LMFDB - Embedded newform 1088.4.a.u.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1088,4,Mod(1,1088)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1088, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1088.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1088 = 2^{6} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1088.a (trivial)

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:	yes
Analytic conductor:	$64.1940780862$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.8396.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 20x + 8 $ x^3 - x^2 - 20*x + 8
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 136)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.395276$ of defining polynomial
Character	$\chi$	$=$	1088.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.11952 q^{3} -11.0296 q^{5} -5.74786 q^{7} +38.9265 q^{9} +O(q^{10})$	$q-8.11952 q^{3} -11.0296 q^{5} -5.74786 q^{7} +38.9265 q^{9} -34.5384 q^{11} +29.8674 q^{13} +89.5549 q^{15} -17.0000 q^{17} -115.136 q^{19} +46.6699 q^{21} +84.9101 q^{23} -3.34827 q^{25} -96.8378 q^{27} +144.293 q^{29} +24.1223 q^{31} +280.435 q^{33} +63.3965 q^{35} +221.197 q^{37} -242.509 q^{39} +345.848 q^{41} +540.768 q^{43} -429.344 q^{45} +354.498 q^{47} -309.962 q^{49} +138.032 q^{51} -66.6193 q^{53} +380.944 q^{55} +934.849 q^{57} -611.482 q^{59} +623.389 q^{61} -223.744 q^{63} -329.425 q^{65} -730.535 q^{67} -689.429 q^{69} -566.733 q^{71} +937.744 q^{73} +27.1863 q^{75} +198.522 q^{77} -707.322 q^{79} -264.741 q^{81} +306.799 q^{83} +187.503 q^{85} -1171.59 q^{87} -707.855 q^{89} -171.674 q^{91} -195.862 q^{93} +1269.90 q^{95} +1304.83 q^{97} -1344.46 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 4 q^{3} + 8 q^{5} + 2 q^{7} - 5 q^{9}+O(q^{10}) $ 3 * q - 4 * q^3 + 8 * q^5 + 2 * q^7 - 5 * q^9	$ 3 q - 4 q^{3} + 8 q^{5} + 2 q^{7} - 5 q^{9} - 84 q^{11} + 50 q^{13} + 148 q^{15} - 51 q^{17} - 224 q^{19} + 16 q^{21} + 234 q^{23} + 161 q^{25} - 292 q^{27} + 72 q^{29} + 2 q^{31} + 148 q^{33} - 428 q^{35} - 100 q^{37} - 156 q^{39} + 218 q^{41} + 44 q^{43} - 768 q^{45} + 16 q^{47} + 403 q^{49} + 68 q^{51} - 462 q^{53} - 460 q^{55} + 688 q^{57} - 68 q^{59} - 460 q^{61} - 586 q^{63} + 408 q^{65} - 1008 q^{67} - 400 q^{69} - 518 q^{71} + 838 q^{73} + 732 q^{75} + 904 q^{77} - 1238 q^{79} + 455 q^{81} - 1148 q^{83} - 136 q^{85} - 1108 q^{87} - 2506 q^{89} - 1416 q^{91} - 648 q^{93} - 40 q^{95} + 2098 q^{97} - 384 q^{99}+O(q^{100}) $ 3 * q - 4 * q^3 + 8 * q^5 + 2 * q^7 - 5 * q^9 - 84 * q^11 + 50 * q^13 + 148 * q^15 - 51 * q^17 - 224 * q^19 + 16 * q^21 + 234 * q^23 + 161 * q^25 - 292 * q^27 + 72 * q^29 + 2 * q^31 + 148 * q^33 - 428 * q^35 - 100 * q^37 - 156 * q^39 + 218 * q^41 + 44 * q^43 - 768 * q^45 + 16 * q^47 + 403 * q^49 + 68 * q^51 - 462 * q^53 - 460 * q^55 + 688 * q^57 - 68 * q^59 - 460 * q^61 - 586 * q^63 + 408 * q^65 - 1008 * q^67 - 400 * q^69 - 518 * q^71 + 838 * q^73 + 732 * q^75 + 904 * q^77 - 1238 * q^79 + 455 * q^81 - 1148 * q^83 - 136 * q^85 - 1108 * q^87 - 2506 * q^89 - 1416 * q^91 - 648 * q^93 - 40 * q^95 + 2098 * q^97 - 384 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−8.11952	−1.56260	−0.781301	−	0.624155i	$-0.785444\pi$
$3$	−8.11952	−1.56260	−0.781301	+	0.624155i	$0.785444\pi$
$4$	0	0
$5$	−11.0296	−0.986516	−0.493258	−	0.869883i	$-0.664194\pi$
$5$	−11.0296	−0.986516	−0.493258	+	0.869883i	$0.664194\pi$
$6$	0	0
$7$	−5.74786	−0.310355	−0.155178	−	0.987887i	$-0.549595\pi$
$7$	−5.74786	−0.310355	−0.155178	+	0.987887i	$0.549595\pi$
$8$	0	0
$9$	38.9265	1.44172
$10$	0	0
$11$	−34.5384	−0.946702	−0.473351	−	0.880874i	$-0.656956\pi$
$11$	−34.5384	−0.946702	−0.473351	+	0.880874i	$0.656956\pi$
$12$	0	0
$13$	29.8674	0.637209	0.318605	−	0.947888i	$-0.396786\pi$
$13$	29.8674	0.637209	0.318605	+	0.947888i	$0.396786\pi$
$14$	0	0
$15$	89.5549	1.54153
$16$	0	0
$17$	−17.0000	−0.242536
$17$	−17.0000	−0.242536
$18$	0	0
$19$	−115.136	−1.39021	−0.695105	−	0.718908i	$-0.744642\pi$
$19$	−115.136	−1.39021	−0.695105	+	0.718908i	$0.744642\pi$
$20$	0	0
$21$	46.6699	0.484962
$22$	0	0
$23$	84.9101	0.769781	0.384891	−	0.922962i	$-0.374239\pi$
$23$	84.9101	0.769781	0.384891	+	0.922962i	$0.374239\pi$
$24$	0	0
$25$	−3.34827	−0.0267861
$26$	0	0
$27$	−96.8378	−0.690239
$28$	0	0
$29$	144.293	0.923950	0.461975	−	0.886893i	$-0.347141\pi$
$29$	144.293	0.923950	0.461975	+	0.886893i	$0.347141\pi$
$30$	0	0
$31$	24.1223	0.139758	0.0698790	−	0.997555i	$-0.477739\pi$
$31$	24.1223	0.139758	0.0698790	+	0.997555i	$0.477739\pi$
$32$	0	0
$33$	280.435	1.47932
$34$	0	0
$35$	63.3965	0.306171
$36$	0	0
$37$	221.197	0.982825	0.491412	−	0.870927i	$-0.336481\pi$
$37$	221.197	0.982825	0.491412	+	0.870927i	$0.336481\pi$
$38$	0	0
$39$	−242.509	−0.995704
$40$	0	0
$41$	345.848	1.31738	0.658688	−	0.752416i	$-0.271112\pi$
$41$	345.848	1.31738	0.658688	+	0.752416i	$0.271112\pi$
$42$	0	0
$43$	540.768	1.91782	0.958911	−	0.283707i	$-0.0915644\pi$
$43$	540.768	1.91782	0.958911	+	0.283707i	$0.0915644\pi$
$44$	0	0
$45$	−429.344	−1.42228
$46$	0	0
$47$	354.498	1.10019	0.550095	−	0.835102i	$-0.314592\pi$
$47$	354.498	1.10019	0.550095	+	0.835102i	$0.314592\pi$
$48$	0	0
$49$	−309.962	−0.903680
$50$	0	0
$51$	138.032	0.378987
$52$	0	0
$53$	−66.6193	−0.172658	−0.0863289	−	0.996267i	$-0.527514\pi$
$53$	−66.6193	−0.172658	−0.0863289	+	0.996267i	$0.527514\pi$
$54$	0	0
$55$	380.944	0.933937
$56$	0	0
$57$	934.849	2.17235
$58$	0	0
$59$	−611.482	−1.34929	−0.674645	−	0.738142i	$-0.735703\pi$
$59$	−611.482	−1.34929	−0.674645	+	0.738142i	$0.735703\pi$
$60$	0	0
$61$	623.389	1.30847	0.654236	−	0.756291i	$-0.272990\pi$
$61$	623.389	1.30847	0.654236	+	0.756291i	$0.272990\pi$
$62$	0	0
$63$	−223.744	−0.447447
$64$	0	0
$65$	−329.425	−0.628617
$66$	0	0
$67$	−730.535	−1.33207	−0.666037	−	0.745918i	$-0.732011\pi$
$67$	−730.535	−1.33207	−0.666037	+	0.745918i	$0.732011\pi$
$68$	0	0
$69$	−689.429	−1.20286
$70$	0	0
$71$	−566.733	−0.947308	−0.473654	−	0.880711i	$-0.657065\pi$
$71$	−566.733	−0.947308	−0.473654	+	0.880711i	$0.657065\pi$
$72$	0	0
$73$	937.744	1.50349	0.751744	−	0.659455i	$-0.229213\pi$
$73$	937.744	1.50349	0.751744	+	0.659455i	$0.229213\pi$
$74$	0	0
$75$	27.1863	0.0418561
$76$	0	0
$77$	198.522	0.293814
$78$	0	0
$79$	−707.322	−1.00734	−0.503670	−	0.863896i	$-0.668017\pi$
$79$	−707.322	−1.00734	−0.503670	+	0.863896i	$0.668017\pi$
$80$	0	0
$81$	−264.741	−0.363156
$82$	0	0
$83$	306.799	0.405729	0.202865	−	0.979207i	$-0.434975\pi$
$83$	306.799	0.405729	0.202865	+	0.979207i	$0.434975\pi$
$84$	0	0
$85$	187.503	0.239265
$86$	0	0
$87$	−1171.59	−1.44377
$88$	0	0
$89$	−707.855	−0.843062	−0.421531	−	0.906814i	$-0.638507\pi$
$89$	−707.855	−0.843062	−0.421531	+	0.906814i	$0.638507\pi$
$90$	0	0
$91$	−171.674	−0.197761
$92$	0	0
$93$	−195.862	−0.218386
$94$	0	0
$95$	1269.90	1.37147
$96$	0	0
$97$	1304.83	1.36583	0.682915	−	0.730497i	$-0.260712\pi$
$97$	1304.83	1.36583	0.682915	+	0.730497i	$0.260712\pi$
$98$	0	0
$99$	−1344.46	−1.36488
$100$	0	0
$101$	1073.48	1.05757	0.528786	−	0.848755i	$-0.322647\pi$
$101$	1073.48	1.05757	0.528786	+	0.848755i	$0.322647\pi$
$102$	0	0
$103$	−1770.68	−1.69389	−0.846944	−	0.531681i	$-0.821560\pi$
$103$	−1770.68	−1.69389	−0.846944	+	0.531681i	$0.821560\pi$
$104$	0	0
$105$	−514.749	−0.478423
$106$	0	0
$107$	−831.371	−0.751137	−0.375568	−	0.926795i	$-0.622553\pi$
$107$	−831.371	−0.751137	−0.375568	+	0.926795i	$0.622553\pi$
$108$	0	0
$109$	−63.8168	−0.0560784	−0.0280392	−	0.999607i	$-0.508926\pi$
$109$	−63.8168	−0.0560784	−0.0280392	+	0.999607i	$0.508926\pi$
$110$	0	0
$111$	−1796.01	−1.53576
$112$	0	0
$113$	−153.642	−0.127906	−0.0639531	−	0.997953i	$-0.520371\pi$
$113$	−153.642	−0.127906	−0.0639531	+	0.997953i	$0.520371\pi$
$114$	0	0
$115$	−936.523	−0.759402
$116$	0	0
$117$	1162.63	0.918680
$118$	0	0
$119$	97.7136	0.0752722
$120$	0	0
$121$	−138.098	−0.103755
$122$	0	0
$123$	−2808.12	−2.05853
$124$	0	0
$125$	1415.63	1.01294
$126$	0	0
$127$	−963.461	−0.673176	−0.336588	−	0.941652i	$-0.609273\pi$
$127$	−963.461	−0.673176	−0.336588	+	0.941652i	$0.609273\pi$
$128$	0	0
$129$	−4390.77	−2.99679
$130$	0	0
$131$	1978.29	1.31942	0.659709	−	0.751521i	$-0.270680\pi$
$131$	1978.29	1.31942	0.659709	+	0.751521i	$0.270680\pi$
$132$	0	0
$133$	661.786	0.431459
$134$	0	0
$135$	1068.08	0.680931
$136$	0	0
$137$	−1512.87	−0.943456	−0.471728	−	0.881744i	$-0.656370\pi$
$137$	−1512.87	−0.943456	−0.471728	+	0.881744i	$0.656370\pi$
$138$	0	0
$139$	626.950	0.382570	0.191285	−	0.981535i	$-0.438735\pi$
$139$	626.950	0.382570	0.191285	+	0.981535i	$0.438735\pi$
$140$	0	0
$141$	−2878.35	−1.71916
$142$	0	0
$143$	−1031.57	−0.603247
$144$	0	0
$145$	−1591.49	−0.911491
$146$	0	0
$147$	2516.74	1.41209
$148$	0	0
$149$	2817.73	1.54925	0.774623	−	0.632423i	$-0.217939\pi$
$149$	2817.73	1.54925	0.774623	+	0.632423i	$0.217939\pi$
$150$	0	0
$151$	−1636.50	−0.881962	−0.440981	−	0.897517i	$-0.645369\pi$
$151$	−1636.50	−0.881962	−0.440981	+	0.897517i	$0.645369\pi$
$152$	0	0
$153$	−661.751	−0.349669
$154$	0	0
$155$	−266.059	−0.137874
$156$	0	0
$157$	−3437.78	−1.74755	−0.873773	−	0.486334i	$-0.838334\pi$
$157$	−3437.78	−1.74755	−0.873773	+	0.486334i	$0.838334\pi$
$158$	0	0
$159$	540.917	0.269796
$160$	0	0
$161$	−488.051	−0.238906
$162$	0	0
$163$	1.79867	0.000864313 0	0.000432157	−	1.00000i	$-0.499862\pi$
$163$	1.79867	0.000864313 0	0.000432157		1.00000i	$0.499862\pi$
$164$	0	0
$165$	−3093.08	−1.45937
$166$	0	0
$167$	−513.531	−0.237953	−0.118977	−	0.992897i	$-0.537961\pi$
$167$	−513.531	−0.237953	−0.118977	+	0.992897i	$0.537961\pi$
$168$	0	0
$169$	−1304.94	−0.593964
$170$	0	0
$171$	−4481.85	−2.00430
$172$	0	0
$173$	−684.285	−0.300724	−0.150362	−	0.988631i	$-0.548044\pi$
$173$	−684.285	−0.300724	−0.150362	+	0.988631i	$0.548044\pi$
$174$	0	0
$175$	19.2454	0.00831322
$176$	0	0
$177$	4964.93	2.10840
$178$	0	0
$179$	2096.90	0.875583	0.437791	−	0.899077i	$-0.355761\pi$
$179$	2096.90	0.875583	0.437791	+	0.899077i	$0.355761\pi$
$180$	0	0
$181$	−3311.41	−1.35986	−0.679932	−	0.733275i	$-0.737991\pi$
$181$	−3311.41	−1.35986	−0.679932	+	0.733275i	$0.737991\pi$
$182$	0	0
$183$	−5061.61	−2.04462
$184$	0	0
$185$	−2439.71	−0.969572
$186$	0	0
$187$	587.153	0.229609
$188$	0	0
$189$	556.610	0.214219
$190$	0	0
$191$	4803.49	1.81973	0.909865	−	0.414904i	$-0.136185\pi$
$191$	4803.49	1.81973	0.909865	+	0.414904i	$0.136185\pi$
$192$	0	0
$193$	37.3851	0.0139432	0.00697160	−	0.999976i	$-0.497781\pi$
$193$	37.3851	0.0139432	0.00697160	+	0.999976i	$0.497781\pi$
$194$	0	0
$195$	2674.77	0.982278
$196$	0	0
$197$	−3436.95	−1.24301	−0.621504	−	0.783411i	$-0.713478\pi$
$197$	−3436.95	−1.24301	−0.621504	+	0.783411i	$0.713478\pi$
$198$	0	0
$199$	2451.54	0.873293	0.436646	−	0.899633i	$-0.356166\pi$
$199$	2451.54	0.873293	0.436646	+	0.899633i	$0.356166\pi$
$200$	0	0
$201$	5931.59	2.08150
$202$	0	0
$203$	−829.376	−0.286753
$204$	0	0
$205$	−3814.56	−1.29961
$206$	0	0
$207$	3305.26	1.10981
$208$	0	0
$209$	3976.61	1.31612
$210$	0	0
$211$	−678.283	−0.221303	−0.110652	−	0.993859i	$-0.535294\pi$
$211$	−678.283	−0.221303	−0.110652	+	0.993859i	$0.535294\pi$
$212$	0	0
$213$	4601.60	1.48027
$214$	0	0
$215$	−5964.45	−1.89196
$216$	0	0
$217$	−138.652	−0.0433747
$218$	0	0
$219$	−7614.03	−2.34935
$220$	0	0
$221$	−507.745	−0.154546
$222$	0	0
$223$	81.3133	0.0244177	0.0122088	−	0.999925i	$-0.496114\pi$
$223$	81.3133	0.0244177	0.0122088	+	0.999925i	$0.496114\pi$
$224$	0	0
$225$	−130.336	−0.0386182
$226$	0	0
$227$	−3622.08	−1.05906	−0.529528	−	0.848292i	$-0.677631\pi$
$227$	−3622.08	−1.05906	−0.529528	+	0.848292i	$0.677631\pi$
$228$	0	0
$229$	345.615	0.0997331	0.0498665	−	0.998756i	$-0.484120\pi$
$229$	345.615	0.0997331	0.0498665	+	0.998756i	$0.484120\pi$
$230$	0	0
$231$	−1611.90	−0.459114
$232$	0	0
$233$	−3869.96	−1.08811	−0.544055	−	0.839049i	$-0.683112\pi$
$233$	−3869.96	−1.08811	−0.544055	+	0.839049i	$0.683112\pi$
$234$	0	0
$235$	−3909.97	−1.08535
$236$	0	0
$237$	5743.11	1.57407
$238$	0	0
$239$	7045.26	1.90678	0.953389	−	0.301743i	$-0.0975685\pi$
$239$	7045.26	1.90678	0.953389	+	0.301743i	$0.0975685\pi$
$240$	0	0
$241$	2959.38	0.790997	0.395499	−	0.918467i	$-0.370572\pi$
$241$	2959.38	0.790997	0.395499	+	0.918467i	$0.370572\pi$
$242$	0	0
$243$	4764.19	1.25771
$244$	0	0
$245$	3418.75	0.891494
$246$	0	0
$247$	−3438.81	−0.885855
$248$	0	0
$249$	−2491.06	−0.633993
$250$	0	0
$251$	2126.30	0.534705	0.267353	−	0.963599i	$-0.413851\pi$
$251$	2126.30	0.534705	0.267353	+	0.963599i	$0.413851\pi$
$252$	0	0
$253$	−2932.66	−0.728754
$254$	0	0
$255$	−1522.43	−0.373876
$256$	0	0
$257$	321.391	0.0780071	0.0390035	−	0.999239i	$-0.487582\pi$
$257$	321.391	0.0780071	0.0390035	+	0.999239i	$0.487582\pi$
$258$	0	0
$259$	−1271.41	−0.305025
$260$	0	0
$261$	5616.83	1.33208
$262$	0	0
$263$	−6796.09	−1.59340	−0.796702	−	0.604373i	$-0.793424\pi$
$263$	−6796.09	−1.59340	−0.796702	+	0.604373i	$0.793424\pi$
$264$	0	0
$265$	734.783	0.170330
$266$	0	0
$267$	5747.44	1.31737
$268$	0	0
$269$	−2893.37	−0.655806	−0.327903	−	0.944711i	$-0.606342\pi$
$269$	−2893.37	−0.655806	−0.327903	+	0.944711i	$0.606342\pi$
$270$	0	0
$271$	884.577	0.198281	0.0991406	−	0.995073i	$-0.468391\pi$
$271$	884.577	0.198281	0.0991406	+	0.995073i	$0.468391\pi$
$272$	0	0
$273$	1393.91	0.309022
$274$	0	0
$275$	115.644	0.0253585
$276$	0	0
$277$	−4185.44	−0.907865	−0.453933	−	0.891036i	$-0.649979\pi$
$277$	−4185.44	−0.907865	−0.453933	+	0.891036i	$0.649979\pi$
$278$	0	0
$279$	938.999	0.201493
$280$	0	0
$281$	−6888.77	−1.46245	−0.731227	−	0.682134i	$-0.761052\pi$
$281$	−6888.77	−1.46245	−0.731227	+	0.682134i	$0.761052\pi$
$282$	0	0
$283$	−7136.66	−1.49905	−0.749524	−	0.661978i	$-0.769717\pi$
$283$	−7136.66	−1.49905	−0.749524	+	0.661978i	$0.769717\pi$
$284$	0	0
$285$	−10311.0	−2.14305
$286$	0	0
$287$	−1987.89	−0.408855
$288$	0	0
$289$	289.000	0.0588235
$290$	0	0
$291$	−10594.6	−2.13425
$292$	0	0
$293$	2507.48	0.499961	0.249981	−	0.968251i	$-0.419576\pi$
$293$	2507.48	0.499961	0.249981	+	0.968251i	$0.419576\pi$
$294$	0	0
$295$	6744.39	1.33110
$296$	0	0
$297$	3344.62	0.653450
$298$	0	0
$299$	2536.04	0.490512
$300$	0	0
$301$	−3108.26	−0.595206
$302$	0	0
$303$	−8716.11	−1.65256
$304$	0	0
$305$	−6875.72	−1.29083
$306$	0	0
$307$	−4328.08	−0.804615	−0.402308	−	0.915505i	$-0.631792\pi$
$307$	−4328.08	−0.804615	−0.402308	+	0.915505i	$0.631792\pi$
$308$	0	0
$309$	14377.1	2.64687
$310$	0	0
$311$	6091.89	1.11074	0.555369	−	0.831604i	$-0.312577\pi$
$311$	6091.89	1.11074	0.555369	+	0.831604i	$0.312577\pi$
$312$	0	0
$313$	885.519	0.159912	0.0799560	−	0.996798i	$-0.474522\pi$
$313$	885.519	0.159912	0.0799560	+	0.996798i	$0.474522\pi$
$314$	0	0
$315$	2467.81	0.441413
$316$	0	0
$317$	8864.22	1.57055	0.785275	−	0.619148i	$-0.212522\pi$
$317$	8864.22	1.57055	0.785275	+	0.619148i	$0.212522\pi$
$318$	0	0
$319$	−4983.65	−0.874705
$320$	0	0
$321$	6750.33	1.17373
$322$	0	0
$323$	1957.31	0.337176
$324$	0	0
$325$	−100.004	−0.0170684
$326$	0	0
$327$	518.162	0.0876282
$328$	0	0
$329$	−2037.61	−0.341450
$330$	0	0
$331$	−3968.47	−0.658994	−0.329497	−	0.944157i	$-0.606879\pi$
$331$	−3968.47	−0.658994	−0.329497	+	0.944157i	$0.606879\pi$
$332$	0	0
$333$	8610.42	1.41696
$334$	0	0
$335$	8057.49	1.31411
$336$	0	0
$337$	5377.09	0.869166	0.434583	−	0.900632i	$-0.356896\pi$
$337$	5377.09	0.869166	0.434583	+	0.900632i	$0.356896\pi$
$338$	0	0
$339$	1247.50	0.199867
$340$	0	0
$341$	−833.147	−0.132309
$342$	0	0
$343$	3753.14	0.590817
$344$	0	0
$345$	7604.11	1.18664
$346$	0	0
$347$	−1870.14	−0.289320	−0.144660	−	0.989481i	$-0.546209\pi$
$347$	−1870.14	−0.289320	−0.144660	+	0.989481i	$0.546209\pi$
$348$	0	0
$349$	11858.8	1.81887	0.909437	−	0.415842i	$-0.136513\pi$
$349$	11858.8	1.81887	0.909437	+	0.415842i	$0.136513\pi$
$350$	0	0
$351$	−2892.29	−0.439826
$352$	0	0
$353$	7742.20	1.16735	0.583677	−	0.811986i	$-0.301614\pi$
$353$	7742.20	1.16735	0.583677	+	0.811986i	$0.301614\pi$
$354$	0	0
$355$	6250.83	0.934535
$356$	0	0
$357$	−793.388	−0.117621
$358$	0	0
$359$	7913.06	1.16333	0.581665	−	0.813429i	$-0.302402\pi$
$359$	7913.06	1.16333	0.581665	+	0.813429i	$0.302402\pi$
$360$	0	0
$361$	6397.30	0.932687
$362$	0	0
$363$	1121.29	0.162128
$364$	0	0
$365$	−10342.9	−1.48322
$366$	0	0
$367$	10181.2	1.44810	0.724049	−	0.689748i	$-0.242279\pi$
$367$	10181.2	1.44810	0.724049	+	0.689748i	$0.242279\pi$
$368$	0	0
$369$	13462.7	1.89929
$370$	0	0
$371$	382.919	0.0535853
$372$	0	0
$373$	4071.36	0.565166	0.282583	−	0.959243i	$-0.408809\pi$
$373$	4071.36	0.565166	0.282583	+	0.959243i	$0.408809\pi$
$374$	0	0
$375$	−11494.2	−1.58282
$376$	0	0
$377$	4309.65	0.588749
$378$	0	0
$379$	8742.67	1.18491	0.592455	−	0.805604i	$-0.298159\pi$
$379$	8742.67	1.18491	0.592455	+	0.805604i	$0.298159\pi$
$380$	0	0
$381$	7822.84	1.05191
$382$	0	0
$383$	−2327.58	−0.310533	−0.155266	−	0.987873i	$-0.549624\pi$
$383$	−2327.58	−0.310533	−0.155266	+	0.987873i	$0.549624\pi$
$384$	0	0
$385$	−2189.62	−0.289852
$386$	0	0
$387$	21050.2	2.76497
$388$	0	0
$389$	−9139.26	−1.19121	−0.595603	−	0.803279i	$-0.703087\pi$
$389$	−9139.26	−1.19121	−0.595603	+	0.803279i	$0.703087\pi$
$390$	0	0
$391$	−1443.47	−0.186699
$392$	0	0
$393$	−16062.7	−2.06172
$394$	0	0
$395$	7801.47	0.993758
$396$	0	0
$397$	−9847.57	−1.24493	−0.622463	−	0.782650i	$-0.713868\pi$
$397$	−9847.57	−1.24493	−0.622463	+	0.782650i	$0.713868\pi$
$398$	0	0
$399$	−5373.38	−0.674199
$400$	0	0
$401$	−7805.47	−0.972036	−0.486018	−	0.873949i	$-0.661551\pi$
$401$	−7805.47	−0.972036	−0.486018	+	0.873949i	$0.661551\pi$
$402$	0	0
$403$	720.471	0.0890551
$404$	0	0
$405$	2919.98	0.358259
$406$	0	0
$407$	−7639.78	−0.930442
$408$	0	0
$409$	−6995.67	−0.845754	−0.422877	−	0.906187i	$-0.638980\pi$
$409$	−6995.67	−0.845754	−0.422877	+	0.906187i	$0.638980\pi$
$410$	0	0
$411$	12283.8	1.47425
$412$	0	0
$413$	3514.71	0.418759
$414$	0	0
$415$	−3383.86	−0.400258
$416$	0	0
$417$	−5090.53	−0.597804
$418$	0	0
$419$	8546.04	0.996423	0.498211	−	0.867056i	$-0.333990\pi$
$419$	8546.04	0.996423	0.498211	+	0.867056i	$0.333990\pi$
$420$	0	0
$421$	−14421.8	−1.66953	−0.834767	−	0.550604i	$-0.814397\pi$
$421$	−14421.8	−1.66953	−0.834767	+	0.550604i	$0.814397\pi$
$422$	0	0
$423$	13799.4	1.58617
$424$	0	0
$425$	56.9205	0.00649659
$426$	0	0
$427$	−3583.15	−0.406091
$428$	0	0
$429$	8375.86	0.942635
$430$	0	0
$431$	6889.34	0.769948	0.384974	−	0.922927i	$-0.374210\pi$
$431$	6889.34	0.769948	0.384974	+	0.922927i	$0.374210\pi$
$432$	0	0
$433$	2395.91	0.265913	0.132956	−	0.991122i	$-0.457553\pi$
$433$	2395.91	0.265913	0.132956	+	0.991122i	$0.457553\pi$
$434$	0	0
$435$	12922.1	1.42430
$436$	0	0
$437$	−9776.21	−1.07016
$438$	0	0
$439$	−5555.40	−0.603975	−0.301987	−	0.953312i	$-0.597650\pi$
$439$	−5555.40	−0.603975	−0.301987	+	0.953312i	$0.597650\pi$
$440$	0	0
$441$	−12065.8	−1.30286
$442$	0	0
$443$	−6702.85	−0.718875	−0.359438	−	0.933169i	$-0.617031\pi$
$443$	−6702.85	−0.718875	−0.359438	+	0.933169i	$0.617031\pi$
$444$	0	0
$445$	7807.35	0.831694
$446$	0	0
$447$	−22878.6	−2.42086
$448$	0	0
$449$	−8596.61	−0.903562	−0.451781	−	0.892129i	$-0.649211\pi$
$449$	−8596.61	−0.903562	−0.451781	+	0.892129i	$0.649211\pi$
$450$	0	0
$451$	−11945.1	−1.24716
$452$	0	0
$453$	13287.6	1.37815
$454$	0	0
$455$	1893.49	0.195095
$456$	0	0
$457$	−5228.76	−0.535210	−0.267605	−	0.963529i	$-0.586232\pi$
$457$	−5228.76	−0.535210	−0.267605	+	0.963529i	$0.586232\pi$
$458$	0	0
$459$	1646.24	0.167407
$460$	0	0
$461$	−13110.9	−1.32459	−0.662296	−	0.749242i	$-0.730418\pi$
$461$	−13110.9	−1.32459	−0.662296	+	0.749242i	$0.730418\pi$
$462$	0	0
$463$	10873.2	1.09141	0.545703	−	0.837979i	$-0.316263\pi$
$463$	10873.2	1.09141	0.545703	+	0.837979i	$0.316263\pi$
$464$	0	0
$465$	2160.27	0.215441
$466$	0	0
$467$	−4938.26	−0.489326	−0.244663	−	0.969608i	$-0.578677\pi$
$467$	−4938.26	−0.489326	−0.244663	+	0.969608i	$0.578677\pi$
$468$	0	0
$469$	4199.01	0.413417
$470$	0	0
$471$	27913.1	2.73072
$472$	0	0
$473$	−18677.3	−1.81561
$474$	0	0
$475$	385.506	0.0372384
$476$	0	0
$477$	−2593.26	−0.248925
$478$	0	0
$479$	3418.51	0.326087	0.163044	−	0.986619i	$-0.447869\pi$
$479$	3418.51	0.326087	0.163044	+	0.986619i	$0.447869\pi$
$480$	0	0
$481$	6606.57	0.626265
$482$	0	0
$483$	3962.74	0.373315
$484$	0	0
$485$	−14391.8	−1.34741
$486$	0	0
$487$	5013.81	0.466524	0.233262	−	0.972414i	$-0.425060\pi$
$487$	5013.81	0.466524	0.233262	+	0.972414i	$0.425060\pi$
$488$	0	0
$489$	−14.6044	−0.00135058
$490$	0	0
$491$	−2050.73	−0.188489	−0.0942446	−	0.995549i	$-0.530044\pi$
$491$	−2050.73	−0.188489	−0.0942446	+	0.995549i	$0.530044\pi$
$492$	0	0
$493$	−2452.98	−0.224091
$494$	0	0
$495$	14828.8	1.34648
$496$	0	0
$497$	3257.50	0.294002
$498$	0	0
$499$	−9394.05	−0.842756	−0.421378	−	0.906885i	$-0.638454\pi$
$499$	−9394.05	−0.842756	−0.421378	+	0.906885i	$0.638454\pi$
$500$	0	0
$501$	4169.62	0.371827
$502$	0	0
$503$	−5304.68	−0.470227	−0.235113	−	0.971968i	$-0.575546\pi$
$503$	−5304.68	−0.470227	−0.235113	+	0.971968i	$0.575546\pi$
$504$	0	0
$505$	−11840.0	−1.04331
$506$	0	0
$507$	10595.5	0.928130
$508$	0	0
$509$	−8530.01	−0.742802	−0.371401	−	0.928473i	$-0.621122\pi$
$509$	−8530.01	−0.742802	−0.371401	+	0.928473i	$0.621122\pi$
$510$	0	0
$511$	−5390.03	−0.466616
$512$	0	0
$513$	11149.5	0.959577
$514$	0	0
$515$	19529.9	1.67105
$516$	0	0
$517$	−12243.8	−1.04155
$518$	0	0
$519$	5556.06	0.469911
$520$	0	0
$521$	18490.4	1.55485	0.777427	−	0.628973i	$-0.216524\pi$
$521$	18490.4	1.55485	0.777427	+	0.628973i	$0.216524\pi$
$522$	0	0
$523$	−18247.7	−1.52566	−0.762828	−	0.646602i	$-0.776190\pi$
$523$	−18247.7	−1.52566	−0.762828	+	0.646602i	$0.776190\pi$
$524$	0	0
$525$	−156.263	−0.0129903
$526$	0	0
$527$	−410.080	−0.0338963
$528$	0	0
$529$	−4957.28	−0.407437
$530$	0	0
$531$	−23802.9	−1.94530
$532$	0	0
$533$	10329.6	0.839445
$534$	0	0
$535$	9169.67	0.741009
$536$	0	0
$537$	−17025.8	−1.36819
$538$	0	0
$539$	10705.6	0.855515
$540$	0	0
$541$	−8186.39	−0.650574	−0.325287	−	0.945615i	$-0.605461\pi$
$541$	−8186.39	−0.650574	−0.325287	+	0.945615i	$0.605461\pi$
$542$	0	0
$543$	26887.1	2.12493
$544$	0	0
$545$	703.873	0.0553222
$546$	0	0
$547$	−16176.9	−1.26449	−0.632244	−	0.774769i	$-0.717866\pi$
$547$	−16176.9	−1.26449	−0.632244	+	0.774769i	$0.717866\pi$
$548$	0	0
$549$	24266.4	1.88645
$550$	0	0
$551$	−16613.3	−1.28449
$552$	0	0
$553$	4065.59	0.312634
$554$	0	0
$555$	19809.2	1.51506
$556$	0	0
$557$	−4013.98	−0.305346	−0.152673	−	0.988277i	$-0.548788\pi$
$557$	−4013.98	−0.305346	−0.152673	+	0.988277i	$0.548788\pi$
$558$	0	0
$559$	16151.3	1.22205
$560$	0	0
$561$	−4767.40	−0.358787
$562$	0	0
$563$	3518.59	0.263394	0.131697	−	0.991290i	$-0.457957\pi$
$563$	3518.59	0.263394	0.131697	+	0.991290i	$0.457957\pi$
$564$	0	0
$565$	1694.61	0.126182
$566$	0	0
$567$	1521.69	0.112707
$568$	0	0
$569$	14769.7	1.08819	0.544093	−	0.839025i	$-0.316874\pi$
$569$	14769.7	1.08819	0.544093	+	0.839025i	$0.316874\pi$
$570$	0	0
$571$	5385.27	0.394688	0.197344	−	0.980334i	$-0.436768\pi$
$571$	5385.27	0.394688	0.197344	+	0.980334i	$0.436768\pi$
$572$	0	0
$573$	−39002.0	−2.84351
$574$	0	0
$575$	−284.302	−0.0206195
$576$	0	0
$577$	14979.0	1.08074	0.540368	−	0.841429i	$-0.318285\pi$
$577$	14979.0	1.08074	0.540368	+	0.841429i	$0.318285\pi$
$578$	0	0
$579$	−303.549	−0.0217877
$580$	0	0
$581$	−1763.44	−0.125920
$582$	0	0
$583$	2300.93	0.163456
$584$	0	0
$585$	−12823.4	−0.906292
$586$	0	0
$587$	21956.0	1.54382	0.771908	−	0.635734i	$-0.219303\pi$
$587$	21956.0	1.54382	0.771908	+	0.635734i	$0.219303\pi$
$588$	0	0
$589$	−2777.35	−0.194293
$590$	0	0
$591$	27906.4	1.94233
$592$	0	0
$593$	15829.7	1.09620	0.548102	−	0.836412i	$-0.315351\pi$
$593$	15829.7	1.09620	0.548102	+	0.836412i	$0.315351\pi$
$594$	0	0
$595$	−1077.74	−0.0742573
$596$	0	0
$597$	−19905.3	−1.36461
$598$	0	0
$599$	−322.721	−0.0220134	−0.0110067	−	0.999939i	$-0.503504\pi$
$599$	−322.721	−0.0220134	−0.0110067	+	0.999939i	$0.503504\pi$
$600$	0	0
$601$	−17054.6	−1.15752	−0.578762	−	0.815496i	$-0.696464\pi$
$601$	−17054.6	−1.15752	−0.578762	+	0.815496i	$0.696464\pi$
$602$	0	0
$603$	−28437.2	−1.92048
$604$	0	0
$605$	1523.16	0.102356
$606$	0	0
$607$	−23467.9	−1.56925	−0.784624	−	0.619972i	$-0.787144\pi$
$607$	−23467.9	−1.56925	−0.784624	+	0.619972i	$0.787144\pi$
$608$	0	0
$609$	6734.13	0.448080
$610$	0	0
$611$	10587.9	0.701051
$612$	0	0
$613$	14770.1	0.973179	0.486590	−	0.873631i	$-0.338241\pi$
$613$	14770.1	0.973179	0.486590	+	0.873631i	$0.338241\pi$
$614$	0	0
$615$	30972.4	2.03078
$616$	0	0
$617$	−4468.29	−0.291550	−0.145775	−	0.989318i	$-0.546568\pi$
$617$	−4468.29	−0.291550	−0.145775	+	0.989318i	$0.546568\pi$
$618$	0	0
$619$	15764.8	1.02365	0.511825	−	0.859090i	$-0.328970\pi$
$619$	15764.8	1.02365	0.511825	+	0.859090i	$0.328970\pi$
$620$	0	0
$621$	−8222.50	−0.531333
$622$	0	0
$623$	4068.65	0.261649
$624$	0	0
$625$	−15195.3	−0.972496
$626$	0	0
$627$	−32288.2	−2.05656
$628$	0	0
$629$	−3760.34	−0.238370
$630$	0	0
$631$	−18841.1	−1.18867	−0.594337	−	0.804216i	$-0.702585\pi$
$631$	−18841.1	−1.18867	−0.594337	+	0.804216i	$0.702585\pi$
$632$	0	0
$633$	5507.33	0.345809
$634$	0	0
$635$	10626.6	0.664099
$636$	0	0
$637$	−9257.75	−0.575833
$638$	0	0
$639$	−22061.0	−1.36576
$640$	0	0
$641$	−19534.5	−1.20369	−0.601846	−	0.798612i	$-0.705568\pi$
$641$	−19534.5	−1.20369	−0.601846	+	0.798612i	$0.705568\pi$
$642$	0	0
$643$	−10019.2	−0.614490	−0.307245	−	0.951630i	$-0.599407\pi$
$643$	−10019.2	−0.614490	−0.307245	+	0.951630i	$0.599407\pi$
$644$	0	0
$645$	48428.4	2.95638
$646$	0	0
$647$	19078.7	1.15929	0.579646	−	0.814868i	$-0.303191\pi$
$647$	19078.7	1.15929	0.579646	+	0.814868i	$0.303191\pi$
$648$	0	0
$649$	21119.6	1.27738
$650$	0	0
$651$	1125.79	0.0677773
$652$	0	0
$653$	−6210.99	−0.372213	−0.186106	−	0.982530i	$-0.559587\pi$
$653$	−6210.99	−0.372213	−0.186106	+	0.982530i	$0.559587\pi$
$654$	0	0
$655$	−21819.7	−1.30163
$656$	0	0
$657$	36503.2	2.16762
$658$	0	0
$659$	4757.72	0.281236	0.140618	−	0.990064i	$-0.455091\pi$
$659$	4757.72	0.281236	0.140618	+	0.990064i	$0.455091\pi$
$660$	0	0
$661$	−25600.6	−1.50643	−0.753213	−	0.657777i	$-0.771497\pi$
$661$	−25600.6	−1.50643	−0.753213	+	0.657777i	$0.771497\pi$
$662$	0	0
$663$	4122.65	0.241494
$664$	0	0
$665$	−7299.22	−0.425642
$666$	0	0
$667$	12251.9	0.711239
$668$	0	0
$669$	−660.224	−0.0381551
$670$	0	0
$671$	−21530.9	−1.23873
$672$	0	0
$673$	−22273.9	−1.27577	−0.637886	−	0.770131i	$-0.720191\pi$
$673$	−22273.9	−1.27577	−0.637886	+	0.770131i	$0.720191\pi$
$674$	0	0
$675$	324.239	0.0184888
$676$	0	0
$677$	25744.4	1.46150	0.730752	−	0.682643i	$-0.239170\pi$
$677$	25744.4	1.46150	0.730752	+	0.682643i	$0.239170\pi$
$678$	0	0
$679$	−7499.99	−0.423893
$680$	0	0
$681$	29409.5	1.65488
$682$	0	0
$683$	−2549.22	−0.142816	−0.0714078	−	0.997447i	$-0.522749\pi$
$683$	−2549.22	−0.142816	−0.0714078	+	0.997447i	$0.522749\pi$
$684$	0	0
$685$	16686.4	0.930735
$686$	0	0
$687$	−2806.23	−0.155843
$688$	0	0
$689$	−1989.74	−0.110019
$690$	0	0
$691$	6030.40	0.331993	0.165996	−	0.986126i	$-0.446916\pi$
$691$	6030.40	0.331993	0.165996	+	0.986126i	$0.446916\pi$
$692$	0	0
$693$	7727.78	0.423599
$694$	0	0
$695$	−6914.99	−0.377411
$696$	0	0
$697$	−5879.42	−0.319511
$698$	0	0
$699$	31422.2	1.70028
$700$	0	0
$701$	36154.3	1.94798	0.973988	−	0.226601i	$-0.0727614\pi$
$701$	36154.3	1.94798	0.973988	+	0.226601i	$0.0727614\pi$
$702$	0	0
$703$	−25467.7	−1.36633
$704$	0	0
$705$	31747.1	1.69598
$706$	0	0
$707$	−6170.19	−0.328223
$708$	0	0
$709$	4624.87	0.244980	0.122490	−	0.992470i	$-0.460912\pi$
$709$	4624.87	0.244980	0.122490	+	0.992470i	$0.460912\pi$
$710$	0	0
$711$	−27533.6	−1.45231
$712$	0	0
$713$	2048.23	0.107583
$714$	0	0
$715$	11377.8	0.595113
$716$	0	0
$717$	−57204.1	−2.97954
$718$	0	0
$719$	−10338.5	−0.536249	−0.268124	−	0.963384i	$-0.586404\pi$
$719$	−10338.5	−0.536249	−0.268124	+	0.963384i	$0.586404\pi$
$720$	0	0
$721$	10177.6	0.525708
$722$	0	0
$723$	−24028.7	−1.23601
$724$	0	0
$725$	−483.132	−0.0247490
$726$	0	0
$727$	31674.8	1.61589	0.807946	−	0.589257i	$-0.200579\pi$
$727$	31674.8	1.61589	0.807946	+	0.589257i	$0.200579\pi$
$728$	0	0
$729$	−31534.9	−1.60214
$730$	0	0
$731$	−9193.05	−0.465140
$732$	0	0
$733$	−38364.9	−1.93321	−0.966604	−	0.256276i	$-0.917504\pi$
$733$	−38364.9	−1.93321	−0.966604	+	0.256276i	$0.917504\pi$
$734$	0	0
$735$	−27758.6	−1.39305
$736$	0	0
$737$	25231.5	1.26108
$738$	0	0
$739$	−16619.9	−0.827299	−0.413649	−	0.910436i	$-0.635746\pi$
$739$	−16619.9	−0.827299	−0.413649	+	0.910436i	$0.635746\pi$
$740$	0	0
$741$	27921.5	1.38424
$742$	0	0
$743$	−11648.3	−0.575149	−0.287574	−	0.957758i	$-0.592849\pi$
$743$	−11648.3	−0.575149	−0.287574	+	0.957758i	$0.592849\pi$
$744$	0	0
$745$	−31078.4	−1.52836
$746$	0	0
$747$	11942.6	0.584949
$748$	0	0
$749$	4778.60	0.233119
$750$	0	0
$751$	23082.4	1.12156	0.560778	−	0.827966i	$-0.310502\pi$
$751$	23082.4	1.12156	0.560778	+	0.827966i	$0.310502\pi$
$752$	0	0
$753$	−17264.6	−0.835532
$754$	0	0
$755$	18049.9	0.870069
$756$	0	0
$757$	18582.0	0.892174	0.446087	−	0.894990i	$-0.352817\pi$
$757$	18582.0	0.892174	0.446087	+	0.894990i	$0.352817\pi$
$758$	0	0
$759$	23811.8	1.13875
$760$	0	0
$761$	−3537.09	−0.168488	−0.0842440	−	0.996445i	$-0.526848\pi$
$761$	−3537.09	−0.168488	−0.0842440	+	0.996445i	$0.526848\pi$
$762$	0	0
$763$	366.810	0.0174042
$764$	0	0
$765$	7298.84	0.344954
$766$	0	0
$767$	−18263.4	−0.859780
$768$	0	0
$769$	26425.8	1.23919	0.619596	−	0.784921i	$-0.287297\pi$
$769$	26425.8	1.23919	0.619596	+	0.784921i	$0.287297\pi$
$770$	0	0
$771$	−2609.54	−0.121894
$772$	0	0
$773$	8224.93	0.382704	0.191352	−	0.981521i	$-0.438713\pi$
$773$	8224.93	0.382704	0.191352	+	0.981521i	$0.438713\pi$
$774$	0	0
$775$	−80.7680	−0.00374358
$776$	0	0
$777$	10323.2	0.476632
$778$	0	0
$779$	−39819.6	−1.83143
$780$	0	0
$781$	19574.1	0.896819
$782$	0	0
$783$	−13973.0	−0.637746
$784$	0	0
$785$	37917.3	1.72398
$786$	0	0
$787$	14980.0	0.678501	0.339251	−	0.940696i	$-0.389827\pi$
$787$	14980.0	0.678501	0.339251	+	0.940696i	$0.389827\pi$
$788$	0	0
$789$	55181.0	2.48985
$790$	0	0
$791$	883.112	0.0396964
$792$	0	0
$793$	18619.0	0.833770
$794$	0	0
$795$	−5966.09	−0.266158
$796$	0	0
$797$	−24969.5	−1.10974	−0.554872	−	0.831936i	$-0.687233\pi$
$797$	−24969.5	−1.10974	−0.554872	+	0.831936i	$0.687233\pi$
$798$	0	0
$799$	−6026.47	−0.266835
$800$	0	0
$801$	−27554.4	−1.21546
$802$	0	0
$803$	−32388.2	−1.42336
$804$	0	0
$805$	5383.00	0.235684
$806$	0	0
$807$	23492.7	1.02476
$808$	0	0
$809$	−42506.6	−1.84728	−0.923641	−	0.383258i	$-0.874802\pi$
$809$	−42506.6	−1.84728	−0.923641	+	0.383258i	$0.874802\pi$
$810$	0	0
$811$	−41332.6	−1.78962	−0.894812	−	0.446443i	$-0.852691\pi$
$811$	−41332.6	−1.78962	−0.894812	+	0.446443i	$0.852691\pi$
$812$	0	0
$813$	−7182.34	−0.309835
$814$	0	0
$815$	−19.8386	−0.000852659 0
$816$	0	0
$817$	−62261.9	−2.66618
$818$	0	0
$819$	−6682.66	−0.285117
$820$	0	0
$821$	16184.5	0.687995	0.343998	−	0.938971i	$-0.388219\pi$
$821$	16184.5	0.687995	0.343998	+	0.938971i	$0.388219\pi$
$822$	0	0
$823$	−20335.7	−0.861310	−0.430655	−	0.902517i	$-0.641717\pi$
$823$	−20335.7	−0.861310	−0.430655	+	0.902517i	$0.641717\pi$
$824$	0	0
$825$	−938.972	−0.0396252
$826$	0	0
$827$	−4710.62	−0.198070	−0.0990352	−	0.995084i	$-0.531576\pi$
$827$	−4710.62	−0.198070	−0.0990352	+	0.995084i	$0.531576\pi$
$828$	0	0
$829$	28681.9	1.20164	0.600822	−	0.799383i	$-0.294840\pi$
$829$	28681.9	1.20164	0.600822	+	0.799383i	$0.294840\pi$
$830$	0	0
$831$	33983.7	1.41863
$832$	0	0
$833$	5269.36	0.219174
$834$	0	0
$835$	5664.03	0.234745
$836$	0	0
$837$	−2335.95	−0.0964664
$838$	0	0
$839$	15678.0	0.645129	0.322565	−	0.946547i	$-0.395455\pi$
$839$	15678.0	0.645129	0.322565	+	0.946547i	$0.395455\pi$
$840$	0	0
$841$	−3568.52	−0.146317
$842$	0	0
$843$	55933.5	2.28523
$844$	0	0
$845$	14392.9	0.585955
$846$	0	0
$847$	793.768	0.0322009
$848$	0	0
$849$	57946.2	2.34241
$850$	0	0
$851$	18781.8	0.756560
$852$	0	0
$853$	20742.8	0.832615	0.416308	−	0.909224i	$-0.363324\pi$
$853$	20742.8	0.832615	0.416308	+	0.909224i	$0.363324\pi$
$854$	0	0
$855$	49432.9	1.97727
$856$	0	0
$857$	−34419.1	−1.37192	−0.685959	−	0.727640i	$-0.740617\pi$
$857$	−34419.1	−1.37192	−0.685959	+	0.727640i	$0.740617\pi$
$858$	0	0
$859$	−11163.7	−0.443424	−0.221712	−	0.975112i	$-0.571164\pi$
$859$	−11163.7	−0.443424	−0.221712	+	0.975112i	$0.571164\pi$
$860$	0	0
$861$	16140.7	0.638877
$862$	0	0
$863$	14692.2	0.579524	0.289762	−	0.957099i	$-0.406424\pi$
$863$	14692.2	0.579524	0.289762	+	0.957099i	$0.406424\pi$
$864$	0	0
$865$	7547.37	0.296669
$866$	0	0
$867$	−2346.54	−0.0919177
$868$	0	0
$869$	24429.8	0.953652
$870$	0	0
$871$	−21819.2	−0.848810
$872$	0	0
$873$	50792.6	1.96915
$874$	0	0
$875$	−8136.83	−0.314372
$876$	0	0
$877$	−42510.4	−1.63680	−0.818400	−	0.574649i	$-0.805139\pi$
$877$	−42510.4	−1.63680	−0.818400	+	0.574649i	$0.805139\pi$
$878$	0	0
$879$	−20359.5	−0.781240
$880$	0	0
$881$	15963.7	0.610476	0.305238	−	0.952276i	$-0.401264\pi$
$881$	15963.7	0.610476	0.305238	+	0.952276i	$0.401264\pi$
$882$	0	0
$883$	219.137	0.00835171	0.00417586	−	0.999991i	$-0.498671\pi$
$883$	219.137	0.00835171	0.00417586	+	0.999991i	$0.498671\pi$
$884$	0	0
$885$	−54761.2	−2.07997
$886$	0	0
$887$	−15562.6	−0.589109	−0.294554	−	0.955635i	$-0.595171\pi$
$887$	−15562.6	−0.589109	−0.294554	+	0.955635i	$0.595171\pi$
$888$	0	0
$889$	5537.84	0.208924
$890$	0	0
$891$	9143.72	0.343801
$892$	0	0
$893$	−40815.5	−1.52950
$894$	0	0
$895$	−23127.9	−0.863777
$896$	0	0
$897$	−20591.4	−0.766475
$898$	0	0
$899$	3480.69	0.129129
$900$	0	0
$901$	1132.53	0.0418757
$902$	0	0
$903$	25237.6	0.930070
$904$	0	0
$905$	36523.5	1.34153
$906$	0	0
$907$	−21.2455	−0.000777780 0	−0.000388890	−	1.00000i	$-0.500124\pi$
$907$	−21.2455	−0.000777780 0	−0.000388890		1.00000i	$0.500124\pi$
$908$	0	0
$909$	41786.7	1.52473
$910$	0	0
$911$	19951.2	0.725589	0.362795	−	0.931869i	$-0.381823\pi$
$911$	19951.2	0.725589	0.362795	+	0.931869i	$0.381823\pi$
$912$	0	0
$913$	−10596.3	−0.384105
$914$	0	0
$915$	55827.5	2.01705
$916$	0	0
$917$	−11370.9	−0.409488
$918$	0	0
$919$	−23938.1	−0.859245	−0.429623	−	0.903009i	$-0.641353\pi$
$919$	−23938.1	−0.859245	−0.429623	+	0.903009i	$0.641353\pi$
$920$	0	0
$921$	35142.0	1.25729
$922$	0	0
$923$	−16926.8	−0.603633
$924$	0	0
$925$	−740.626	−0.0263261
$926$	0	0
$927$	−68926.6	−2.44212
$928$	0	0
$929$	−50164.7	−1.77164	−0.885818	−	0.464032i	$-0.846402\pi$
$929$	−50164.7	−1.77164	−0.885818	+	0.464032i	$0.846402\pi$
$930$	0	0
$931$	35687.8	1.25631
$932$	0	0
$933$	−49463.2	−1.73564
$934$	0	0
$935$	−6476.05	−0.226513
$936$	0	0
$937$	−52228.3	−1.82094	−0.910472	−	0.413570i	$-0.864282\pi$
$937$	−52228.3	−1.82094	−0.910472	+	0.413570i	$0.864282\pi$
$938$	0	0
$939$	−7189.98	−0.249879
$940$	0	0
$941$	−13304.5	−0.460909	−0.230454	−	0.973083i	$-0.574021\pi$
$941$	−13304.5	−0.460909	−0.230454	+	0.973083i	$0.574021\pi$
$942$	0	0
$943$	29366.0	1.01409
$944$	0	0
$945$	−6139.18	−0.211331
$946$	0	0
$947$	−45228.0	−1.55197	−0.775984	−	0.630753i	$-0.782746\pi$
$947$	−45228.0	−1.55197	−0.775984	+	0.630753i	$0.782746\pi$
$948$	0	0
$949$	28008.0	0.958037
$950$	0	0
$951$	−71973.2	−2.45414
$952$	0	0
$953$	−13303.3	−0.452189	−0.226095	−	0.974105i	$-0.572596\pi$
$953$	−13303.3	−0.452189	−0.226095	+	0.974105i	$0.572596\pi$
$954$	0	0
$955$	−52980.5	−1.79519
$956$	0	0
$957$	40464.8	1.36682
$958$	0	0
$959$	8695.79	0.292807
$960$	0	0
$961$	−29209.1	−0.980468
$962$	0	0
$963$	−32362.4	−1.08293
$964$	0	0
$965$	−412.342	−0.0137552
$966$	0	0
$967$	−37951.7	−1.26209	−0.631047	−	0.775744i	$-0.717375\pi$
$967$	−37951.7	−1.26209	−0.631047	+	0.775744i	$0.717375\pi$
$968$	0	0
$969$	−15892.4	−0.526871
$970$	0	0
$971$	18488.3	0.611038	0.305519	−	0.952186i	$-0.401170\pi$
$971$	18488.3	0.611038	0.305519	+	0.952186i	$0.401170\pi$
$972$	0	0
$973$	−3603.62	−0.118733
$974$	0	0
$975$	811.984	0.0266711
$976$	0	0
$977$	873.224	0.0285946	0.0142973	−	0.999898i	$-0.495449\pi$
$977$	873.224	0.0285946	0.0142973	+	0.999898i	$0.495449\pi$
$978$	0	0
$979$	24448.2	0.798128
$980$	0	0
$981$	−2484.17	−0.0808495
$982$	0	0
$983$	29554.2	0.958934	0.479467	−	0.877560i	$-0.340830\pi$
$983$	29554.2	0.958934	0.479467	+	0.877560i	$0.340830\pi$
$984$	0	0
$985$	37908.1	1.22625
$986$	0	0
$987$	16544.4	0.533550
$988$	0	0
$989$	45916.6	1.47630
$990$	0	0
$991$	−44156.1	−1.41540	−0.707702	−	0.706512i	$-0.750268\pi$
$991$	−44156.1	−1.41540	−0.707702	+	0.706512i	$0.750268\pi$
$992$	0	0
$993$	32222.1	1.02974
$994$	0	0
$995$	−27039.5	−0.861517
$996$	0	0
$997$	25915.3	0.823216	0.411608	−	0.911361i	$-0.364967\pi$
$997$	25915.3	0.823216	0.411608	+	0.911361i	$0.364967\pi$
$998$	0	0
$999$	−21420.2	−0.678384

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1088.4.a.u.1.1		3
4.3	odd	2		1088.4.a.y.1.3		3
8.3	odd	2		136.4.a.b.1.1	✓	3
8.5	even	2		272.4.a.j.1.3		3
24.5	odd	2		2448.4.a.bj.1.1		3
24.11	even	2		1224.4.a.i.1.1		3
136.67	odd	2		2312.4.a.d.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.4.a.b.1.1	✓	3	8.3	odd	2
272.4.a.j.1.3		3	8.5	even	2
1088.4.a.u.1.1		3	1.1	even	1	trivial
1088.4.a.y.1.3		3	4.3	odd	2
1224.4.a.i.1.1		3	24.11	even	2
2312.4.a.d.1.3		3	136.67	odd	2
2448.4.a.bj.1.1		3	24.5	odd	2

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:	\( N \)	\(=\)	\( 1088 = 2^{6} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1088.a (trivial)

\(f(q)\)	\(=\)	\(q-8.11952 q^{3} -11.0296 q^{5} -5.74786 q^{7} +38.9265 q^{9} +O(q^{10})\)	\(q-8.11952 q^{3} -11.0296 q^{5} -5.74786 q^{7} +38.9265 q^{9} -34.5384 q^{11} +29.8674 q^{13} +89.5549 q^{15} -17.0000 q^{17} -115.136 q^{19} +46.6699 q^{21} +84.9101 q^{23} -3.34827 q^{25} -96.8378 q^{27} +144.293 q^{29} +24.1223 q^{31} +280.435 q^{33} +63.3965 q^{35} +221.197 q^{37} -242.509 q^{39} +345.848 q^{41} +540.768 q^{43} -429.344 q^{45} +354.498 q^{47} -309.962 q^{49} +138.032 q^{51} -66.6193 q^{53} +380.944 q^{55} +934.849 q^{57} -611.482 q^{59} +623.389 q^{61} -223.744 q^{63} -329.425 q^{65} -730.535 q^{67} -689.429 q^{69} -566.733 q^{71} +937.744 q^{73} +27.1863 q^{75} +198.522 q^{77} -707.322 q^{79} -264.741 q^{81} +306.799 q^{83} +187.503 q^{85} -1171.59 q^{87} -707.855 q^{89} -171.674 q^{91} -195.862 q^{93} +1269.90 q^{95} +1304.83 q^{97} -1344.46 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 4 q^{3} + 8 q^{5} + 2 q^{7} - 5 q^{9}+O(q^{10}) \) 3 * q - 4 * q^3 + 8 * q^5 + 2 * q^7 - 5 * q^9	\( 3 q - 4 q^{3} + 8 q^{5} + 2 q^{7} - 5 q^{9} - 84 q^{11} + 50 q^{13} + 148 q^{15} - 51 q^{17} - 224 q^{19} + 16 q^{21} + 234 q^{23} + 161 q^{25} - 292 q^{27} + 72 q^{29} + 2 q^{31} + 148 q^{33} - 428 q^{35} - 100 q^{37} - 156 q^{39} + 218 q^{41} + 44 q^{43} - 768 q^{45} + 16 q^{47} + 403 q^{49} + 68 q^{51} - 462 q^{53} - 460 q^{55} + 688 q^{57} - 68 q^{59} - 460 q^{61} - 586 q^{63} + 408 q^{65} - 1008 q^{67} - 400 q^{69} - 518 q^{71} + 838 q^{73} + 732 q^{75} + 904 q^{77} - 1238 q^{79} + 455 q^{81} - 1148 q^{83} - 136 q^{85} - 1108 q^{87} - 2506 q^{89} - 1416 q^{91} - 648 q^{93} - 40 q^{95} + 2098 q^{97} - 384 q^{99}+O(q^{100}) \) 3 * q - 4 * q^3 + 8 * q^5 + 2 * q^7 - 5 * q^9 - 84 * q^11 + 50 * q^13 + 148 * q^15 - 51 * q^17 - 224 * q^19 + 16 * q^21 + 234 * q^23 + 161 * q^25 - 292 * q^27 + 72 * q^29 + 2 * q^31 + 148 * q^33 - 428 * q^35 - 100 * q^37 - 156 * q^39 + 218 * q^41 + 44 * q^43 - 768 * q^45 + 16 * q^47 + 403 * q^49 + 68 * q^51 - 462 * q^53 - 460 * q^55 + 688 * q^57 - 68 * q^59 - 460 * q^61 - 586 * q^63 + 408 * q^65 - 1008 * q^67 - 400 * q^69 - 518 * q^71 + 838 * q^73 + 732 * q^75 + 904 * q^77 - 1238 * q^79 + 455 * q^81 - 1148 * q^83 - 136 * q^85 - 1108 * q^87 - 2506 * q^89 - 1416 * q^91 - 648 * q^93 - 40 * q^95 + 2098 * q^97 - 384 * q^99

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