LMFDB - Embedded newform 693.4.a.n.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [693,4,Mod(1,693)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(693, 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("693.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 693 = 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	693.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:	$40.8883236340$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 28x^{3} - 11x^{2} + 108x - 64 $ x^5 - 28x^3 - 11x^2 + 108*x - 64
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.767088$ of defining polynomial
Character	$\chi$	$=$	693.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.232912 q^{2} -7.94575 q^{4} -7.75746 q^{5} -7.00000 q^{7} +3.71396 q^{8} +1.80680 q^{10} -11.0000 q^{11} +59.4737 q^{13} +1.63038 q^{14} +62.7010 q^{16} +45.4400 q^{17} +111.752 q^{19} +61.6388 q^{20} +2.56203 q^{22} -105.326 q^{23} -64.8219 q^{25} -13.8521 q^{26} +55.6203 q^{28} -10.0244 q^{29} +315.364 q^{31} -44.3154 q^{32} -10.5835 q^{34} +54.3022 q^{35} -182.176 q^{37} -26.0283 q^{38} -28.8108 q^{40} -487.944 q^{41} -358.348 q^{43} +87.4033 q^{44} +24.5317 q^{46} -205.857 q^{47} +49.0000 q^{49} +15.0978 q^{50} -472.563 q^{52} -134.518 q^{53} +85.3320 q^{55} -25.9977 q^{56} +2.33481 q^{58} +891.997 q^{59} +654.613 q^{61} -73.4520 q^{62} -491.286 q^{64} -461.364 q^{65} -102.298 q^{67} -361.055 q^{68} -12.6476 q^{70} -119.126 q^{71} -346.254 q^{73} +42.4310 q^{74} -887.951 q^{76} +77.0000 q^{77} -774.834 q^{79} -486.400 q^{80} +113.648 q^{82} -1040.89 q^{83} -352.499 q^{85} +83.4636 q^{86} -40.8535 q^{88} -502.925 q^{89} -416.316 q^{91} +836.896 q^{92} +47.9465 q^{94} -866.908 q^{95} -939.285 q^{97} -11.4127 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{2} + 21 q^{4} - 7 q^{5} - 35 q^{7} - 60 q^{8} + 55 q^{10} - 55 q^{11} + 111 q^{13} + 35 q^{14} + 201 q^{16} - 136 q^{17} + 111 q^{19} - 219 q^{20} + 55 q^{22} + 28 q^{23} + 190 q^{25} + q^{26}+ \cdots - 245 q^{98}+O(q^{100}) $ 5 * q - 5 * q^2 + 21 * q^4 - 7 * q^5 - 35 * q^7 - 60 * q^8 + 55 * q^10 - 55 * q^11 + 111 * q^13 + 35 * q^14 + 201 * q^16 - 136 * q^17 + 111 * q^19 - 219 * q^20 + 55 * q^22 + 28 * q^23 + 190 * q^25 + q^26 - 147 * q^28 - 61 * q^29 - 280 * q^31 - 535 * q^32 - 572 * q^34 + 49 * q^35 - 41 * q^37 - 267 * q^38 - 336 * q^40 - 426 * q^41 + 424 * q^43 - 231 * q^44 + 140 * q^46 - 75 * q^47 + 245 * q^49 - 490 * q^50 - 269 * q^52 - 1500 * q^53 + 77 * q^55 + 420 * q^56 - 1767 * q^58 - 757 * q^59 + 658 * q^61 + 568 * q^62 - 748 * q^64 - 537 * q^65 - 583 * q^67 + 1650 * q^68 - 385 * q^70 + 764 * q^71 + 875 * q^73 + 825 * q^74 + 213 * q^76 + 385 * q^77 - 244 * q^79 + 2577 * q^80 - 2006 * q^82 - 924 * q^83 - 1402 * q^85 - 1272 * q^86 + 660 * q^88 + 1110 * q^89 - 777 * q^91 + 2046 * q^92 - 3349 * q^94 - 1923 * q^95 - 852 * q^97 - 245 * q^98

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 $

$2$	−0.232912	−0.0823468	−0.0411734	−	0.999152i	$-0.513110\pi$
$2$	−0.232912	−0.0823468	−0.0411734	+	0.999152i	$0.513110\pi$
$3$	0	0
$3$	0	0
$4$	−7.94575	−0.993219
$5$	−7.75746	−0.693848	−0.346924	−	0.937893i	$-0.612774\pi$
$5$	−7.75746	−0.693848	−0.346924	+	0.937893i	$0.612774\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	3.71396	0.164135
$9$	0	0
$10$	1.80680	0.0571362
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	59.4737	1.26885	0.634424	−	0.772985i	$-0.281237\pi$
$13$	59.4737	1.26885	0.634424	+	0.772985i	$0.281237\pi$
$14$	1.63038	0.0311242
$15$	0	0
$16$	62.7010	0.979703
$17$	45.4400	0.648284	0.324142	−	0.946008i	$-0.394924\pi$
$17$	45.4400	0.648284	0.324142	+	0.946008i	$0.394924\pi$
$18$	0	0
$19$	111.752	1.34935	0.674673	−	0.738117i	$-0.264285\pi$
$19$	111.752	1.34935	0.674673	+	0.738117i	$0.264285\pi$
$20$	61.6388	0.689143
$21$	0	0
$22$	2.56203	0.0248285
$23$	−105.326	−0.954871	−0.477435	−	0.878667i	$-0.658433\pi$
$23$	−105.326	−0.954871	−0.477435	+	0.878667i	$0.658433\pi$
$24$	0	0
$25$	−64.8219	−0.518575
$26$	−13.8521	−0.104486
$27$	0	0
$28$	55.6203	0.375401
$29$	−10.0244	−0.0641892	−0.0320946	−	0.999485i	$-0.510218\pi$
$29$	−10.0244	−0.0641892	−0.0320946	+	0.999485i	$0.510218\pi$
$30$	0	0
$31$	315.364	1.82713	0.913565	−	0.406694i	$-0.133318\pi$
$31$	315.364	1.82713	0.913565	+	0.406694i	$0.133318\pi$
$32$	−44.3154	−0.244811
$33$	0	0
$34$	−10.5835	−0.0533841
$35$	54.3022	0.262250
$36$	0	0
$37$	−182.176	−0.809448	−0.404724	−	0.914439i	$-0.632632\pi$
$37$	−182.176	−0.809448	−0.404724	+	0.914439i	$0.632632\pi$
$38$	−26.0283	−0.111114
$39$	0	0
$40$	−28.8108	−0.113885
$41$	−487.944	−1.85864	−0.929318	−	0.369280i	$-0.879604\pi$
$41$	−487.944	−1.85864	−0.929318	+	0.369280i	$0.879604\pi$
$42$	0	0
$43$	−358.348	−1.27087	−0.635437	−	0.772152i	$-0.719180\pi$
$43$	−358.348	−1.27087	−0.635437	+	0.772152i	$0.719180\pi$
$44$	87.4033	0.299467
$45$	0	0
$46$	24.5317	0.0786305
$47$	−205.857	−0.638879	−0.319439	−	0.947607i	$-0.603495\pi$
$47$	−205.857	−0.638879	−0.319439	+	0.947607i	$0.603495\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	15.0978	0.0427030
$51$	0	0
$52$	−472.563	−1.26024
$53$	−134.518	−0.348632	−0.174316	−	0.984690i	$-0.555771\pi$
$53$	−134.518	−0.348632	−0.174316	+	0.984690i	$0.555771\pi$
$54$	0	0
$55$	85.3320	0.209203
$56$	−25.9977	−0.0620373
$57$	0	0
$58$	2.33481	0.00528578
$59$	891.997	1.96827	0.984137	−	0.177410i	$-0.0567719\pi$
$59$	891.997	1.96827	0.984137	+	0.177410i	$0.0567719\pi$
$60$	0	0
$61$	654.613	1.37401	0.687005	−	0.726653i	$-0.258925\pi$
$61$	654.613	1.37401	0.687005	+	0.726653i	$0.258925\pi$
$62$	−73.4520	−0.150458
$63$	0	0
$64$	−491.286	−0.959544
$65$	−461.364	−0.880388
$66$	0	0
$67$	−102.298	−0.186532	−0.0932659	−	0.995641i	$-0.529731\pi$
$67$	−102.298	−0.186532	−0.0932659	+	0.995641i	$0.529731\pi$
$68$	−361.055	−0.643888
$69$	0	0
$70$	−12.6476	−0.0215954
$71$	−119.126	−0.199122	−0.0995608	−	0.995031i	$-0.531744\pi$
$71$	−119.126	−0.199122	−0.0995608	+	0.995031i	$0.531744\pi$
$72$	0	0
$73$	−346.254	−0.555151	−0.277575	−	0.960704i	$-0.589531\pi$
$73$	−346.254	−0.555151	−0.277575	+	0.960704i	$0.589531\pi$
$74$	42.4310	0.0666554
$75$	0	0
$76$	−887.951	−1.34020
$77$	77.0000	0.113961
$78$	0	0
$79$	−774.834	−1.10349	−0.551744	−	0.834013i	$-0.686038\pi$
$79$	−774.834	−1.10349	−0.551744	+	0.834013i	$0.686038\pi$
$80$	−486.400	−0.679765
$81$	0	0
$82$	113.648	0.153053
$83$	−1040.89	−1.37653	−0.688265	−	0.725459i	$-0.741627\pi$
$83$	−1040.89	−1.37653	−0.688265	+	0.725459i	$0.741627\pi$
$84$	0	0
$85$	−352.499	−0.449811
$86$	83.4636	0.104652
$87$	0	0
$88$	−40.8535	−0.0494886
$89$	−502.925	−0.598988	−0.299494	−	0.954098i	$-0.596818\pi$
$89$	−502.925	−0.598988	−0.299494	+	0.954098i	$0.596818\pi$
$90$	0	0
$91$	−416.316	−0.479579
$92$	836.896	0.948396
$93$	0	0
$94$	47.9465	0.0526096
$95$	−866.908	−0.936241
$96$	0	0
$97$	−939.285	−0.983195	−0.491598	−	0.870822i	$-0.663587\pi$
$97$	−939.285	−0.983195	−0.491598	+	0.870822i	$0.663587\pi$
$98$	−11.4127	−0.0117638
$99$	0	0
$100$	515.058	0.515058
$101$	664.250	0.654409	0.327205	−	0.944953i	$-0.393893\pi$
$101$	664.250	0.654409	0.327205	+	0.944953i	$0.393893\pi$
$102$	0	0
$103$	−1500.66	−1.43558	−0.717789	−	0.696261i	$-0.754846\pi$
$103$	−1500.66	−1.43558	−0.717789	+	0.696261i	$0.754846\pi$
$104$	220.882	0.208263
$105$	0	0
$106$	31.3309	0.0287087
$107$	1090.60	0.985353	0.492676	−	0.870213i	$-0.336019\pi$
$107$	1090.60	0.985353	0.492676	+	0.870213i	$0.336019\pi$
$108$	0	0
$109$	525.673	0.461930	0.230965	−	0.972962i	$-0.425812\pi$
$109$	525.673	0.461930	0.230965	+	0.972962i	$0.425812\pi$
$110$	−19.8748	−0.0172272
$111$	0	0
$112$	−438.907	−0.370293
$113$	1978.09	1.64675	0.823377	−	0.567494i	$-0.192087\pi$
$113$	1978.09	1.64675	0.823377	+	0.567494i	$0.192087\pi$
$114$	0	0
$115$	817.063	0.662535
$116$	79.6515	0.0637539
$117$	0	0
$118$	−207.757	−0.162081
$119$	−318.080	−0.245028
$120$	0	0
$121$	121.000	0.0909091
$122$	−152.467	−0.113145
$123$	0	0
$124$	−2505.80	−1.81474
$125$	1472.53	1.05366
$126$	0	0
$127$	−2464.93	−1.72226	−0.861131	−	0.508383i	$-0.830243\pi$
$127$	−2464.93	−1.72226	−0.861131	+	0.508383i	$0.830243\pi$
$128$	468.950	0.323826
$129$	0	0
$130$	107.457	0.0724971
$131$	552.047	0.368188	0.184094	−	0.982909i	$-0.441065\pi$
$131$	552.047	0.368188	0.184094	+	0.982909i	$0.441065\pi$
$132$	0	0
$133$	−782.261	−0.510005
$134$	23.8263	0.0153603
$135$	0	0
$136$	168.762	0.106406
$137$	−794.265	−0.495318	−0.247659	−	0.968847i	$-0.579661\pi$
$137$	−794.265	−0.495318	−0.247659	+	0.968847i	$0.579661\pi$
$138$	0	0
$139$	430.970	0.262982	0.131491	−	0.991317i	$-0.458024\pi$
$139$	430.970	0.262982	0.131491	+	0.991317i	$0.458024\pi$
$140$	−431.472	−0.260472
$141$	0	0
$142$	27.7458	0.0163970
$143$	−654.210	−0.382572
$144$	0	0
$145$	77.7640	0.0445376
$146$	80.6468	0.0457149
$147$	0	0
$148$	1447.53	0.803959
$149$	−2645.94	−1.45479	−0.727397	−	0.686217i	$-0.759270\pi$
$149$	−2645.94	−1.45479	−0.727397	+	0.686217i	$0.759270\pi$
$150$	0	0
$151$	−538.071	−0.289984	−0.144992	−	0.989433i	$-0.546316\pi$
$151$	−538.071	−0.289984	−0.144992	+	0.989433i	$0.546316\pi$
$152$	415.041	0.221475
$153$	0	0
$154$	−17.9342	−0.00938429
$155$	−2446.42	−1.26775
$156$	0	0
$157$	2052.43	1.04332	0.521661	−	0.853153i	$-0.325313\pi$
$157$	2052.43	1.04332	0.521661	+	0.853153i	$0.325313\pi$
$158$	180.468	0.0908688
$159$	0	0
$160$	343.775	0.169861
$161$	737.283	0.360907
$162$	0	0
$163$	2191.90	1.05327	0.526634	−	0.850092i	$-0.323454\pi$
$163$	2191.90	1.05327	0.526634	+	0.850092i	$0.323454\pi$
$164$	3877.08	1.84603
$165$	0	0
$166$	242.435	0.113353
$167$	−40.0010	−0.0185351	−0.00926757	−	0.999957i	$-0.502950\pi$
$167$	−40.0010	−0.0185351	−0.00926757	+	0.999957i	$0.502950\pi$
$168$	0	0
$169$	1340.12	0.609975
$170$	82.1012	0.0370405
$171$	0	0
$172$	2847.35	1.26226
$173$	−4251.43	−1.86838	−0.934192	−	0.356770i	$-0.883878\pi$
$173$	−4251.43	−1.86838	−0.934192	+	0.356770i	$0.883878\pi$
$174$	0	0
$175$	453.753	0.196003
$176$	−689.711	−0.295392
$177$	0	0
$178$	117.137	0.0493247
$179$	−4768.34	−1.99108	−0.995538	−	0.0943649i	$-0.969918\pi$
$179$	−4768.34	−1.99108	−0.995538	+	0.0943649i	$0.969918\pi$
$180$	0	0
$181$	−2955.03	−1.21351	−0.606756	−	0.794888i	$-0.707530\pi$
$181$	−2955.03	−1.21351	−0.606756	+	0.794888i	$0.707530\pi$
$182$	96.9648	0.0394918
$183$	0	0
$184$	−391.177	−0.156728
$185$	1413.22	0.561634
$186$	0	0
$187$	−499.840	−0.195465
$188$	1635.69	0.634547
$189$	0	0
$190$	201.913	0.0770965
$191$	−715.502	−0.271057	−0.135529	−	0.990773i	$-0.543273\pi$
$191$	−715.502	−0.271057	−0.135529	+	0.990773i	$0.543273\pi$
$192$	0	0
$193$	1799.38	0.671098	0.335549	−	0.942023i	$-0.391078\pi$
$193$	1799.38	0.671098	0.335549	+	0.942023i	$0.391078\pi$
$194$	218.771	0.0809630
$195$	0	0
$196$	−389.342	−0.141888
$197$	−2978.55	−1.07722	−0.538612	−	0.842554i	$-0.681051\pi$
$197$	−2978.55	−1.07722	−0.538612	+	0.842554i	$0.681051\pi$
$198$	0	0
$199$	−2303.16	−0.820434	−0.410217	−	0.911988i	$-0.634547\pi$
$199$	−2303.16	−0.820434	−0.410217	+	0.911988i	$0.634547\pi$
$200$	−240.745	−0.0851164
$201$	0	0
$202$	−154.712	−0.0538885
$203$	70.1709	0.0242612
$204$	0	0
$205$	3785.21	1.28961
$206$	349.522	0.118215
$207$	0	0
$208$	3729.06	1.24309
$209$	−1229.27	−0.406843
$210$	0	0
$211$	−4258.59	−1.38945	−0.694724	−	0.719277i	$-0.744473\pi$
$211$	−4258.59	−1.38945	−0.694724	+	0.719277i	$0.744473\pi$
$212$	1068.85	0.346268
$213$	0	0
$214$	−254.015	−0.0811406
$215$	2779.87	0.881794
$216$	0	0
$217$	−2207.55	−0.690590
$218$	−122.435	−0.0380384
$219$	0	0
$220$	−678.027	−0.207784
$221$	2702.49	0.822574
$222$	0	0
$223$	−2739.54	−0.822660	−0.411330	−	0.911486i	$-0.634936\pi$
$223$	−2739.54	−0.822660	−0.411330	+	0.911486i	$0.634936\pi$
$224$	310.208	0.0925297
$225$	0	0
$226$	−460.721	−0.135605
$227$	−3373.57	−0.986395	−0.493198	−	0.869917i	$-0.664172\pi$
$227$	−3373.57	−0.986395	−0.493198	+	0.869917i	$0.664172\pi$
$228$	0	0
$229$	349.100	0.100739	0.0503693	−	0.998731i	$-0.483960\pi$
$229$	349.100	0.100739	0.0503693	+	0.998731i	$0.483960\pi$
$230$	−190.304	−0.0545576
$231$	0	0
$232$	−37.2302	−0.0105357
$233$	−2210.59	−0.621548	−0.310774	−	0.950484i	$-0.600588\pi$
$233$	−2210.59	−0.621548	−0.310774	+	0.950484i	$0.600588\pi$
$234$	0	0
$235$	1596.93	0.443285
$236$	−7087.59	−1.95493
$237$	0	0
$238$	74.0847	0.0201773
$239$	4073.04	1.10236	0.551178	−	0.834388i	$-0.314179\pi$
$239$	4073.04	1.10236	0.551178	+	0.834388i	$0.314179\pi$
$240$	0	0
$241$	1967.60	0.525910	0.262955	−	0.964808i	$-0.415303\pi$
$241$	1967.60	0.525910	0.262955	+	0.964808i	$0.415303\pi$
$242$	−28.1823	−0.00748607
$243$	0	0
$244$	−5201.39	−1.36469
$245$	−380.115	−0.0991211
$246$	0	0
$247$	6646.28	1.71212
$248$	1171.25	0.299896
$249$	0	0
$250$	−342.971	−0.0867655
$251$	7019.95	1.76532	0.882660	−	0.470011i	$-0.155750\pi$
$251$	7019.95	1.76532	0.882660	+	0.470011i	$0.155750\pi$
$252$	0	0
$253$	1158.59	0.287904
$254$	574.112	0.141823
$255$	0	0
$256$	3821.07	0.932878
$257$	2810.13	0.682066	0.341033	−	0.940051i	$-0.389223\pi$
$257$	2810.13	0.682066	0.341033	+	0.940051i	$0.389223\pi$
$258$	0	0
$259$	1275.23	0.305943
$260$	3665.89	0.874418
$261$	0	0
$262$	−128.578	−0.0303191
$263$	−5388.34	−1.26334	−0.631672	−	0.775236i	$-0.717631\pi$
$263$	−5388.34	−1.26334	−0.631672	+	0.775236i	$0.717631\pi$
$264$	0	0
$265$	1043.52	0.241898
$266$	182.198	0.0419973
$267$	0	0
$268$	812.831	0.185267
$269$	839.014	0.190169	0.0950847	−	0.995469i	$-0.469688\pi$
$269$	839.014	0.190169	0.0950847	+	0.995469i	$0.469688\pi$
$270$	0	0
$271$	3831.43	0.858830	0.429415	−	0.903107i	$-0.358720\pi$
$271$	3831.43	0.858830	0.429415	+	0.903107i	$0.358720\pi$
$272$	2849.14	0.635126
$273$	0	0
$274$	184.994	0.0407879
$275$	713.040	0.156356
$276$	0	0
$277$	−800.595	−0.173657	−0.0868287	−	0.996223i	$-0.527673\pi$
$277$	−800.595	−0.173657	−0.0868287	+	0.996223i	$0.527673\pi$
$278$	−100.378	−0.0216557
$279$	0	0
$280$	201.676	0.0430444
$281$	−3988.10	−0.846655	−0.423327	−	0.905977i	$-0.639138\pi$
$281$	−3988.10	−0.846655	−0.423327	+	0.905977i	$0.639138\pi$
$282$	0	0
$283$	218.036	0.0457982	0.0228991	−	0.999738i	$-0.492710\pi$
$283$	218.036	0.0457982	0.0228991	+	0.999738i	$0.492710\pi$
$284$	946.544	0.197771
$285$	0	0
$286$	152.373	0.0315036
$287$	3415.61	0.702498
$288$	0	0
$289$	−2848.20	−0.579728
$290$	−18.1122	−0.00366752
$291$	0	0
$292$	2751.25	0.551386
$293$	−2886.47	−0.575527	−0.287763	−	0.957702i	$-0.592912\pi$
$293$	−2886.47	−0.575527	−0.287763	+	0.957702i	$0.592912\pi$
$294$	0	0
$295$	−6919.63	−1.36568
$296$	−676.594	−0.132859
$297$	0	0
$298$	616.272	0.119798
$299$	−6264.13	−1.21159
$300$	0	0
$301$	2508.44	0.480345
$302$	125.323	0.0238792
$303$	0	0
$304$	7006.94	1.32196
$305$	−5078.13	−0.953354
$306$	0	0
$307$	1453.05	0.270130	0.135065	−	0.990837i	$-0.456876\pi$
$307$	1453.05	0.270130	0.135065	+	0.990837i	$0.456876\pi$
$308$	−611.823	−0.113188
$309$	0	0
$310$	569.800	0.104395
$311$	10772.3	1.96412	0.982062	−	0.188556i	$-0.0603806\pi$
$311$	10772.3	1.96412	0.982062	+	0.188556i	$0.0603806\pi$
$312$	0	0
$313$	−2382.11	−0.430175	−0.215087	−	0.976595i	$-0.569004\pi$
$313$	−2382.11	−0.430175	−0.215087	+	0.976595i	$0.569004\pi$
$314$	−478.035	−0.0859142
$315$	0	0
$316$	6156.64	1.09601
$317$	7870.28	1.39444	0.697222	−	0.716855i	$-0.254419\pi$
$317$	7870.28	1.39444	0.697222	+	0.716855i	$0.254419\pi$
$318$	0	0
$319$	110.269	0.0193538
$320$	3811.13	0.665777
$321$	0	0
$322$	−171.722	−0.0297195
$323$	5078.00	0.874760
$324$	0	0
$325$	−3855.19	−0.657993
$326$	−510.519	−0.0867333
$327$	0	0
$328$	−1812.20	−0.305068
$329$	1441.00	0.241474
$330$	0	0
$331$	−6237.15	−1.03572	−0.517862	−	0.855464i	$-0.673272\pi$
$331$	−6237.15	−1.03572	−0.517862	+	0.855464i	$0.673272\pi$
$332$	8270.62	1.36720
$333$	0	0
$334$	9.31670	0.00152631
$335$	793.569	0.129425
$336$	0	0
$337$	−5496.22	−0.888422	−0.444211	−	0.895922i	$-0.646516\pi$
$337$	−5496.22	−0.888422	−0.444211	+	0.895922i	$0.646516\pi$
$338$	−312.129	−0.0502295
$339$	0	0
$340$	2800.87	0.446760
$341$	−3469.00	−0.550900
$342$	0	0
$343$	−343.000	−0.0539949
$344$	−1330.89	−0.208595
$345$	0	0
$346$	990.210	0.153855
$347$	10566.3	1.63466	0.817332	−	0.576167i	$-0.195452\pi$
$347$	10566.3	1.63466	0.817332	+	0.576167i	$0.195452\pi$
$348$	0	0
$349$	−2553.15	−0.391596	−0.195798	−	0.980644i	$-0.562730\pi$
$349$	−2553.15	−0.391596	−0.195798	+	0.980644i	$0.562730\pi$
$350$	−105.684	−0.0161402
$351$	0	0
$352$	487.470	0.0738132
$353$	−1724.64	−0.260037	−0.130019	−	0.991512i	$-0.541504\pi$
$353$	−1724.64	−0.260037	−0.130019	+	0.991512i	$0.541504\pi$
$354$	0	0
$355$	924.113	0.138160
$356$	3996.11	0.594926
$357$	0	0
$358$	1110.60	0.163959
$359$	−2686.78	−0.394994	−0.197497	−	0.980303i	$-0.563281\pi$
$359$	−2686.78	−0.394994	−0.197497	+	0.980303i	$0.563281\pi$
$360$	0	0
$361$	5629.43	0.820736
$362$	688.262	0.0999289
$363$	0	0
$364$	3307.94	0.476327
$365$	2686.05	0.385190
$366$	0	0
$367$	−2379.21	−0.338403	−0.169201	−	0.985582i	$-0.554119\pi$
$367$	−2379.21	−0.338403	−0.169201	+	0.985582i	$0.554119\pi$
$368$	−6604.05	−0.935490
$369$	0	0
$370$	−329.157	−0.0462487
$371$	941.627	0.131770
$372$	0	0
$373$	9982.30	1.38569	0.692847	−	0.721085i	$-0.256356\pi$
$373$	9982.30	1.38569	0.692847	+	0.721085i	$0.256356\pi$
$374$	116.419	0.0160959
$375$	0	0
$376$	−764.543	−0.104863
$377$	−596.189	−0.0814464
$378$	0	0
$379$	−1197.67	−0.162323	−0.0811613	−	0.996701i	$-0.525863\pi$
$379$	−1197.67	−0.162323	−0.0811613	+	0.996701i	$0.525863\pi$
$380$	6888.24	0.929893
$381$	0	0
$382$	166.649	0.0223207
$383$	3954.76	0.527621	0.263811	−	0.964575i	$-0.415021\pi$
$383$	3954.76	0.527621	0.263811	+	0.964575i	$0.415021\pi$
$384$	0	0
$385$	−597.324	−0.0790713
$386$	−419.096	−0.0552627
$387$	0	0
$388$	7463.33	0.976528
$389$	14191.8	1.84976	0.924878	−	0.380263i	$-0.124167\pi$
$389$	14191.8	1.84976	0.924878	+	0.380263i	$0.124167\pi$
$390$	0	0
$391$	−4786.03	−0.619027
$392$	181.984	0.0234479
$393$	0	0
$394$	693.740	0.0887059
$395$	6010.74	0.765654
$396$	0	0
$397$	−8724.90	−1.10300	−0.551499	−	0.834176i	$-0.685944\pi$
$397$	−8724.90	−1.10300	−0.551499	+	0.834176i	$0.685944\pi$
$398$	536.432	0.0675601
$399$	0	0
$400$	−4064.40	−0.508049
$401$	−9736.18	−1.21247	−0.606236	−	0.795285i	$-0.707321\pi$
$401$	−9736.18	−1.21247	−0.606236	+	0.795285i	$0.707321\pi$
$402$	0	0
$403$	18755.8	2.31835
$404$	−5277.97	−0.649972
$405$	0	0
$406$	−16.3436	−0.00199784
$407$	2003.94	0.244058
$408$	0	0
$409$	−10284.9	−1.24342	−0.621708	−	0.783249i	$-0.713561\pi$
$409$	−10284.9	−1.24342	−0.621708	+	0.783249i	$0.713561\pi$
$410$	−881.620	−0.106195
$411$	0	0
$412$	11923.9	1.42584
$413$	−6243.98	−0.743938
$414$	0	0
$415$	8074.62	0.955103
$416$	−2635.60	−0.310627
$417$	0	0
$418$	286.311	0.0335022
$419$	−12837.5	−1.49679	−0.748393	−	0.663255i	$-0.769174\pi$
$419$	−12837.5	−1.49679	−0.748393	+	0.663255i	$0.769174\pi$
$420$	0	0
$421$	3108.45	0.359849	0.179925	−	0.983680i	$-0.442415\pi$
$421$	3108.45	0.359849	0.179925	+	0.983680i	$0.442415\pi$
$422$	991.876	0.114417
$423$	0	0
$424$	−499.594	−0.0572227
$425$	−2945.51	−0.336184
$426$	0	0
$427$	−4582.29	−0.519327
$428$	−8665.67	−0.978671
$429$	0	0
$430$	−647.465	−0.0726129
$431$	414.489	0.0463231	0.0231615	−	0.999732i	$-0.492627\pi$
$431$	414.489	0.0463231	0.0231615	+	0.999732i	$0.492627\pi$
$432$	0	0
$433$	7591.87	0.842591	0.421296	−	0.906923i	$-0.361575\pi$
$433$	7591.87	0.842591	0.421296	+	0.906923i	$0.361575\pi$
$434$	514.164	0.0568679
$435$	0	0
$436$	−4176.87	−0.458797
$437$	−11770.4	−1.28845
$438$	0	0
$439$	−1585.71	−0.172396	−0.0861982	−	0.996278i	$-0.527472\pi$
$439$	−1585.71	−0.172396	−0.0861982	+	0.996278i	$0.527472\pi$
$440$	316.919	0.0343376
$441$	0	0
$442$	−629.441	−0.0677363
$443$	−3305.39	−0.354501	−0.177250	−	0.984166i	$-0.556720\pi$
$443$	−3305.39	−0.354501	−0.177250	+	0.984166i	$0.556720\pi$
$444$	0	0
$445$	3901.42	0.415606
$446$	638.072	0.0677434
$447$	0	0
$448$	3439.00	0.362673
$449$	2470.77	0.259694	0.129847	−	0.991534i	$-0.458551\pi$
$449$	2470.77	0.259694	0.129847	+	0.991534i	$0.458551\pi$
$450$	0	0
$451$	5367.39	0.560400
$452$	−15717.4	−1.63559
$453$	0	0
$454$	785.745	0.0812265
$455$	3229.55	0.332755
$456$	0	0
$457$	2916.85	0.298566	0.149283	−	0.988795i	$-0.452303\pi$
$457$	2916.85	0.298566	0.149283	+	0.988795i	$0.452303\pi$
$458$	−81.3095	−0.00829551
$459$	0	0
$460$	−6492.18	−0.658042
$461$	6185.85	0.624954	0.312477	−	0.949925i	$-0.398841\pi$
$461$	6185.85	0.624954	0.312477	+	0.949925i	$0.398841\pi$
$462$	0	0
$463$	−19037.0	−1.91086	−0.955428	−	0.295224i	$-0.904606\pi$
$463$	−19037.0	−1.91086	−0.955428	+	0.295224i	$0.904606\pi$
$464$	−628.541	−0.0628864
$465$	0	0
$466$	514.873	0.0511825
$467$	3241.03	0.321150	0.160575	−	0.987024i	$-0.448665\pi$
$467$	3241.03	0.321150	0.160575	+	0.987024i	$0.448665\pi$
$468$	0	0
$469$	716.083	0.0705024
$470$	−371.943	−0.0365031
$471$	0	0
$472$	3312.84	0.323063
$473$	3941.83	0.383183
$474$	0	0
$475$	−7243.95	−0.699737
$476$	2527.39	0.243367
$477$	0	0
$478$	−948.659	−0.0907754
$479$	−17662.0	−1.68475	−0.842377	−	0.538889i	$-0.818844\pi$
$479$	−17662.0	−1.68475	−0.842377	+	0.538889i	$0.818844\pi$
$480$	0	0
$481$	−10834.7	−1.02707
$482$	−458.278	−0.0433070
$483$	0	0
$484$	−961.436	−0.0902926
$485$	7286.46	0.682188
$486$	0	0
$487$	−11143.6	−1.03689	−0.518445	−	0.855111i	$-0.673489\pi$
$487$	−11143.6	−1.03689	−0.518445	+	0.855111i	$0.673489\pi$
$488$	2431.20	0.225523
$489$	0	0
$490$	88.5334	0.00816231
$491$	−7223.60	−0.663944	−0.331972	−	0.943289i	$-0.607714\pi$
$491$	−7223.60	−0.663944	−0.331972	+	0.943289i	$0.607714\pi$
$492$	0	0
$493$	−455.510	−0.0416128
$494$	−1548.00	−0.140987
$495$	0	0
$496$	19773.6	1.79004
$497$	833.880	0.0752609
$498$	0	0
$499$	−13771.3	−1.23545	−0.617725	−	0.786394i	$-0.711946\pi$
$499$	−13771.3	−1.23545	−0.617725	+	0.786394i	$0.711946\pi$
$500$	−11700.4	−1.04652
$501$	0	0
$502$	−1635.03	−0.145368
$503$	16826.6	1.49157	0.745787	−	0.666185i	$-0.232074\pi$
$503$	16826.6	1.49157	0.745787	+	0.666185i	$0.232074\pi$
$504$	0	0
$505$	−5152.89	−0.454061
$506$	−269.849	−0.0237080
$507$	0	0
$508$	19585.7	1.71058
$509$	−17447.3	−1.51932	−0.759662	−	0.650318i	$-0.774636\pi$
$509$	−17447.3	−1.51932	−0.759662	+	0.650318i	$0.774636\pi$
$510$	0	0
$511$	2423.78	0.209827
$512$	−4641.57	−0.400645
$513$	0	0
$514$	−654.513	−0.0561660
$515$	11641.3	0.996073
$516$	0	0
$517$	2264.43	0.192629
$518$	−297.017	−0.0251934
$519$	0	0
$520$	−1713.49	−0.144503
$521$	1688.35	0.141973	0.0709866	−	0.997477i	$-0.477385\pi$
$521$	1688.35	0.141973	0.0709866	+	0.997477i	$0.477385\pi$
$522$	0	0
$523$	−21089.3	−1.76323	−0.881616	−	0.471967i	$-0.843544\pi$
$523$	−21089.3	−1.76323	−0.881616	+	0.471967i	$0.843544\pi$
$524$	−4386.43	−0.365691
$525$	0	0
$526$	1255.01	0.104032
$527$	14330.1	1.18450
$528$	0	0
$529$	−1073.40	−0.0882223
$530$	−243.048	−0.0199195
$531$	0	0
$532$	6215.66	0.506547
$533$	−29019.8	−2.35833
$534$	0	0
$535$	−8460.32	−0.683685
$536$	−379.928	−0.0306164
$537$	0	0
$538$	−195.416	−0.0156598
$539$	−539.000	−0.0430730
$540$	0	0
$541$	15205.5	1.20838	0.604191	−	0.796840i	$-0.293496\pi$
$541$	15205.5	1.20838	0.604191	+	0.796840i	$0.293496\pi$
$542$	−892.386	−0.0707219
$543$	0	0
$544$	−2013.70	−0.158707
$545$	−4077.88	−0.320509
$546$	0	0
$547$	−6915.32	−0.540544	−0.270272	−	0.962784i	$-0.587114\pi$
$547$	−6915.32	−0.540544	−0.270272	+	0.962784i	$0.587114\pi$
$548$	6311.03	0.491960
$549$	0	0
$550$	−166.076	−0.0128754
$551$	−1120.24	−0.0866135
$552$	0	0
$553$	5423.84	0.417080
$554$	186.468	0.0143001
$555$	0	0
$556$	−3424.38	−0.261198
$557$	2243.06	0.170631	0.0853156	−	0.996354i	$-0.472810\pi$
$557$	2243.06	0.170631	0.0853156	+	0.996354i	$0.472810\pi$
$558$	0	0
$559$	−21312.3	−1.61255
$560$	3404.80	0.256927
$561$	0	0
$562$	928.875	0.0697193
$563$	1533.10	0.114765	0.0573824	−	0.998352i	$-0.481725\pi$
$563$	1533.10	0.114765	0.0573824	+	0.998352i	$0.481725\pi$
$564$	0	0
$565$	−15345.0	−1.14260
$566$	−50.7832	−0.00377134
$567$	0	0
$568$	−442.428	−0.0326828
$569$	−8226.60	−0.606110	−0.303055	−	0.952973i	$-0.598007\pi$
$569$	−8226.60	−0.606110	−0.303055	+	0.952973i	$0.598007\pi$
$570$	0	0
$571$	565.628	0.0414550	0.0207275	−	0.999785i	$-0.493402\pi$
$571$	565.628	0.0414550	0.0207275	+	0.999785i	$0.493402\pi$
$572$	5198.19	0.379978
$573$	0	0
$574$	−795.536	−0.0578485
$575$	6827.44	0.495172
$576$	0	0
$577$	−7258.21	−0.523680	−0.261840	−	0.965111i	$-0.584329\pi$
$577$	−7258.21	−0.523680	−0.261840	+	0.965111i	$0.584329\pi$
$578$	663.380	0.0477387
$579$	0	0
$580$	−617.893	−0.0442356
$581$	7286.20	0.520279
$582$	0	0
$583$	1479.70	0.105116
$584$	−1285.97	−0.0911198
$585$	0	0
$586$	672.293	0.0473928
$587$	−17458.6	−1.22758	−0.613792	−	0.789468i	$-0.710357\pi$
$587$	−17458.6	−1.22758	−0.613792	+	0.789468i	$0.710357\pi$
$588$	0	0
$589$	35242.4	2.46543
$590$	1611.66	0.112460
$591$	0	0
$592$	−11422.6	−0.793019
$593$	13482.3	0.933648	0.466824	−	0.884350i	$-0.345398\pi$
$593$	13482.3	0.933648	0.466824	+	0.884350i	$0.345398\pi$
$594$	0	0
$595$	2467.49	0.170012
$596$	21024.0	1.44493
$597$	0	0
$598$	1458.99	0.0997702
$599$	5040.45	0.343819	0.171909	−	0.985113i	$-0.445006\pi$
$599$	5040.45	0.343819	0.171909	+	0.985113i	$0.445006\pi$
$600$	0	0
$601$	13587.9	0.922234	0.461117	−	0.887339i	$-0.347449\pi$
$601$	13587.9	0.922234	0.461117	+	0.887339i	$0.347449\pi$
$602$	−584.245	−0.0395549
$603$	0	0
$604$	4275.38	0.288018
$605$	−938.652	−0.0630771
$606$	0	0
$607$	−24238.0	−1.62074	−0.810371	−	0.585917i	$-0.800735\pi$
$607$	−24238.0	−1.62074	−0.810371	+	0.585917i	$0.800735\pi$
$608$	−4952.32	−0.330334
$609$	0	0
$610$	1182.76	0.0785056
$611$	−12243.1	−0.810640
$612$	0	0
$613$	−7239.20	−0.476979	−0.238490	−	0.971145i	$-0.576652\pi$
$613$	−7239.20	−0.476979	−0.238490	+	0.971145i	$0.576652\pi$
$614$	−338.433	−0.0222444
$615$	0	0
$616$	285.975	0.0187049
$617$	−25095.8	−1.63747	−0.818736	−	0.574170i	$-0.805325\pi$
$617$	−25095.8	−1.63747	−0.818736	+	0.574170i	$0.805325\pi$
$618$	0	0
$619$	9983.31	0.648244	0.324122	−	0.946015i	$-0.394931\pi$
$619$	9983.31	0.648244	0.324122	+	0.946015i	$0.394931\pi$
$620$	19438.7	1.25915
$621$	0	0
$622$	−2509.00	−0.161739
$623$	3520.47	0.226396
$624$	0	0
$625$	−3320.39	−0.212505
$626$	554.821	0.0354235
$627$	0	0
$628$	−16308.1	−1.03625
$629$	−8278.09	−0.524752
$630$	0	0
$631$	10735.0	0.677264	0.338632	−	0.940919i	$-0.390036\pi$
$631$	10735.0	0.677264	0.338632	+	0.940919i	$0.390036\pi$
$632$	−2877.70	−0.181121
$633$	0	0
$634$	−1833.08	−0.114828
$635$	19121.6	1.19499
$636$	0	0
$637$	2914.21	0.181264
$638$	−25.6829	−0.00159372
$639$	0	0
$640$	−3637.86	−0.224686
$641$	−12831.2	−0.790643	−0.395322	−	0.918543i	$-0.629367\pi$
$641$	−12831.2	−0.790643	−0.395322	+	0.918543i	$0.629367\pi$
$642$	0	0
$643$	−16536.8	−1.01423	−0.507113	−	0.861880i	$-0.669287\pi$
$643$	−16536.8	−1.01423	−0.507113	+	0.861880i	$0.669287\pi$
$644$	−5858.27	−0.358460
$645$	0	0
$646$	−1182.73	−0.0720337
$647$	6600.54	0.401073	0.200536	−	0.979686i	$-0.435732\pi$
$647$	6600.54	0.401073	0.200536	+	0.979686i	$0.435732\pi$
$648$	0	0
$649$	−9811.97	−0.593457
$650$	897.920	0.0541836
$651$	0	0
$652$	−17416.3	−1.04613
$653$	22720.9	1.36162	0.680810	−	0.732460i	$-0.261628\pi$
$653$	22720.9	1.36162	0.680810	+	0.732460i	$0.261628\pi$
$654$	0	0
$655$	−4282.48	−0.255466
$656$	−30594.6	−1.82091
$657$	0	0
$658$	−335.626	−0.0198846
$659$	−916.372	−0.0541681	−0.0270841	−	0.999633i	$-0.508622\pi$
$659$	−916.372	−0.0541681	−0.0270841	+	0.999633i	$0.508622\pi$
$660$	0	0
$661$	−10592.7	−0.623312	−0.311656	−	0.950195i	$-0.600884\pi$
$661$	−10592.7	−0.623312	−0.311656	+	0.950195i	$0.600884\pi$
$662$	1452.71	0.0852886
$663$	0	0
$664$	−3865.80	−0.225937
$665$	6068.36	0.353866
$666$	0	0
$667$	1055.83	0.0612924
$668$	317.838	0.0184095
$669$	0	0
$670$	−184.832	−0.0106577
$671$	−7200.74	−0.414279
$672$	0	0
$673$	10778.2	0.617336	0.308668	−	0.951170i	$-0.400117\pi$
$673$	10778.2	0.617336	0.308668	+	0.951170i	$0.400117\pi$
$674$	1280.14	0.0731587
$675$	0	0
$676$	−10648.2	−0.605839
$677$	−24654.1	−1.39960	−0.699802	−	0.714336i	$-0.746729\pi$
$677$	−24654.1	−1.39960	−0.699802	+	0.714336i	$0.746729\pi$
$678$	0	0
$679$	6575.00	0.371613
$680$	−1309.17	−0.0738298
$681$	0	0
$682$	807.972	0.0453649
$683$	21152.8	1.18505	0.592524	−	0.805553i	$-0.298132\pi$
$683$	21152.8	1.18505	0.592524	+	0.805553i	$0.298132\pi$
$684$	0	0
$685$	6161.48	0.343676
$686$	79.8888	0.00444631
$687$	0	0
$688$	−22468.8	−1.24508
$689$	−8000.29	−0.442361
$690$	0	0
$691$	3359.89	0.184973	0.0924865	−	0.995714i	$-0.470519\pi$
$691$	3359.89	0.184973	0.0924865	+	0.995714i	$0.470519\pi$
$692$	33780.8	1.85572
$693$	0	0
$694$	−2461.02	−0.134609
$695$	−3343.23	−0.182469
$696$	0	0
$697$	−22172.2	−1.20492
$698$	594.659	0.0322467
$699$	0	0
$700$	−3605.41	−0.194674
$701$	20003.4	1.07777	0.538887	−	0.842378i	$-0.318845\pi$
$701$	20003.4	1.07777	0.538887	+	0.842378i	$0.318845\pi$
$702$	0	0
$703$	−20358.5	−1.09223
$704$	5404.15	0.289313
$705$	0	0
$706$	401.688	0.0214132
$707$	−4649.75	−0.247344
$708$	0	0
$709$	6330.56	0.335331	0.167665	−	0.985844i	$-0.446377\pi$
$709$	6330.56	0.335331	0.167665	+	0.985844i	$0.446377\pi$
$710$	−215.237	−0.0113770
$711$	0	0
$712$	−1867.84	−0.0983149
$713$	−33216.1	−1.74467
$714$	0	0
$715$	5075.01	0.265447
$716$	37888.0	1.97757
$717$	0	0
$718$	625.784	0.0325265
$719$	111.791	0.00579845	0.00289923	−	0.999996i	$-0.499077\pi$
$719$	111.791	0.00579845	0.00289923	+	0.999996i	$0.499077\pi$
$720$	0	0
$721$	10504.6	0.542597
$722$	−1311.16	−0.0675849
$723$	0	0
$724$	23480.0	1.20528
$725$	649.801	0.0332869
$726$	0	0
$727$	−17266.8	−0.880867	−0.440433	−	0.897785i	$-0.645175\pi$
$727$	−17266.8	−0.880867	−0.440433	+	0.897785i	$0.645175\pi$
$728$	−1546.18	−0.0787159
$729$	0	0
$730$	−625.614	−0.0317192
$731$	−16283.4	−0.823888
$732$	0	0
$733$	−858.639	−0.0432668	−0.0216334	−	0.999766i	$-0.506887\pi$
$733$	−858.639	−0.0432668	−0.0216334	+	0.999766i	$0.506887\pi$
$734$	554.146	0.0278664
$735$	0	0
$736$	4667.58	0.233762
$737$	1125.27	0.0562415
$738$	0	0
$739$	16141.2	0.803471	0.401736	−	0.915756i	$-0.368407\pi$
$739$	16141.2	0.803471	0.401736	+	0.915756i	$0.368407\pi$
$740$	−11229.1	−0.557825
$741$	0	0
$742$	−219.316	−0.0108509
$743$	−1249.50	−0.0616953	−0.0308476	−	0.999524i	$-0.509821\pi$
$743$	−1249.50	−0.0616953	−0.0308476	+	0.999524i	$0.509821\pi$
$744$	0	0
$745$	20525.8	1.00941
$746$	−2325.00	−0.114107
$747$	0	0
$748$	3971.61	0.194140
$749$	−7634.23	−0.372428
$750$	0	0
$751$	82.9863	0.00403224	0.00201612	−	0.999998i	$-0.499358\pi$
$751$	82.9863	0.00403224	0.00201612	+	0.999998i	$0.499358\pi$
$752$	−12907.4	−0.625912
$753$	0	0
$754$	138.859	0.00670685
$755$	4174.06	0.201205
$756$	0	0
$757$	−3439.65	−0.165147	−0.0825735	−	0.996585i	$-0.526314\pi$
$757$	−3439.65	−0.165147	−0.0825735	+	0.996585i	$0.526314\pi$
$758$	278.952	0.0133667
$759$	0	0
$760$	−3219.66	−0.153670
$761$	−5651.24	−0.269195	−0.134597	−	0.990900i	$-0.542974\pi$
$761$	−5651.24	−0.269195	−0.134597	+	0.990900i	$0.542974\pi$
$762$	0	0
$763$	−3679.71	−0.174593
$764$	5685.20	0.269219
$765$	0	0
$766$	−921.111	−0.0434479
$767$	53050.3	2.49744
$768$	0	0
$769$	14381.1	0.674378	0.337189	−	0.941437i	$-0.390524\pi$
$769$	14381.1	0.674378	0.337189	+	0.941437i	$0.390524\pi$
$770$	139.124	0.00651127
$771$	0	0
$772$	−14297.4	−0.666547
$773$	31316.0	1.45713	0.728563	−	0.684979i	$-0.240189\pi$
$773$	31316.0	1.45713	0.728563	+	0.684979i	$0.240189\pi$
$774$	0	0
$775$	−20442.5	−0.947503
$776$	−3488.46	−0.161377
$777$	0	0
$778$	−3305.45	−0.152321
$779$	−54528.6	−2.50794
$780$	0	0
$781$	1310.38	0.0600374
$782$	1114.72	0.0509749
$783$	0	0
$784$	3072.35	0.139958
$785$	−15921.6	−0.723907
$786$	0	0
$787$	29847.6	1.35191	0.675954	−	0.736944i	$-0.263732\pi$
$787$	29847.6	1.35191	0.675954	+	0.736944i	$0.263732\pi$
$788$	23666.8	1.06992
$789$	0	0
$790$	−1399.97	−0.0630491
$791$	−13846.6	−0.622415
$792$	0	0
$793$	38932.2	1.74341
$794$	2032.13	0.0908283
$795$	0	0
$796$	18300.3	0.814871
$797$	−7581.40	−0.336947	−0.168474	−	0.985706i	$-0.553884\pi$
$797$	−7581.40	−0.336947	−0.168474	+	0.985706i	$0.553884\pi$
$798$	0	0
$799$	−9354.15	−0.414175
$800$	2872.61	0.126953
$801$	0	0
$802$	2267.67	0.0998432
$803$	3808.80	0.167384
$804$	0	0
$805$	−5719.44	−0.250415
$806$	−4368.46	−0.190909
$807$	0	0
$808$	2467.00	0.107412
$809$	25260.6	1.09779	0.548896	−	0.835891i	$-0.315048\pi$
$809$	25260.6	1.09779	0.548896	+	0.835891i	$0.315048\pi$
$810$	0	0
$811$	−25594.8	−1.10821	−0.554103	−	0.832448i	$-0.686939\pi$
$811$	−25594.8	−1.10821	−0.554103	+	0.832448i	$0.686939\pi$
$812$	−557.561	−0.0240967
$813$	0	0
$814$	−466.741	−0.0200974
$815$	−17003.6	−0.730808
$816$	0	0
$817$	−40046.0	−1.71485
$818$	2395.48	0.102391
$819$	0	0
$820$	−30076.3	−1.28087
$821$	19077.2	0.810959	0.405480	−	0.914104i	$-0.367105\pi$
$821$	19077.2	0.810959	0.405480	+	0.914104i	$0.367105\pi$
$822$	0	0
$823$	13889.2	0.588270	0.294135	−	0.955764i	$-0.404969\pi$
$823$	13889.2	0.588270	0.294135	+	0.955764i	$0.404969\pi$
$824$	−5573.39	−0.235629
$825$	0	0
$826$	1454.30	0.0612609
$827$	−1354.92	−0.0569714	−0.0284857	−	0.999594i	$-0.509069\pi$
$827$	−1354.92	−0.0569714	−0.0284857	+	0.999594i	$0.509069\pi$
$828$	0	0
$829$	29760.9	1.24685	0.623426	−	0.781882i	$-0.285740\pi$
$829$	29760.9	1.24685	0.623426	+	0.781882i	$0.285740\pi$
$830$	−1880.68	−0.0786496
$831$	0	0
$832$	−29218.6	−1.21752
$833$	2226.56	0.0926120
$834$	0	0
$835$	310.306	0.0128606
$836$	9767.46	0.404084
$837$	0	0
$838$	2990.01	0.123256
$839$	−33789.1	−1.39038	−0.695191	−	0.718825i	$-0.744680\pi$
$839$	−33789.1	−1.39038	−0.695191	+	0.718825i	$0.744680\pi$
$840$	0	0
$841$	−24288.5	−0.995880
$842$	−723.995	−0.0296324
$843$	0	0
$844$	33837.7	1.38003
$845$	−10395.9	−0.423230
$846$	0	0
$847$	−847.000	−0.0343604
$848$	−8434.42	−0.341556
$849$	0	0
$850$	686.044	0.0276837
$851$	19187.9	0.772918
$852$	0	0
$853$	22293.3	0.894851	0.447426	−	0.894321i	$-0.352341\pi$
$853$	22293.3	0.894851	0.447426	+	0.894321i	$0.352341\pi$
$854$	1067.27	0.0427649
$855$	0	0
$856$	4050.46	0.161731
$857$	43972.3	1.75270	0.876352	−	0.481672i	$-0.159970\pi$
$857$	43972.3	1.75270	0.876352	+	0.481672i	$0.159970\pi$
$858$	0	0
$859$	−34582.5	−1.37362	−0.686810	−	0.726837i	$-0.740990\pi$
$859$	−34582.5	−1.37362	−0.686810	+	0.726837i	$0.740990\pi$
$860$	−22088.2	−0.875814
$861$	0	0
$862$	−96.5395	−0.00381456
$863$	−48984.4	−1.93215	−0.966077	−	0.258254i	$-0.916853\pi$
$863$	−48984.4	−1.93215	−0.966077	+	0.258254i	$0.916853\pi$
$864$	0	0
$865$	32980.3	1.29638
$866$	−1768.24	−0.0693847
$867$	0	0
$868$	17540.6	0.685907
$869$	8523.17	0.332714
$870$	0	0
$871$	−6084.01	−0.236681
$872$	1952.33	0.0758189
$873$	0	0
$874$	2741.46	0.106100
$875$	−10307.7	−0.398246
$876$	0	0
$877$	3914.98	0.150741	0.0753703	−	0.997156i	$-0.475986\pi$
$877$	3914.98	0.150741	0.0753703	+	0.997156i	$0.475986\pi$
$878$	369.332	0.0141963
$879$	0	0
$880$	5350.40	0.204957
$881$	−4908.77	−0.187719	−0.0938597	−	0.995585i	$-0.529921\pi$
$881$	−4908.77	−0.187719	−0.0938597	+	0.995585i	$0.529921\pi$
$882$	0	0
$883$	8791.82	0.335072	0.167536	−	0.985866i	$-0.446419\pi$
$883$	8791.82	0.335072	0.167536	+	0.985866i	$0.446419\pi$
$884$	−21473.3	−0.816996
$885$	0	0
$886$	769.865	0.0291920
$887$	10011.0	0.378958	0.189479	−	0.981885i	$-0.439320\pi$
$887$	10011.0	0.378958	0.189479	+	0.981885i	$0.439320\pi$
$888$	0	0
$889$	17254.5	0.650954
$890$	−908.686	−0.0342239
$891$	0	0
$892$	21767.7	0.817082
$893$	−23004.8	−0.862069
$894$	0	0
$895$	36990.2	1.38150
$896$	−3282.65	−0.122395
$897$	0	0
$898$	−575.471	−0.0213850
$899$	−3161.34	−0.117282
$900$	0	0
$901$	−6112.51	−0.226012
$902$	−1250.13	−0.0461471
$903$	0	0
$904$	7346.55	0.270290
$905$	22923.5	0.841993
$906$	0	0
$907$	−44587.3	−1.63230	−0.816150	−	0.577840i	$-0.803896\pi$
$907$	−44587.3	−1.63230	−0.816150	+	0.577840i	$0.803896\pi$
$908$	26805.5	0.979706
$909$	0	0
$910$	−752.201	−0.0274013
$911$	−22201.6	−0.807434	−0.403717	−	0.914884i	$-0.632282\pi$
$911$	−22201.6	−0.807434	−0.403717	+	0.914884i	$0.632282\pi$
$912$	0	0
$913$	11449.7	0.415039
$914$	−679.370	−0.0245860
$915$	0	0
$916$	−2773.86	−0.100056
$917$	−3864.33	−0.139162
$918$	0	0
$919$	−4755.83	−0.170708	−0.0853538	−	0.996351i	$-0.527202\pi$
$919$	−4755.83	−0.170708	−0.0853538	+	0.996351i	$0.527202\pi$
$920$	3034.54	0.108745
$921$	0	0
$922$	−1440.76	−0.0514630
$923$	−7084.84	−0.252655
$924$	0	0
$925$	11809.0	0.419759
$926$	4433.95	0.157353
$927$	0	0
$928$	444.236	0.0157142
$929$	22020.9	0.777699	0.388849	−	0.921301i	$-0.372873\pi$
$929$	22020.9	0.777699	0.388849	+	0.921301i	$0.372873\pi$
$930$	0	0
$931$	5475.83	0.192764
$932$	17564.8	0.617333
$933$	0	0
$934$	−754.874	−0.0264457
$935$	3877.49	0.135623
$936$	0	0
$937$	29862.2	1.04115	0.520573	−	0.853817i	$-0.325718\pi$
$937$	29862.2	1.04115	0.520573	+	0.853817i	$0.325718\pi$
$938$	−166.784	−0.00580565
$939$	0	0
$940$	−12688.8	−0.440279
$941$	8090.56	0.280281	0.140141	−	0.990132i	$-0.455245\pi$
$941$	8090.56	0.280281	0.140141	+	0.990132i	$0.455245\pi$
$942$	0	0
$943$	51393.3	1.77476
$944$	55929.1	1.92832
$945$	0	0
$946$	−918.099	−0.0315539
$947$	−4398.75	−0.150940	−0.0754700	−	0.997148i	$-0.524046\pi$
$947$	−4398.75	−0.150940	−0.0754700	+	0.997148i	$0.524046\pi$
$948$	0	0
$949$	−20593.0	−0.704402
$950$	1687.20	0.0576211
$951$	0	0
$952$	−1181.34	−0.0402178
$953$	4288.91	0.145783	0.0728917	−	0.997340i	$-0.476777\pi$
$953$	4288.91	0.145783	0.0728917	+	0.997340i	$0.476777\pi$
$954$	0	0
$955$	5550.48	0.188073
$956$	−32363.3	−1.09488
$957$	0	0
$958$	4113.69	0.138734
$959$	5559.85	0.187213
$960$	0	0
$961$	69663.3	2.33840
$962$	2523.53	0.0845756
$963$	0	0
$964$	−15634.1	−0.522344
$965$	−13958.6	−0.465640
$966$	0	0
$967$	34271.8	1.13972	0.569858	−	0.821743i	$-0.306998\pi$
$967$	34271.8	1.13972	0.569858	+	0.821743i	$0.306998\pi$
$968$	449.389	0.0149214
$969$	0	0
$970$	−1697.10	−0.0561760
$971$	−36628.3	−1.21056	−0.605282	−	0.796011i	$-0.706940\pi$
$971$	−36628.3	−1.21056	−0.605282	+	0.796011i	$0.706940\pi$
$972$	0	0
$973$	−3016.79	−0.0993977
$974$	2595.48	0.0853845
$975$	0	0
$976$	41044.9	1.34612
$977$	−29087.7	−0.952504	−0.476252	−	0.879309i	$-0.658005\pi$
$977$	−29087.7	−0.952504	−0.476252	+	0.879309i	$0.658005\pi$
$978$	0	0
$979$	5532.17	0.180602
$980$	3020.30	0.0984490
$981$	0	0
$982$	1682.46	0.0546737
$983$	32900.7	1.06752	0.533758	−	0.845637i	$-0.320779\pi$
$983$	32900.7	1.06752	0.533758	+	0.845637i	$0.320779\pi$
$984$	0	0
$985$	23106.0	0.747429
$986$	106.094	0.00342668
$987$	0	0
$988$	−52809.7	−1.70051
$989$	37743.5	1.21352
$990$	0	0
$991$	7288.46	0.233628	0.116814	−	0.993154i	$-0.462732\pi$
$991$	7288.46	0.233628	0.116814	+	0.993154i	$0.462732\pi$
$992$	−13975.5	−0.447301
$993$	0	0
$994$	−194.221	−0.00619749
$995$	17866.6	0.569257
$996$	0	0
$997$	2983.80	0.0947822	0.0473911	−	0.998876i	$-0.484909\pi$
$997$	2983.80	0.0947822	0.0473911	+	0.998876i	$0.484909\pi$
$998$	3207.51	0.101735
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.n.1.3		5
3.2	odd	2		231.4.a.l.1.3	✓	5
21.20	even	2		1617.4.a.p.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.4.a.l.1.3	✓	5	3.2	odd	2
693.4.a.n.1.3		5	1.1	even	1	trivial
1617.4.a.p.1.3		5	21.20	even	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:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	693.a (trivial)

\(f(q)\)	\(=\)	\(q-0.232912 q^{2} -7.94575 q^{4} -7.75746 q^{5} -7.00000 q^{7} +3.71396 q^{8} +1.80680 q^{10} -11.0000 q^{11} +59.4737 q^{13} +1.63038 q^{14} +62.7010 q^{16} +45.4400 q^{17} +111.752 q^{19} +61.6388 q^{20} +2.56203 q^{22} -105.326 q^{23} -64.8219 q^{25} -13.8521 q^{26} +55.6203 q^{28} -10.0244 q^{29} +315.364 q^{31} -44.3154 q^{32} -10.5835 q^{34} +54.3022 q^{35} -182.176 q^{37} -26.0283 q^{38} -28.8108 q^{40} -487.944 q^{41} -358.348 q^{43} +87.4033 q^{44} +24.5317 q^{46} -205.857 q^{47} +49.0000 q^{49} +15.0978 q^{50} -472.563 q^{52} -134.518 q^{53} +85.3320 q^{55} -25.9977 q^{56} +2.33481 q^{58} +891.997 q^{59} +654.613 q^{61} -73.4520 q^{62} -491.286 q^{64} -461.364 q^{65} -102.298 q^{67} -361.055 q^{68} -12.6476 q^{70} -119.126 q^{71} -346.254 q^{73} +42.4310 q^{74} -887.951 q^{76} +77.0000 q^{77} -774.834 q^{79} -486.400 q^{80} +113.648 q^{82} -1040.89 q^{83} -352.499 q^{85} +83.4636 q^{86} -40.8535 q^{88} -502.925 q^{89} -416.316 q^{91} +836.896 q^{92} +47.9465 q^{94} -866.908 q^{95} -939.285 q^{97} -11.4127 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{2} + 21 q^{4} - 7 q^{5} - 35 q^{7} - 60 q^{8} + 55 q^{10} - 55 q^{11} + 111 q^{13} + 35 q^{14} + 201 q^{16} - 136 q^{17} + 111 q^{19} - 219 q^{20} + 55 q^{22} + 28 q^{23} + 190 q^{25} + q^{26}+ \cdots - 245 q^{98}+O(q^{100}) \) 5 * q - 5 * q^2 + 21 * q^4 - 7 * q^5 - 35 * q^7 - 60 * q^8 + 55 * q^10 - 55 * q^11 + 111 * q^13 + 35 * q^14 + 201 * q^16 - 136 * q^17 + 111 * q^19 - 219 * q^20 + 55 * q^22 + 28 * q^23 + 190 * q^25 + q^26 - 147 * q^28 - 61 * q^29 - 280 * q^31 - 535 * q^32 - 572 * q^34 + 49 * q^35 - 41 * q^37 - 267 * q^38 - 336 * q^40 - 426 * q^41 + 424 * q^43 - 231 * q^44 + 140 * q^46 - 75 * q^47 + 245 * q^49 - 490 * q^50 - 269 * q^52 - 1500 * q^53 + 77 * q^55 + 420 * q^56 - 1767 * q^58 - 757 * q^59 + 658 * q^61 + 568 * q^62 - 748 * q^64 - 537 * q^65 - 583 * q^67 + 1650 * q^68 - 385 * q^70 + 764 * q^71 + 875 * q^73 + 825 * q^74 + 213 * q^76 + 385 * q^77 - 244 * q^79 + 2577 * q^80 - 2006 * q^82 - 924 * q^83 - 1402 * q^85 - 1272 * q^86 + 660 * q^88 + 1110 * q^89 - 777 * q^91 + 2046 * q^92 - 3349 * q^94 - 1923 * q^95 - 852 * q^97 - 245 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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