$	−36712.5	−1.48406
$850$	0	0
$851$	−30868.7	−1.24344
$852$	0	0
$853$	− 33073.6i	− 1.32757i	−0.747922	−	0.663786i	$-0.768948\pi$
$853$	− 33073.6i	− 1.32757i	0.747922	−	0.663786i	$-0.231052\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	8618.95i	0.343545i	0.985137	+	0.171772i	$0.0549494\pi$
$857$	8618.95i	0.343545i	−0.985137	+	0.171772i	$0.945051\pi$
$858$	0	0
$859$	−1081.04	−0.0429389	−0.0214694	−	0.999770i	$-0.506834\pi$
$859$	−1081.04	−0.0429389	−0.0214694	+	0.999770i	$0.506834\pi$
$860$	0	0
$861$	−11105.0	−0.439557
$862$	0	0
$863$	− 36824.8i	− 1.45253i	−0.687416	−	0.726264i	$-0.741255\pi$
$863$	− 36824.8i	− 1.45253i	0.687416	−	0.726264i	$-0.258745\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 50976.7i	− 1.99684i
$868$	0	0
$869$	−5207.11	−0.203267
$870$	0	0
$871$	−62402.8	−2.42760
$872$	0	0
$873$	34357.5i	1.33199i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	37485.7i	1.44333i	0.692242	+	0.721666i	$0.256623\pi$
$877$	37485.7i	1.44333i	−0.692242	+	0.721666i	$0.743377\pi$
$878$	0	0
$879$	22729.6	0.872184
$880$	0	0
$881$	−31944.9	−1.22162	−0.610811	−	0.791776i	$-0.709157\pi$
$881$	−31944.9	−1.22162	−0.610811	+	0.791776i	$0.709157\pi$
$882$	0	0
$883$	− 15735.2i	− 0.599697i	−0.953987	−	0.299848i	$-0.903064\pi$
$883$	− 15735.2i	− 0.599697i	0.953987	−	0.299848i	$-0.0969361\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 20898.8i	− 0.791109i	−0.918442	−	0.395555i	$-0.870552\pi$
$887$	− 20898.8i	− 0.791109i	0.918442	−	0.395555i	$-0.129448\pi$
$888$	0	0
$889$	20554.2	0.775438
$890$	0	0
$891$	16004.2	0.601753
$892$	0	0
$893$	− 33460.2i	− 1.25387i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 56038.5i	− 2.08592i
$898$	0	0
$899$	−44732.7	−1.65953
$900$	0	0
$901$	33595.4	1.24220
$902$	0	0
$903$	− 63344.7i	− 2.33442i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	40403.5i	1.47914i	0.673082	+	0.739568i	$0.264970\pi$
$907$	40403.5i	1.47914i	−0.673082	+	0.739568i	$0.735030\pi$
$908$	0	0
$909$	51510.0	1.87951
$910$	0	0
$911$	29611.6	1.07692	0.538460	−	0.842651i	$-0.319006\pi$
$911$	29611.6	1.07692	0.538460	+	0.842651i	$0.319006\pi$
$912$	0	0
$913$	− 11654.0i	− 0.422443i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	69976.4i	2.51998i
$918$	0	0
$919$	38281.0	1.37407	0.687036	−	0.726623i	$-0.258911\pi$
$919$	38281.0	1.37407	0.687036	+	0.726623i	$0.258911\pi$
$920$	0	0
$921$	−37256.8	−1.33296
$922$	0	0
$923$	− 52054.4i	− 1.85633i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 24734.4i	− 0.876360i
$928$	0	0
$929$	27299.7	0.964127	0.482063	−	0.876136i	$-0.339887\pi$
$929$	27299.7	0.964127	0.482063	+	0.876136i	$0.339887\pi$
$930$	0	0
$931$	−48927.3	−1.72237
$932$	0	0
$933$	− 3366.61i	− 0.118133i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	3011.09i	0.104982i	0.998621	+	0.0524910i	$0.0167161\pi$
$937$	3011.09i	0.104982i	−0.998621	+	0.0524910i	$0.983284\pi$
$938$	0	0
$939$	−55087.4	−1.91449
$940$	0	0
$941$	−37442.4	−1.29712	−0.648558	−	0.761165i	$-0.724628\pi$
$941$	−37442.4	−1.29712	−0.648558	+	0.761165i	$0.724628\pi$
$942$	0	0
$943$	− 4009.81i	− 0.138470i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 45648.9i	− 1.56641i	−0.621764	−	0.783204i	$-0.713584\pi$
$947$	− 45648.9i	− 1.56641i	0.621764	−	0.783204i	$-0.286416\pi$
$948$	0	0
$949$	−21893.1	−0.748871
$950$	0	0
$951$	−5238.51	−0.178623
$952$	0	0
$953$	− 20898.9i	− 0.710370i	−0.934796	−	0.355185i	$-0.884418\pi$
$953$	− 20898.9i	− 0.710370i	0.934796	−	0.355185i	$-0.115582\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 19168.5i	− 0.647472i
$958$	0	0
$959$	−29051.7	−0.978237
$960$	0	0
$961$	28950.8	0.971796
$962$	0	0
$963$	57840.0i	1.93548i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	20494.2i	0.681539i	0.940147	+	0.340770i	$0.110688\pi$
$967$	20494.2i	0.681539i	−0.940147	+	0.340770i	$0.889312\pi$
$968$	0	0
$969$	−109075.	−3.61610
$970$	0	0
$971$	37338.2	1.23403	0.617013	−	0.786953i	$-0.288343\pi$
$971$	37338.2	1.23403	0.617013	+	0.786953i	$0.288343\pi$
$972$	0	0
$973$	38188.6i	1.25824i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1220.37i	0.0399623i	0.999800	+	0.0199811i	$0.00636062\pi$
$977$	1220.37i	0.0399623i	−0.999800	+	0.0199811i	$0.993639\pi$
$978$	0	0
$979$	8143.50	0.265850
$980$	0	0
$981$	104690.	3.40724
$982$	0	0
$983$	31417.9i	1.01941i	0.860350	+	0.509703i	$0.170245\pi$
$983$	31417.9i	1.01941i	−0.860350	+	0.509703i	$0.829755\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 77210.8i	− 2.49002i
$988$	0	0
$989$	22872.5	0.735392
$990$	0	0
$991$	6320.79	0.202610	0.101305	−	0.994855i	$-0.467698\pi$
$991$	6320.79	0.202610	0.101305	+	0.994855i	$0.467698\pi$
$992$	0	0
$993$	− 26376.7i	− 0.842941i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1067.05i	0.0338954i	0.999856	+	0.0169477i	$0.00539488\pi$
$997$	1067.05i	0.0338954i	−0.999856	+	0.0169477i	$0.994605\pi$
$998$	0	0
$999$	108056.	3.42216

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1100.4.b.j.749.10		10
5.2	odd	4		1100.4.a.m.1.5	yes	5
5.3	odd	4		1100.4.a.j.1.1	✓	5
5.4	even	2	inner	1100.4.b.j.749.1		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1100.4.a.j.1.1	✓	5	5.3	odd	4
1100.4.a.m.1.5	yes	5	5.2	odd	4
1100.4.b.j.749.1		10	5.4	even	2	inner
1100.4.b.j.749.10		10	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1100.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+9.44158i q^{3} -27.8042i q^{7} -62.1434 q^{9} +11.0000 q^{11} +62.6157i q^{13} +101.549i q^{17} +113.764 q^{19} +262.516 q^{21} +94.7892i q^{23} -331.809i q^{27} -184.566 q^{29} +242.367 q^{31} +103.857i q^{33} +325.657i q^{37} -591.191 q^{39} -42.3024 q^{41} -241.298i q^{43} -294.119i q^{47} -430.075 q^{49} -958.781 q^{51} -330.830i q^{53} +1074.12i q^{57} -365.656 q^{59} -368.332 q^{61} +1727.85i q^{63} +996.600i q^{67} -894.959 q^{69} -831.332 q^{71} +349.642i q^{73} -305.847i q^{77} -473.373 q^{79} +1454.93 q^{81} -1059.45i q^{83} -1742.59i q^{87} +740.318 q^{89} +1740.98 q^{91} +2288.33i q^{93} -552.874i q^{97} -683.577 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 118 q^{9} + 110 q^{11} - 6 q^{19} + 316 q^{21} - 314 q^{29} + 114 q^{31} - 1036 q^{39} + 148 q^{41} - 1054 q^{49} - 2292 q^{51} - 1384 q^{59} - 1996 q^{61} - 2492 q^{69} - 3290 q^{71} - 2640 q^{79}+ \cdots - 1298 q^{99}+O(q^{100}) \) 10 * q - 118 * q^9 + 110 * q^11 - 6 * q^19 + 316 * q^21 - 314 * q^29 + 114 * q^31 - 1036 * q^39 + 148 * q^41 - 1054 * q^49 - 2292 * q^51 - 1384 * q^59 - 1996 * q^61 - 2492 * q^69 - 3290 * q^71 - 2640 * q^79 - 1550 * q^81 - 2482 * q^89 - 2432 * q^91 - 1298 * q^99

Introduction

L-functions

Modular forms

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Database

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