LMFDB - Embedded newform 704.4.e.g.703.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [704,4,Mod(703,704)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(704, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("704.703");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 704 = 2^{6} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	704.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$41.5373446440$
Analytic rank:	$0$
Dimension:	$36$
Twist minimal:	no (minimal twist has level 352)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			703.6
Character	$\chi$	$=$	704.703
Dual form			704.4.e.g.703.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.43252i q^{3} -17.0450 q^{5} -13.3052 q^{7} +21.0829 q^{9} +(30.7486 + 19.6347i) q^{11} +66.3043i q^{13} -41.4623i q^{15} -10.3897i q^{17} -86.7262 q^{19} -32.3650i q^{21} +134.362i q^{23} +165.533 q^{25} +116.962i q^{27} +69.9754i q^{29} -79.2776i q^{31} +(-47.7618 + 74.7966i) q^{33} +226.787 q^{35} +111.615 q^{37} -161.286 q^{39} -261.845i q^{41} -128.924 q^{43} -359.358 q^{45} -422.628i q^{47} -165.973 q^{49} +25.2731 q^{51} -727.354 q^{53} +(-524.111 - 334.674i) q^{55} -210.963i q^{57} +573.045i q^{59} -889.645i q^{61} -280.511 q^{63} -1130.16i q^{65} +137.512i q^{67} -326.838 q^{69} -35.6256i q^{71} -845.012i q^{73} +402.662i q^{75} +(-409.115 - 261.243i) q^{77} +603.045 q^{79} +284.724 q^{81} +1071.60 q^{83} +177.093i q^{85} -170.216 q^{87} -532.570 q^{89} -882.189i q^{91} +192.844 q^{93} +1478.25 q^{95} -371.900 q^{97} +(648.269 + 413.956i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q - 364 q^{9} + 1236 q^{25} + 592 q^{33} - 984 q^{45} + 2372 q^{49} - 376 q^{53} + 2824 q^{69} + 2592 q^{77} + 6756 q^{81} + 512 q^{89} + 872 q^{93} + 3904 q^{97}+O(q^{100}) $ 36 * q - 364 * q^9 + 1236 * q^25 + 592 * q^33 - 984 * q^45 + 2372 * q^49 - 376 * q^53 + 2824 * q^69 + 2592 * q^77 + 6756 * q^81 + 512 * q^89 + 872 * q^93 + 3904 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/704\mathbb{Z}\right)^\times$.

$n$	$133$	$321$	$639$
$\chi(n)$	$1$	$-1$	$-1$

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
$3$			2.43252i			0.468138i	0.972220	+	0.234069i	$0.0752042\pi$
$3$			2.43252i			0.468138i	−0.972220	+	0.234069i	$0.924796\pi$
$4$		0			0
$5$	−17.0450			−1.52455			−0.762277	−	0.647251i	$-0.775919\pi$
$5$	−17.0450			−1.52455			−0.762277	+	0.647251i	$0.775919\pi$
$6$		0			0
$7$	−13.3052			−0.718411			−0.359205	−	0.933259i	$-0.616952\pi$
$7$	−13.3052			−0.718411			−0.359205	+	0.933259i	$0.616952\pi$
$8$		0			0
$9$	21.0829			0.780846
$10$		0			0
$11$	30.7486	+	19.6347i	0.842824	+	0.538190i
$11$	30.7486	+	19.6347i	0.842824	+	0.538190i
$12$		0			0
$13$			66.3043i			1.41458i	0.706925	+	0.707289i	$0.250082\pi$
$13$			66.3043i			1.41458i	−0.706925	+	0.707289i	$0.749918\pi$
$14$		0			0
$15$		−	41.4623i		−	0.713702i
$16$		0			0
$17$		−	10.3897i		−	0.148228i	−0.997250	−	0.0741138i	$-0.976387\pi$
$17$		−	10.3897i		−	0.148228i	0.997250	−	0.0741138i	$-0.0236128\pi$
$18$		0			0
$19$	−86.7262			−1.04718			−0.523588	−	0.851971i	$-0.675407\pi$
$19$	−86.7262			−1.04718			−0.523588	+	0.851971i	$0.675407\pi$
$20$		0			0
$21$		−	32.3650i		−	0.336316i
$22$		0			0
$23$			134.362i			1.21811i	0.793129	+	0.609053i	$0.208450\pi$
$23$			134.362i			1.21811i	−0.793129	+	0.609053i	$0.791550\pi$
$24$		0			0
$25$	165.533			1.32426
$26$		0			0
$27$			116.962i			0.833683i
$28$		0			0
$29$			69.9754i			0.448072i	0.974581	+	0.224036i	$0.0719234\pi$
$29$			69.9754i			0.448072i	−0.974581	+	0.224036i	$0.928077\pi$
$30$		0			0
$31$		−	79.2776i		−	0.459312i	−0.973272	−	0.229656i	$-0.926240\pi$
$31$		−	79.2776i		−	0.459312i	0.973272	−	0.229656i	$-0.0737601\pi$
$32$		0			0
$33$	−47.7618	+	74.7966i	−0.251947	+	0.394558i
$34$		0			0
$35$	226.787			1.09526
$36$		0			0
$37$	111.615			0.495930			0.247965	−	0.968769i	$-0.420238\pi$
$37$	111.615			0.495930			0.247965	+	0.968769i	$0.420238\pi$
$38$		0			0
$39$	−161.286			−0.662218
$40$		0			0
$41$		−	261.845i		−	0.997398i	−0.866775	−	0.498699i	$-0.833811\pi$
$41$		−	261.845i		−	0.997398i	0.866775	−	0.498699i	$-0.166189\pi$
$42$		0			0
$43$	−128.924			−0.457225			−0.228612	−	0.973518i	$-0.573419\pi$
$43$	−128.924			−0.457225			−0.228612	+	0.973518i	$0.573419\pi$
$44$		0			0
$45$	−359.358			−1.19044
$46$		0			0
$47$		−	422.628i		−	1.31163i	−0.754921	−	0.655815i	$-0.772325\pi$
$47$		−	422.628i		−	1.31163i	0.754921	−	0.655815i	$-0.227675\pi$
$48$		0			0
$49$	−165.973			−0.483886
$50$		0			0
$51$	25.2731			0.0693910
$52$		0			0
$53$	−727.354			−1.88509			−0.942545	−	0.334081i	$-0.891574\pi$
$53$	−727.354			−1.88509			−0.942545	+	0.334081i	$0.891574\pi$
$54$		0			0
$55$	−524.111	−	334.674i	−1.28493	−	0.820499i
$56$		0			0
$57$		−	210.963i		−	0.490224i
$58$		0			0
$59$			573.045i			1.26448i	0.774774	+	0.632238i	$0.217863\pi$
$59$			573.045i			1.26448i	−0.774774	+	0.632238i	$0.782137\pi$
$60$		0			0
$61$		−	889.645i		−	1.86733i	−0.358142	−	0.933667i	$-0.616590\pi$
$61$		−	889.645i		−	1.86733i	0.358142	−	0.933667i	$-0.383410\pi$
$62$		0			0
$63$	−280.511			−0.560968
$64$		0			0
$65$		−	1130.16i		−	2.15660i
$66$		0			0
$67$			137.512i			0.250743i	0.992110	+	0.125371i	$0.0400122\pi$
$67$			137.512i			0.250743i	−0.992110	+	0.125371i	$0.959988\pi$
$68$		0			0
$69$	−326.838			−0.570242
$70$		0			0
$71$		−	35.6256i		−	0.0595491i	−0.999557	−	0.0297745i	$-0.990521\pi$
$71$		−	35.6256i		−	0.0595491i	0.999557	−	0.0297745i	$-0.00947893\pi$
$72$		0			0
$73$		−	845.012i		−	1.35481i	−0.735610	−	0.677406i	$-0.763104\pi$
$73$		−	845.012i		−	1.35481i	0.735610	−	0.677406i	$-0.236896\pi$
$74$		0			0
$75$			402.662i			0.619939i
$76$		0			0
$77$	−409.115	−	261.243i	−0.605493	−	0.386641i
$78$		0			0
$79$	603.045			0.858833			0.429417	−	0.903106i	$-0.358719\pi$
$79$	603.045			0.858833			0.429417	+	0.903106i	$0.358719\pi$
$80$		0			0
$81$	284.724			0.390568
$82$		0			0
$83$	1071.60			1.41715			0.708574	−	0.705637i	$-0.249339\pi$
$83$	1071.60			1.41715			0.708574	+	0.705637i	$0.249339\pi$
$84$		0			0
$85$			177.093i			0.225981i
$86$		0			0
$87$	−170.216			−0.209760
$88$		0			0
$89$	−532.570			−0.634296			−0.317148	−	0.948376i	$-0.602725\pi$
$89$	−532.570			−0.634296			−0.317148	+	0.948376i	$0.602725\pi$
$90$		0			0
$91$		−	882.189i		−	1.01625i
$92$		0			0
$93$	192.844			0.215022
$94$		0			0
$95$	1478.25			1.59648
$96$		0			0
$97$	−371.900			−0.389286			−0.194643	−	0.980874i	$-0.562355\pi$
$97$	−371.900			−0.389286			−0.194643	+	0.980874i	$0.562355\pi$
$98$		0			0
$99$	648.269	+	413.956i	0.658116	+	0.420244i
$100$		0			0
$101$		−	1273.91i		−	1.25503i	−0.778603	−	0.627517i	$-0.784071\pi$
$101$		−	1273.91i		−	1.25503i	0.778603	−	0.627517i	$-0.215929\pi$
$102$		0			0
$103$			1121.47i			1.07284i	0.843952	+	0.536418i	$0.180223\pi$
$103$			1121.47i			1.07284i	−0.843952	+	0.536418i	$0.819777\pi$
$104$		0			0
$105$			551.663i			0.512731i
$106$		0			0
$107$	−2103.03			−1.90007			−0.950035	−	0.312145i	$-0.898953\pi$
$107$	−2103.03			−1.90007			−0.950035	+	0.312145i	$0.898953\pi$
$108$		0			0
$109$		−	699.322i		−	0.614522i	−0.951625	−	0.307261i	$-0.900588\pi$
$109$		−	699.322i		−	0.614522i	0.951625	−	0.307261i	$-0.0994124\pi$
$110$		0			0
$111$			271.506i			0.232164i
$112$		0			0
$113$	−1292.70			−1.07617			−0.538086	−	0.842890i	$-0.680852\pi$
$113$	−1292.70			−1.07617			−0.538086	+	0.842890i	$0.680852\pi$
$114$		0			0
$115$		−	2290.21i		−	1.85707i
$116$		0			0
$117$			1397.88i			1.10457i
$118$		0			0
$119$			138.236i			0.106488i
$120$		0			0
$121$	559.957	+	1207.48i	0.420704	+	0.907198i
$122$		0			0
$123$	636.943			0.466920
$124$		0			0
$125$	−690.886			−0.494358
$126$		0			0
$127$	−790.580			−0.552383			−0.276192	−	0.961103i	$-0.589072\pi$
$127$	−790.580			−0.552383			−0.276192	+	0.961103i	$0.589072\pi$
$128$		0			0
$129$		−	313.609i		−	0.214044i
$130$		0			0
$131$	263.168			0.175520			0.0877601	−	0.996142i	$-0.472029\pi$
$131$	263.168			0.175520			0.0877601	+	0.996142i	$0.472029\pi$
$132$		0			0
$133$	1153.91			0.752303
$134$		0			0
$135$		−	1993.63i		−	1.27099i
$136$		0			0
$137$	−278.091			−0.173423			−0.0867113	−	0.996233i	$-0.527636\pi$
$137$	−278.091			−0.173423			−0.0867113	+	0.996233i	$0.527636\pi$
$138$		0			0
$139$	−1129.28			−0.689098			−0.344549	−	0.938768i	$-0.611968\pi$
$139$	−1129.28			−0.689098			−0.344549	+	0.938768i	$0.611968\pi$
$140$		0			0
$141$	1028.05			0.614025
$142$		0			0
$143$	−1301.87	+	2038.77i	−0.761311	+	1.19224i
$144$		0			0
$145$		−	1192.73i		−	0.683110i
$146$		0			0
$147$		−	403.732i		−	0.226526i
$148$		0			0
$149$		−	761.663i		−	0.418778i	−0.977832	−	0.209389i	$-0.932853\pi$
$149$		−	761.663i		−	0.418778i	0.977832	−	0.209389i	$-0.0671475\pi$
$150$		0			0
$151$	2528.60			1.36275			0.681374	−	0.731936i	$-0.261383\pi$
$151$	2528.60			1.36275			0.681374	+	0.731936i	$0.261383\pi$
$152$		0			0
$153$		−	219.044i		−	0.115743i
$154$		0			0
$155$			1351.29i			0.700246i
$156$		0			0
$157$	−1085.35			−0.551722			−0.275861	−	0.961198i	$-0.588963\pi$
$157$	−1085.35			−0.551722			−0.275861	+	0.961198i	$0.588963\pi$
$158$		0			0
$159$		−	1769.30i		−	0.882482i
$160$		0			0
$161$		−	1787.71i		−	0.875101i
$162$		0			0
$163$			1268.44i			0.609522i	0.952429	+	0.304761i	$0.0985765\pi$
$163$			1268.44i			0.609522i	−0.952429	+	0.304761i	$0.901423\pi$
$164$		0			0
$165$	814.101	−	1274.91i	0.384107	−	0.601525i
$166$		0			0
$167$	136.790			0.0633839			0.0316920	−	0.999498i	$-0.489910\pi$
$167$	136.790			0.0633839			0.0316920	+	0.999498i	$0.489910\pi$
$168$		0			0
$169$	−2199.26			−1.00103
$170$		0			0
$171$	−1828.44			−0.817684
$172$		0			0
$173$		−	4202.29i		−	1.84679i	−0.383855	−	0.923393i	$-0.625404\pi$
$173$		−	4202.29i		−	1.84679i	0.383855	−	0.923393i	$-0.374596\pi$
$174$		0			0
$175$	−2202.44			−0.951365
$176$		0			0
$177$	−1393.94			−0.591949
$178$		0			0
$179$		−	4018.22i		−	1.67785i	−0.544245	−	0.838926i	$-0.683184\pi$
$179$		−	4018.22i		−	1.67785i	0.544245	−	0.838926i	$-0.316816\pi$
$180$		0			0
$181$	777.837			0.319426			0.159713	−	0.987163i	$-0.448943\pi$
$181$	777.837			0.319426			0.159713	+	0.987163i	$0.448943\pi$
$182$		0			0
$183$	2164.08			0.874171
$184$		0			0
$185$	−1902.48			−0.756071
$186$		0			0
$187$	203.999	−	319.469i	0.0797746	−	0.124930i
$188$		0			0
$189$		−	1556.20i		−	0.598926i
$190$		0			0
$191$			3512.22i			1.33055i	0.746598	+	0.665275i	$0.231686\pi$
$191$			3512.22i			1.33055i	−0.746598	+	0.665275i	$0.768314\pi$
$192$		0			0
$193$			3725.97i			1.38964i	0.719181	+	0.694822i	$0.244517\pi$
$193$			3725.97i			1.38964i	−0.719181	+	0.694822i	$0.755483\pi$
$194$		0			0
$195$	2749.13			1.00959
$196$		0			0
$197$		−	1998.64i		−	0.722830i	−0.932405	−	0.361415i	$-0.882294\pi$
$197$		−	1998.64i		−	0.722830i	0.932405	−	0.361415i	$-0.117706\pi$
$198$		0			0
$199$			3982.51i			1.41866i	0.704878	+	0.709328i	$0.251002\pi$
$199$			3982.51i			1.41866i	−0.704878	+	0.709328i	$0.748998\pi$
$200$		0			0
$201$	−334.501			−0.117382
$202$		0			0
$203$		−	931.033i		−	0.321900i
$204$		0			0
$205$			4463.16i			1.52059i
$206$		0			0
$207$			2832.74i			0.951154i
$208$		0			0
$209$	−2666.71	−	1702.84i	−0.882585	−	0.563580i
$210$		0			0
$211$	−3168.73			−1.03386			−0.516930	−	0.856028i	$-0.672925\pi$
$211$	−3168.73			−1.03386			−0.516930	+	0.856028i	$0.672925\pi$
$212$		0			0
$213$	86.6600			0.0278772
$214$		0			0
$215$	2197.51			0.697064
$216$		0			0
$217$			1054.80i			0.329975i
$218$		0			0
$219$	2055.51			0.634239
$220$		0			0
$221$	688.881			0.209680
$222$		0			0
$223$		−	2416.57i		−	0.725675i	−0.931852	−	0.362837i	$-0.881808\pi$
$223$		−	2416.57i		−	0.725675i	0.931852	−	0.362837i	$-0.118192\pi$
$224$		0			0
$225$	3489.91			1.03405
$226$		0			0
$227$	−4183.20			−1.22312			−0.611561	−	0.791197i	$-0.709458\pi$
$227$	−4183.20			−1.22312			−0.611561	+	0.791197i	$0.709458\pi$
$228$		0			0
$229$	−1820.35			−0.525292			−0.262646	−	0.964892i	$-0.584595\pi$
$229$	−1820.35			−0.525292			−0.262646	+	0.964892i	$0.584595\pi$
$230$		0			0
$231$	635.478	−	995.180i	0.181002	−	0.283455i
$232$		0			0
$233$		−	1596.36i		−	0.448844i	−0.974492	−	0.224422i	$-0.927951\pi$
$233$		−	1596.36i		−	0.448844i	0.974492	−	0.224422i	$-0.0720495\pi$
$234$		0			0
$235$			7203.71i			1.99965i
$236$		0			0
$237$			1466.92i			0.402053i
$238$		0			0
$239$	−4509.16			−1.22039			−0.610195	−	0.792251i	$-0.708909\pi$
$239$	−4509.16			−1.22039			−0.610195	+	0.792251i	$0.708909\pi$
$240$		0			0
$241$			254.715i			0.0680815i	0.999420	+	0.0340408i	$0.0108376\pi$
$241$			254.715i			0.0680815i	−0.999420	+	0.0340408i	$0.989162\pi$
$242$		0			0
$243$			3850.58i			1.01652i
$244$		0			0
$245$	2829.01			0.737710
$246$		0			0
$247$		−	5750.32i		−	1.48131i
$248$		0			0
$249$			2606.68i			0.663421i
$250$		0			0
$251$		−	6788.06i		−	1.70701i	−0.521087	−	0.853503i	$-0.674473\pi$
$251$		−	6788.06i		−	1.70701i	0.521087	−	0.853503i	$-0.325527\pi$
$252$		0			0
$253$	−2638.16	+	4131.45i	−0.655572	+	1.02665i
$254$		0			0
$255$	−430.781			−0.105790
$256$		0			0
$257$	2103.18			0.510477			0.255238	−	0.966878i	$-0.417846\pi$
$257$	2103.18			0.510477			0.255238	+	0.966878i	$0.417846\pi$
$258$		0			0
$259$	−1485.05			−0.356281
$260$		0			0
$261$			1475.28i			0.349876i
$262$		0			0
$263$	1909.67			0.447740			0.223870	−	0.974619i	$-0.428131\pi$
$263$	1909.67			0.447740			0.223870	+	0.974619i	$0.428131\pi$
$264$		0			0
$265$	12397.8			2.87392
$266$		0			0
$267$		−	1295.49i		−	0.296938i
$268$		0			0
$269$	750.932			0.170205			0.0851025	−	0.996372i	$-0.472878\pi$
$269$	750.932			0.170205			0.0851025	+	0.996372i	$0.472878\pi$
$270$		0			0
$271$	−6715.28			−1.50525			−0.752627	−	0.658447i	$-0.771214\pi$
$271$	−6715.28			−1.50525			−0.752627	+	0.658447i	$0.771214\pi$
$272$		0			0
$273$	2145.94			0.475744
$274$		0			0
$275$	5089.91	+	3250.19i	1.11612	+	0.712705i
$276$		0			0
$277$			8735.13i			1.89474i	0.320139	+	0.947370i	$0.396270\pi$
$277$			8735.13i			1.89474i	−0.320139	+	0.947370i	$0.603730\pi$
$278$		0			0
$279$		−	1671.40i		−	0.358652i
$280$		0			0
$281$		−	4509.01i		−	0.957242i	−0.878022	−	0.478621i	$-0.841137\pi$
$281$		−	4509.01i		−	0.957242i	0.878022	−	0.478621i	$-0.158863\pi$
$282$		0			0
$283$	5688.09			1.19478			0.597388	−	0.801952i	$-0.296205\pi$
$283$	5688.09			1.19478			0.597388	+	0.801952i	$0.296205\pi$
$284$		0			0
$285$			3595.87i			0.747372i
$286$		0			0
$287$			3483.89i			0.716541i
$288$		0			0
$289$	4805.05			0.978029
$290$		0			0
$291$		−	904.654i		−	0.182240i
$292$		0			0
$293$		−	1125.73i		−	0.224457i	−0.993682	−	0.112228i	$-0.964201\pi$
$293$		−	1125.73i		−	0.224457i	0.993682	−	0.112228i	$-0.0357988\pi$
$294$		0			0
$295$		−	9767.56i		−	1.92776i
$296$		0			0
$297$	−2296.52	+	3596.43i	−0.448679	+	0.702647i
$298$		0			0
$299$	−8908.79			−1.72311
$300$		0			0
$301$	1715.35			0.328475
$302$		0			0
$303$	3098.80			0.587530
$304$		0			0
$305$			15164.0i			2.84685i
$306$		0			0
$307$	8803.56			1.63663			0.818316	−	0.574769i	$-0.194908\pi$
$307$	8803.56			1.63663			0.818316	+	0.574769i	$0.194908\pi$
$308$		0			0
$309$	−2728.01			−0.502236
$310$		0			0
$311$		−	1526.95i		−	0.278410i	−0.990264	−	0.139205i	$-0.955545\pi$
$311$		−	1526.95i		−	0.278410i	0.990264	−	0.139205i	$-0.0444547\pi$
$312$		0			0
$313$	729.366			0.131713			0.0658565	−	0.997829i	$-0.479022\pi$
$313$	729.366			0.131713			0.0658565	+	0.997829i	$0.479022\pi$
$314$		0			0
$315$	4781.31			0.855226
$316$		0			0
$317$	896.384			0.158820			0.0794100	−	0.996842i	$-0.474696\pi$
$317$	896.384			0.158820			0.0794100	+	0.996842i	$0.474696\pi$
$318$		0			0
$319$	−1373.95	+	2151.65i	−0.241148	+	0.377646i
$320$		0			0
$321$		−	5115.65i		−	0.889495i
$322$		0			0
$323$			901.058i			0.155221i
$324$		0			0
$325$			10975.6i			1.87327i
$326$		0			0
$327$	1701.11			0.287681
$328$		0			0
$329$			5623.13i			0.942289i
$330$		0			0
$331$			10009.7i			1.66219i	0.556131	+	0.831095i	$0.312285\pi$
$331$			10009.7i			1.66219i	−0.556131	+	0.831095i	$0.687715\pi$
$332$		0			0
$333$	2353.16			0.387245
$334$		0			0
$335$		−	2343.90i		−	0.382271i
$336$		0			0
$337$			1013.10i			0.163759i	0.996642	+	0.0818797i	$0.0260923\pi$
$337$			1013.10i			0.163759i	−0.996642	+	0.0818797i	$0.973908\pi$
$338$		0			0
$339$		−	3144.53i		−	0.503797i
$340$		0			0
$341$	1556.59	−	2437.68i	0.247197	−	0.387119i
$342$		0			0
$343$	6771.96			1.06604
$344$		0			0
$345$	5570.97			0.869365
$346$		0			0
$347$	−3920.87			−0.606581			−0.303290	−	0.952898i	$-0.598085\pi$
$347$	−3920.87			−0.606581			−0.303290	+	0.952898i	$0.598085\pi$
$348$		0			0
$349$		−	5079.23i		−	0.779040i	−0.921018	−	0.389520i	$-0.872641\pi$
$349$		−	5079.23i		−	0.779040i	0.921018	−	0.389520i	$-0.127359\pi$
$350$		0			0
$351$	−7755.11			−1.17931
$352$		0			0
$353$	−6450.24			−0.972554			−0.486277	−	0.873805i	$-0.661645\pi$
$353$	−6450.24			−0.972554			−0.486277	+	0.873805i	$0.661645\pi$
$354$		0			0
$355$			607.240i			0.0907858i
$356$		0			0
$357$	−336.263			−0.0498513
$358$		0			0
$359$	335.400			0.0493085			0.0246542	−	0.999696i	$-0.492152\pi$
$359$	335.400			0.0493085			0.0246542	+	0.999696i	$0.492152\pi$
$360$		0			0
$361$	662.436			0.0965791
$362$		0			0
$363$	−2937.22	+	1362.10i	−0.424694	+	0.196947i
$364$		0			0
$365$			14403.3i			2.06548i
$366$		0			0
$367$			1346.75i			0.191552i	0.995403	+	0.0957762i	$0.0305333\pi$
$367$			1346.75i			0.191552i	−0.995403	+	0.0957762i	$0.969467\pi$
$368$		0			0
$369$		−	5520.44i		−	0.778815i
$370$		0			0
$371$	9677.55			1.35427
$372$		0			0
$373$		−	2779.96i		−	0.385900i	−0.981209	−	0.192950i	$-0.938194\pi$
$373$		−	2779.96i		−	0.385900i	0.981209	−	0.192950i	$-0.0618055\pi$
$374$		0			0
$375$		−	1680.59i		−	0.231428i
$376$		0			0
$377$	−4639.67			−0.633833
$378$		0			0
$379$		−	6083.66i		−	0.824530i	−0.911064	−	0.412265i	$-0.864738\pi$
$379$		−	6083.66i		−	0.824530i	0.911064	−	0.412265i	$-0.135262\pi$
$380$		0			0
$381$		−	1923.10i		−	0.258592i
$382$		0			0
$383$			522.352i			0.0696891i	0.999393	+	0.0348445i	$0.0110936\pi$
$383$			522.352i			0.0696891i	−0.999393	+	0.0348445i	$0.988906\pi$
$384$		0			0
$385$	6973.38	+	4452.89i	0.923107	+	0.589455i
$386$		0			0
$387$	−2718.08			−0.357022
$388$		0			0
$389$	6237.60			0.813004			0.406502	−	0.913650i	$-0.366748\pi$
$389$	6237.60			0.813004			0.406502	+	0.913650i	$0.366748\pi$
$390$		0			0
$391$	1395.98			0.180557
$392$		0			0
$393$			640.162i			0.0821677i
$394$		0			0
$395$	−10278.9			−1.30934
$396$		0			0
$397$	9886.74			1.24988			0.624938	−	0.780674i	$-0.285124\pi$
$397$	9886.74			1.24988			0.624938	+	0.780674i	$0.285124\pi$
$398$		0			0
$399$			2806.90i			0.352182i
$400$		0			0
$401$	7463.90			0.929499			0.464750	−	0.885442i	$-0.346144\pi$
$401$	7463.90			0.929499			0.464750	+	0.885442i	$0.346144\pi$
$402$		0			0
$403$	5256.45			0.649733
$404$		0			0
$405$	−4853.13			−0.595441
$406$		0			0
$407$	3432.01	+	2191.53i	0.417981	+	0.266904i
$408$		0			0
$409$		−	4293.30i		−	0.519047i	−0.965737	−	0.259523i	$-0.916435\pi$
$409$		−	4293.30i		−	0.519047i	0.965737	−	0.259523i	$-0.0835655\pi$
$410$		0			0
$411$		−	676.461i		−	0.0811858i
$412$		0			0
$413$		−	7624.44i		−	0.908413i
$414$		0			0
$415$	−18265.4			−2.16052
$416$		0			0
$417$		−	2747.00i		−	0.322593i
$418$		0			0
$419$		−	9104.73i		−	1.06156i	−0.847509	−	0.530782i	$-0.821898\pi$
$419$		−	9104.73i		−	1.06156i	0.847509	−	0.530782i	$-0.178102\pi$
$420$		0			0
$421$	−17033.4			−1.97186			−0.985932	−	0.167147i	$-0.946544\pi$
$421$	−17033.4			−1.97186			−0.985932	+	0.167147i	$0.946544\pi$
$422$		0			0
$423$		−	8910.21i		−	1.02418i
$424$		0			0
$425$		−	1719.84i		−	0.196293i
$426$		0			0
$427$			11836.9i			1.34151i
$428$		0			0
$429$	−4959.34	−	3166.81i	−0.558133	−	0.356399i
$430$		0			0
$431$	17431.7			1.94816			0.974081	−	0.226200i	$-0.0726303\pi$
$431$	17431.7			1.94816			0.974081	+	0.226200i	$0.0726303\pi$
$432$		0			0
$433$	2796.27			0.310347			0.155174	−	0.987887i	$-0.450406\pi$
$433$	2796.27			0.310347			0.155174	+	0.987887i	$0.450406\pi$
$434$		0			0
$435$	2901.34			0.319790
$436$		0			0
$437$		−	11652.7i		−	1.27557i
$438$		0			0
$439$	3459.06			0.376063			0.188032	−	0.982163i	$-0.439789\pi$
$439$	3459.06			0.376063			0.188032	+	0.982163i	$0.439789\pi$
$440$		0			0
$441$	−3499.18			−0.377841
$442$		0			0
$443$			14339.9i			1.53794i	0.639282	+	0.768972i	$0.279232\pi$
$443$			14339.9i			1.53794i	−0.639282	+	0.768972i	$0.720768\pi$
$444$		0			0
$445$	9077.67			0.967018
$446$		0			0
$447$	1852.76			0.196046
$448$		0			0
$449$	−14781.7			−1.55365			−0.776827	−	0.629715i	$-0.783172\pi$
$449$	−14781.7			−1.55365			−0.776827	+	0.629715i	$0.783172\pi$
$450$		0			0
$451$	5141.25	−	8051.37i	0.536789	−	0.840631i
$452$		0			0
$453$			6150.87i			0.637954i
$454$		0			0
$455$			15036.9i			1.54932i
$456$		0			0
$457$		−	6979.66i		−	0.714431i	−0.934022	−	0.357215i	$-0.883726\pi$
$457$		−	6979.66i		−	0.714431i	0.934022	−	0.357215i	$-0.116274\pi$
$458$		0			0
$459$	1215.20			0.123575
$460$		0			0
$461$			3695.48i			0.373353i	0.982421	+	0.186677i	$0.0597716\pi$
$461$			3695.48i			0.373353i	−0.982421	+	0.186677i	$0.940228\pi$
$462$		0			0
$463$			4226.03i			0.424191i	0.977249	+	0.212095i	$0.0680287\pi$
$463$			4226.03i			0.424191i	−0.977249	+	0.212095i	$0.931971\pi$
$464$		0			0
$465$	−3287.04			−0.327812
$466$		0			0
$467$		−	15465.2i		−	1.53243i	−0.642587	−	0.766213i	$-0.722139\pi$
$467$		−	15465.2i		−	1.53243i	0.642587	−	0.766213i	$-0.277861\pi$
$468$		0			0
$469$		−	1829.62i		−	0.180136i
$470$		0			0
$471$		−	2640.13i		−	0.258282i
$472$		0			0
$473$	−3964.22	−	2531.38i	−0.385360	−	0.246074i
$474$		0			0
$475$	−14356.0			−1.38674
$476$		0			0
$477$	−15334.7			−1.47197
$478$		0			0
$479$	2237.40			0.213423			0.106711	−	0.994290i	$-0.465968\pi$
$479$	2237.40			0.213423			0.106711	+	0.994290i	$0.465968\pi$
$480$		0			0
$481$			7400.56i			0.701531i
$482$		0			0
$483$	4348.63			0.409668
$484$		0			0
$485$	6339.05			0.593487
$486$		0			0
$487$			16047.9i			1.49322i	0.665261	+	0.746611i	$0.268320\pi$
$487$			16047.9i			1.49322i	−0.665261	+	0.746611i	$0.731680\pi$
$488$		0			0
$489$	−3085.51			−0.285340
$490$		0			0
$491$	−17992.0			−1.65371			−0.826853	−	0.562419i	$-0.809871\pi$
$491$	−17992.0			−1.65371			−0.826853	+	0.562419i	$0.809871\pi$
$492$		0			0
$493$	727.022			0.0664167
$494$		0			0
$495$	−11049.8	−	7055.89i	−1.00333	−	0.640684i
$496$		0			0
$497$			474.005i			0.0427807i
$498$		0			0
$499$		−	4355.95i		−	0.390780i	−0.980726	−	0.195390i	$-0.937403\pi$
$499$		−	4355.95i		−	0.390780i	0.980726	−	0.195390i	$-0.0625972\pi$
$500$		0			0
$501$			332.744i			0.0296725i
$502$		0			0
$503$	−18175.1			−1.61111			−0.805556	−	0.592520i	$-0.798133\pi$
$503$	−18175.1			−1.61111			−0.805556	+	0.592520i	$0.798133\pi$
$504$		0			0
$505$			21713.8i			1.91337i
$506$		0			0
$507$		−	5349.75i		−	0.468621i
$508$		0			0
$509$	15698.5			1.36704			0.683521	−	0.729931i	$-0.260448\pi$
$509$	15698.5			1.36704			0.683521	+	0.729931i	$0.260448\pi$
$510$		0			0
$511$			11243.0i			0.973311i
$512$		0			0
$513$		−	10143.7i		−	0.873013i
$514$		0			0
$515$		−	19115.6i		−	1.63560i
$516$		0			0
$517$	8298.18	−	12995.2i	0.705906	−	1.10547i
$518$		0			0
$519$	10222.1			0.864552
$520$		0			0
$521$	−16640.5			−1.39930			−0.699650	−	0.714485i	$-0.746661\pi$
$521$	−16640.5			−1.39930			−0.699650	+	0.714485i	$0.746661\pi$
$522$		0			0
$523$	−14801.3			−1.23751			−0.618754	−	0.785585i	$-0.712362\pi$
$523$	−14801.3			−1.23751			−0.618754	+	0.785585i	$0.712362\pi$
$524$		0			0
$525$		−	5357.48i		−	0.445371i
$526$		0			0
$527$	−823.670			−0.0680828
$528$		0			0
$529$	−5886.19			−0.483783
$530$		0			0
$531$			12081.4i			0.987361i
$532$		0			0
$533$	17361.5			1.41090
$534$		0			0
$535$	35846.2			2.89676
$536$		0			0
$537$	9774.39			0.785467
$538$		0			0
$539$	−5103.44	−	3258.83i	−0.407831	−	0.260423i
$540$		0			0
$541$			2608.25i			0.207278i	0.994615	+	0.103639i	$0.0330487\pi$
$541$			2608.25i			0.207278i	−0.994615	+	0.103639i	$0.966951\pi$
$542$		0			0
$543$			1892.10i			0.149536i
$544$		0			0
$545$			11920.0i			0.936871i
$546$		0			0
$547$	872.953			0.0682354			0.0341177	−	0.999418i	$-0.489138\pi$
$547$	872.953			0.0682354			0.0341177	+	0.999418i	$0.489138\pi$
$548$		0			0
$549$		−	18756.3i		−	1.45810i
$550$		0			0
$551$		−	6068.70i		−	0.469211i
$552$		0			0
$553$	−8023.60			−0.616995
$554$		0			0
$555$		−	4627.82i		−	0.353946i
$556$		0			0
$557$			8753.54i			0.665887i	0.942947	+	0.332944i	$0.108042\pi$
$557$			8753.54i			0.665887i	−0.942947	+	0.332944i	$0.891958\pi$
$558$		0			0
$559$		−	8548.19i		−	0.646780i
$560$		0			0
$561$	777.114	+	496.230i	0.0584844	+	0.0373455i
$562$		0			0
$563$	12393.3			0.927733			0.463867	−	0.885905i	$-0.346462\pi$
$563$	12393.3			0.927733			0.463867	+	0.885905i	$0.346462\pi$
$564$		0			0
$565$	22034.2			1.64068
$566$		0			0
$567$	−3788.29			−0.280588
$568$		0			0
$569$		−	1169.19i		−	0.0861425i	−0.999072	−	0.0430713i	$-0.986286\pi$
$569$		−	1169.19i		−	0.0861425i	0.999072	−	0.0430713i	$-0.0137142\pi$
$570$		0			0
$571$	5255.47			0.385175			0.192587	−	0.981280i	$-0.438312\pi$
$571$	5255.47			0.385175			0.192587	+	0.981280i	$0.438312\pi$
$572$		0			0
$573$	−8543.53			−0.622882
$574$		0			0
$575$			22241.4i			1.61309i
$576$		0			0
$577$	−418.397			−0.0301873			−0.0150937	−	0.999886i	$-0.504805\pi$
$577$	−418.397			−0.0301873			−0.0150937	+	0.999886i	$0.504805\pi$
$578$		0			0
$579$	−9063.50			−0.650546
$580$		0			0
$581$	−14257.8			−1.01809
$582$		0			0
$583$	−22365.1	−	14281.4i	−1.58880	−	1.01454i
$584$		0			0
$585$		−	23827.0i		−	1.68397i
$586$		0			0
$587$			19051.0i			1.33955i	0.742562	+	0.669777i	$0.233610\pi$
$587$			19051.0i			1.33955i	−0.742562	+	0.669777i	$0.766390\pi$
$588$		0			0
$589$			6875.45i			0.480981i
$590$		0			0
$591$	4861.74			0.338384
$592$		0			0
$593$			12215.5i			0.845918i	0.906149	+	0.422959i	$0.139009\pi$
$593$			12215.5i			0.845918i	−0.906149	+	0.422959i	$0.860991\pi$
$594$		0			0
$595$		−	2356.24i		−	0.162347i
$596$		0			0
$597$	−9687.53			−0.664127
$598$		0			0
$599$		−	2579.14i		−	0.175928i	−0.996124	−	0.0879640i	$-0.971964\pi$
$599$		−	2579.14i		−	0.175928i	0.996124	−	0.0879640i	$-0.0280360\pi$
$600$		0			0
$601$		−	21548.9i		−	1.46256i	−0.682079	−	0.731278i	$-0.738924\pi$
$601$		−	21548.9i		−	1.46256i	0.682079	−	0.731278i	$-0.261076\pi$
$602$		0			0
$603$			2899.15i			0.195792i
$604$		0			0
$605$	−9544.47	−	20581.5i	−0.641385	−	1.38307i
$606$		0			0
$607$	−6555.28			−0.438337			−0.219168	−	0.975687i	$-0.570334\pi$
$607$	−6555.28			−0.438337			−0.219168	+	0.975687i	$0.570334\pi$
$608$		0			0
$609$	2264.75			0.150694
$610$		0			0
$611$	28022.1			1.85540
$612$		0			0
$613$			9731.22i			0.641175i	0.947219	+	0.320587i	$0.103880\pi$
$613$			9731.22i			0.641175i	−0.947219	+	0.320587i	$0.896120\pi$
$614$		0			0
$615$	−10856.7			−0.711845
$616$		0			0
$617$	13257.7			0.865048			0.432524	−	0.901622i	$-0.357623\pi$
$617$	13257.7			0.865048			0.432524	+	0.901622i	$0.357623\pi$
$618$		0			0
$619$			5265.08i			0.341876i	0.985282	+	0.170938i	$0.0546798\pi$
$619$			5265.08i			0.341876i	−0.985282	+	0.170938i	$0.945320\pi$
$620$		0			0
$621$	−15715.3			−1.01551
$622$		0			0
$623$	7085.92			0.455685
$624$		0			0
$625$	−8915.45			−0.570589
$626$		0			0
$627$	4142.20	−	6486.83i	0.263833	−	0.413172i
$628$		0			0
$629$		−	1159.65i		−	0.0735105i
$630$		0			0
$631$		−	6525.46i		−	0.411687i	−0.978585	−	0.205844i	$-0.934006\pi$
$631$		−	6525.46i		−	0.411687i	0.978585	−	0.205844i	$-0.0659938\pi$
$632$		0			0
$633$		−	7708.00i		−	0.483990i
$634$		0			0
$635$	13475.5			0.842138
$636$		0			0
$637$		−	11004.7i		−	0.684495i
$638$		0			0
$639$		−	751.090i		−	0.0464987i
$640$		0			0
$641$	10522.0			0.648354			0.324177	−	0.945996i	$-0.394913\pi$
$641$	10522.0			0.648354			0.324177	+	0.945996i	$0.394913\pi$
$642$		0			0
$643$			25594.8i			1.56977i	0.619641	+	0.784885i	$0.287278\pi$
$643$			25594.8i			1.56977i	−0.619641	+	0.784885i	$0.712722\pi$
$644$		0			0
$645$			5345.47i			0.326322i
$646$		0			0
$647$			7636.35i			0.464012i	0.972714	+	0.232006i	$0.0745289\pi$
$647$			7636.35i			0.464012i	−0.972714	+	0.232006i	$0.925471\pi$
$648$		0			0
$649$	−11251.6	+	17620.3i	−0.680528	+	1.06573i
$650$		0			0
$651$	−2565.82			−0.154474
$652$		0			0
$653$	−24020.7			−1.43951			−0.719757	−	0.694226i	$-0.755747\pi$
$653$	−24020.7			−1.43951			−0.719757	+	0.694226i	$0.755747\pi$
$654$		0			0
$655$	−4485.71			−0.267590
$656$		0			0
$657$		−	17815.3i		−	1.05790i
$658$		0			0
$659$	−8775.80			−0.518751			−0.259375	−	0.965777i	$-0.583517\pi$
$659$	−8775.80			−0.518751			−0.259375	+	0.965777i	$0.583517\pi$
$660$		0			0
$661$	−29689.7			−1.74705			−0.873523	−	0.486783i	$-0.838170\pi$
$661$	−29689.7			−1.74705			−0.873523	+	0.486783i	$0.838170\pi$
$662$		0			0
$663$			1675.72i			0.0981590i
$664$		0			0
$665$	−19668.3			−1.14693
$666$		0			0
$667$	−9402.04			−0.545800
$668$		0			0
$669$	5878.35			0.339716
$670$		0			0
$671$	17467.9	−	27355.4i	1.00498	−	1.57383i
$672$		0			0
$673$		−	19440.7i		−	1.11350i	−0.830680	−	0.556750i	$-0.812048\pi$
$673$		−	19440.7i		−	1.11350i	0.830680	−	0.556750i	$-0.187952\pi$
$674$		0			0
$675$			19361.1i			1.10402i
$676$		0			0
$677$		−	25818.9i		−	1.46573i	−0.680373	−	0.732866i	$-0.738182\pi$
$677$		−	25818.9i		−	1.46573i	0.680373	−	0.732866i	$-0.261818\pi$
$678$		0			0
$679$	4948.19			0.279667
$680$		0			0
$681$		−	10175.7i		−	0.572590i
$682$		0			0
$683$		−	26132.5i		−	1.46403i	−0.681290	−	0.732014i	$-0.738581\pi$
$683$		−	26132.5i		−	1.46403i	0.681290	−	0.732014i	$-0.261419\pi$
$684$		0			0
$685$	4740.06			0.264392
$686$		0			0
$687$		−	4428.03i		−	0.245909i
$688$		0			0
$689$		−	48226.7i		−	2.66660i
$690$		0			0
$691$		−	12265.4i		−	0.675251i	−0.941281	−	0.337626i	$-0.890376\pi$
$691$		−	12265.4i		−	0.675251i	0.941281	−	0.337626i	$-0.109624\pi$
$692$		0			0
$693$	−8625.32	−	5507.74i	−0.472797	−	0.301907i
$694$		0			0
$695$	19248.7			1.05057
$696$		0			0
$697$	−2720.49			−0.147842
$698$		0			0
$699$	3883.17			0.210121
$700$		0			0
$701$		−	27576.1i		−	1.48578i	−0.669412	−	0.742891i	$-0.733454\pi$
$701$		−	27576.1i		−	1.48578i	0.669412	−	0.742891i	$-0.266546\pi$
$702$		0			0
$703$	−9679.95			−0.519326
$704$		0			0
$705$	−17523.1			−0.936113
$706$		0			0
$707$			16949.5i			0.901630i
$708$		0			0
$709$	−17099.4			−0.905755			−0.452878	−	0.891573i	$-0.649603\pi$
$709$	−17099.4			−0.905755			−0.452878	+	0.891573i	$0.649603\pi$
$710$		0			0
$711$	12713.9			0.670617
$712$		0			0
$713$	10651.9			0.559491
$714$		0			0
$715$	22190.3	−	34750.8i	1.16066	−	1.81763i
$716$		0			0
$717$		−	10968.6i		−	0.571312i
$718$		0			0
$719$			20718.6i			1.07465i	0.843375	+	0.537326i	$0.180566\pi$
$719$			20718.6i			1.07465i	−0.843375	+	0.537326i	$0.819434\pi$
$720$		0			0
$721$		−	14921.4i		−	0.770737i
$722$		0			0
$723$	−619.599			−0.0318716
$724$		0			0
$725$			11583.2i			0.593366i
$726$		0			0
$727$		−	28072.1i		−	1.43210i	−0.698050	−	0.716049i	$-0.745949\pi$
$727$		−	28072.1i		−	1.43210i	0.698050	−	0.716049i	$-0.254051\pi$
$728$		0			0
$729$	−1679.07			−0.0853054
$730$		0			0
$731$			1339.48i			0.0677733i
$732$		0			0
$733$			26418.1i			1.33121i	0.746306	+	0.665604i	$0.231826\pi$
$733$			26418.1i			1.33121i	−0.746306	+	0.665604i	$0.768174\pi$
$734$		0			0
$735$			6881.63i			0.345351i
$736$		0			0
$737$	−2700.01	+	4228.31i	−0.134947	+	0.211332i
$738$		0			0
$739$	−9710.73			−0.483376			−0.241688	−	0.970354i	$-0.577701\pi$
$739$	−9710.73			−0.483376			−0.241688	+	0.970354i	$0.577701\pi$
$740$		0			0
$741$	13987.8			0.693459
$742$		0			0
$743$	−16103.3			−0.795119			−0.397559	−	0.917576i	$-0.630143\pi$
$743$	−16103.3			−0.795119			−0.397559	+	0.917576i	$0.630143\pi$
$744$		0			0
$745$			12982.6i			0.638449i
$746$		0			0
$747$	22592.4			1.10657
$748$		0			0
$749$	27981.1			1.36503
$750$		0			0
$751$			4159.76i			0.202119i	0.994880	+	0.101060i	$0.0322233\pi$
$751$			4159.76i			0.202119i	−0.994880	+	0.101060i	$0.967777\pi$
$752$		0			0
$753$	16512.1			0.799115
$754$		0			0
$755$	−43100.1			−2.07758
$756$		0			0
$757$	−10791.8			−0.518142			−0.259071	−	0.965858i	$-0.583416\pi$
$757$	−10791.8			−0.518142			−0.259071	+	0.965858i	$0.583416\pi$
$758$		0			0
$759$	−10049.8	−	6417.38i	−0.480614	−	0.306899i
$760$		0			0
$761$			8976.40i			0.427588i	0.976879	+	0.213794i	$0.0685821\pi$
$761$			8976.40i			0.427588i	−0.976879	+	0.213794i	$0.931418\pi$
$762$		0			0
$763$			9304.58i			0.441479i
$764$		0			0
$765$			3733.62i			0.176456i
$766$		0			0
$767$	−37995.3			−1.78870
$768$		0			0
$769$		−	1696.25i		−	0.0795429i	−0.999209	−	0.0397714i	$-0.987337\pi$
$769$		−	1696.25i		−	0.0795429i	0.999209	−	0.0397714i	$-0.0126630\pi$
$770$		0			0
$771$			5116.01i			0.238974i
$772$		0			0
$773$	−11930.9			−0.555140			−0.277570	−	0.960705i	$-0.589529\pi$
$773$	−11930.9			−0.555140			−0.277570	+	0.960705i	$0.589529\pi$
$774$		0			0
$775$		−	13123.1i		−	0.608251i
$776$		0			0
$777$		−	3612.42i		−	0.166789i
$778$		0			0
$779$			22708.8i			1.04445i
$780$		0			0
$781$	699.499	−	1095.44i	0.0320487	−	0.0501894i
$782$		0			0
$783$	−8184.49			−0.373550
$784$		0			0
$785$	18499.8			0.841130
$786$		0			0
$787$	4960.04			0.224659			0.112329	−	0.993671i	$-0.464169\pi$
$787$	4960.04			0.224659			0.112329	+	0.993671i	$0.464169\pi$
$788$		0			0
$789$			4645.31i			0.209604i
$790$		0			0
$791$	17199.6			0.773133
$792$		0			0
$793$	58987.3			2.64149
$794$		0			0
$795$			30157.8i			1.34539i
$796$		0			0
$797$	−29754.3			−1.32240			−0.661200	−	0.750210i	$-0.729952\pi$
$797$	−29754.3			−1.32240			−0.661200	+	0.750210i	$0.729952\pi$
$798$		0			0
$799$	−4390.97			−0.194420
$800$		0			0
$801$	−11228.1			−0.495287
$802$		0			0
$803$	16591.6	−	25983.0i	0.729145	−	1.14187i
$804$		0			0
$805$			30471.5i			1.33414i
$806$		0			0
$807$			1826.66i			0.0796795i
$808$		0			0
$809$			26889.9i			1.16860i	0.811537	+	0.584301i	$0.198631\pi$
$809$			26889.9i			1.16860i	−0.811537	+	0.584301i	$0.801369\pi$
$810$		0			0
$811$	32017.3			1.38629			0.693143	−	0.720800i	$-0.256225\pi$
$811$	32017.3			1.38629			0.693143	+	0.720800i	$0.256225\pi$
$812$		0			0
$813$		−	16335.0i		−	0.704667i
$814$		0			0
$815$		−	21620.6i		−	0.929248i
$816$		0			0
$817$	11181.1			0.478795
$818$		0			0
$819$		−	18599.1i		−	0.793533i
$820$		0			0
$821$			27289.8i			1.16007i	0.814590	+	0.580037i	$0.196962\pi$
$821$			27289.8i			1.16007i	−0.814590	+	0.580037i	$0.803038\pi$
$822$		0			0
$823$		−	22033.4i		−	0.933215i	−0.884465	−	0.466607i	$-0.845476\pi$
$823$		−	22033.4i		−	0.933215i	0.884465	−	0.466607i	$-0.154524\pi$
$824$		0			0
$825$	−7906.15	+	12381.3i	−0.333645	+	0.522499i
$826$		0			0
$827$	−33737.0			−1.41856			−0.709280	−	0.704927i	$-0.750980\pi$
$827$	−33737.0			−1.41856			−0.709280	+	0.704927i	$0.750980\pi$
$828$		0			0
$829$	−16510.5			−0.691716			−0.345858	−	0.938287i	$-0.612412\pi$
$829$	−16510.5			−0.691716			−0.345858	+	0.938287i	$0.612412\pi$
$830$		0			0
$831$	−21248.4			−0.887001
$832$		0			0
$833$			1724.41i			0.0717253i
$834$		0			0
$835$	−2331.59			−0.0966322
$836$		0			0
$837$	9272.50			0.382921
$838$		0			0
$839$			45871.0i			1.88754i	0.330607	+	0.943769i	$0.392747\pi$
$839$			45871.0i			1.88754i	−0.330607	+	0.943769i	$0.607253\pi$
$840$		0			0
$841$	19492.4			0.799231
$842$		0			0
$843$	10968.2			0.448122
$844$		0			0
$845$	37486.5			1.52612
$846$		0			0
$847$	−7450.31	−	16065.7i	−0.302238	−	0.651741i
$848$		0			0
$849$			13836.4i			0.559321i
$850$		0			0
$851$			14996.8i			0.604095i
$852$		0			0
$853$			15785.3i			0.633619i	0.948489	+	0.316810i	$0.102612\pi$
$853$			15785.3i			0.633619i	−0.948489	+	0.316810i	$0.897388\pi$
$854$		0			0
$855$	31165.7			1.24660
$856$		0			0
$857$		−	34256.8i		−	1.36545i	−0.730676	−	0.682724i	$-0.760795\pi$
$857$		−	34256.8i		−	1.36545i	0.730676	−	0.682724i	$-0.239205\pi$
$858$		0			0
$859$			17672.6i			0.701959i	0.936383	+	0.350980i	$0.114151\pi$
$859$			17672.6i			0.701959i	−0.936383	+	0.350980i	$0.885849\pi$
$860$		0			0
$861$	−8474.62			−0.335440
$862$		0			0
$863$		−	42470.1i		−	1.67520i	−0.546282	−	0.837602i	$-0.683957\pi$
$863$		−	42470.1i		−	1.67520i	0.546282	−	0.837602i	$-0.316043\pi$
$864$		0			0
$865$			71628.1i			2.81553i
$866$		0			0
$867$			11688.4i			0.457853i
$868$		0			0
$869$	18542.8	+	11840.6i	0.723845	+	0.462215i
$870$		0			0
$871$	−9117.64			−0.354695
$872$		0			0
$873$	−7840.72			−0.303973
$874$		0			0
$875$	9192.34			0.355152
$876$		0			0
$877$			35718.5i			1.37529i	0.726048	+	0.687644i	$0.241355\pi$
$877$			35718.5i			1.37529i	−0.726048	+	0.687644i	$0.758645\pi$
$878$		0			0
$879$	2738.35			0.105077
$880$		0			0
$881$	−50710.8			−1.93926			−0.969631	−	0.244574i	$-0.921352\pi$
$881$	−50710.8			−1.93926			−0.969631	+	0.244574i	$0.921352\pi$
$882$		0			0
$883$			45180.8i			1.72192i	0.508674	+	0.860959i	$0.330136\pi$
$883$			45180.8i			1.72192i	−0.508674	+	0.860959i	$0.669864\pi$
$884$		0			0
$885$	23759.8			0.902459
$886$		0			0
$887$	−4529.58			−0.171464			−0.0857318	−	0.996318i	$-0.527323\pi$
$887$	−4529.58			−0.171464			−0.0857318	+	0.996318i	$0.527323\pi$
$888$		0			0
$889$	10518.8			0.396838
$890$		0			0
$891$	8754.87	+	5590.47i	0.329180	+	0.210200i
$892$		0			0
$893$			36652.9i			1.37351i
$894$		0			0
$895$			68490.6i			2.55798i
$896$		0			0
$897$		−	21670.8i		−	0.806652i
$898$		0			0
$899$	5547.48			0.205805
$900$		0			0
$901$			7556.98i			0.279422i
$902$		0			0
$903$			4172.61i			0.153772i
$904$		0			0
$905$	−13258.3			−0.486983
$906$		0			0
$907$		−	18622.6i		−	0.681757i	−0.940107	−	0.340879i	$-0.889276\pi$
$907$		−	18622.6i		−	0.681757i	0.940107	−	0.340879i	$-0.110724\pi$
$908$		0			0
$909$		−	26857.6i		−	0.979989i
$910$		0			0
$911$			8308.78i			0.302176i	0.988520	+	0.151088i	$0.0482777\pi$
$911$			8308.78i			0.302176i	−0.988520	+	0.151088i	$0.951722\pi$
$912$		0			0
$913$	32950.2	+	21040.5i	1.19441	+	0.762694i
$914$		0			0
$915$	−36886.8			−1.33272
$916$		0			0
$917$	−3501.50			−0.126096
$918$		0			0
$919$	30375.5			1.09031			0.545155	−	0.838335i	$-0.316471\pi$
$919$	30375.5			1.09031			0.545155	+	0.838335i	$0.316471\pi$
$920$		0			0
$921$			21414.8i			0.766170i
$922$		0			0
$923$	2362.13			0.0842368
$924$		0			0
$925$	18476.0			0.656742
$926$		0			0
$927$			23643.9i			0.837720i
$928$		0			0
$929$	−8885.01			−0.313787			−0.156893	−	0.987616i	$-0.550148\pi$
$929$	−8885.01			−0.313787			−0.156893	+	0.987616i	$0.550148\pi$
$930$		0			0
$931$	14394.2			0.506714
$932$		0			0
$933$	3714.34			0.130334
$934$		0			0
$935$	−3477.16	+	5445.35i	−0.121621	+	0.190462i
$936$		0			0
$937$		−	17067.2i		−	0.595048i	−0.954714	−	0.297524i	$-0.903839\pi$
$937$		−	17067.2i		−	0.595048i	0.954714	−	0.297524i	$-0.0961609\pi$
$938$		0			0
$939$			1774.20i			0.0616599i
$940$		0			0
$941$		−	8029.30i		−	0.278159i	−0.990281	−	0.139080i	$-0.955586\pi$
$941$		−	8029.30i		−	0.278159i	0.990281	−	0.139080i	$-0.0444144\pi$
$942$		0			0
$943$	35182.1			1.21494
$944$		0			0
$945$			26525.5i			0.913095i
$946$		0			0
$947$			2363.09i			0.0810878i	0.999178	+	0.0405439i	$0.0129091\pi$
$947$			2363.09i			0.0810878i	−0.999178	+	0.0405439i	$0.987091\pi$
$948$		0			0
$949$	56028.0			1.91649
$950$		0			0
$951$			2180.47i			0.0743498i
$952$		0			0
$953$			4486.13i			0.152487i	0.997089	+	0.0762435i	$0.0242926\pi$
$953$			4486.13i			0.152487i	−0.997089	+	0.0762435i	$0.975707\pi$
$954$		0			0
$955$		−	59865.8i		−	2.02850i
$956$		0			0
$957$	−5233.92	−	3342.15i	−0.176791	−	0.112891i
$958$		0			0
$959$	3700.04			0.124589
$960$		0			0
$961$	23506.1			0.789032
$962$		0			0
$963$	−44337.8			−1.48366
$964$		0			0
$965$		−	63509.3i		−	2.11859i
$966$		0			0
$967$	−30172.1			−1.00338			−0.501690	−	0.865047i	$-0.667288\pi$
$967$	−30172.1			−1.00338			−0.501690	+	0.865047i	$0.667288\pi$
$968$		0			0
$969$	−2191.84			−0.0726647
$970$		0			0
$971$		−	22408.0i		−	0.740585i	−0.928915	−	0.370292i	$-0.879257\pi$
$971$		−	22408.0i		−	0.740585i	0.928915	−	0.370292i	$-0.120743\pi$
$972$		0			0
$973$	15025.3			0.495055
$974$		0			0
$975$	−26698.2			−0.876951
$976$		0			0
$977$	2866.87			0.0938784			0.0469392	−	0.998898i	$-0.485053\pi$
$977$	2866.87			0.0938784			0.0469392	+	0.998898i	$0.485053\pi$
$978$		0			0
$979$	−16375.8	−	10456.9i	−0.534599	−	0.341371i
$980$		0			0
$981$		−	14743.7i		−	0.479847i
$982$		0			0
$983$		−	2310.73i		−	0.0749754i	−0.999297	−	0.0374877i	$-0.988064\pi$
$983$		−	2310.73i		−	0.0749754i	0.999297	−	0.0374877i	$-0.0119355\pi$
$984$		0			0
$985$			34066.9i			1.10199i
$986$		0			0
$987$	−13678.4			−0.441122
$988$		0			0
$989$		−	17322.5i		−	0.556948i
$990$		0			0
$991$		−	7176.98i		−	0.230055i	−0.993362	−	0.115027i	$-0.963304\pi$
$991$		−	7176.98i		−	0.230055i	0.993362	−	0.115027i	$-0.0366956\pi$
$992$		0			0
$993$	−24348.9			−0.778135
$994$		0			0
$995$		−	67882.0i		−	2.16282i
$996$		0			0
$997$			13428.4i			0.426561i	0.976991	+	0.213280i	$0.0684148\pi$
$997$			13428.4i			0.426561i	−0.976991	+	0.213280i	$0.931585\pi$
$998$		0			0
$999$			13054.8i			0.413448i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	704.4.e.g.703.6		36
4.3	odd	2	inner	704.4.e.g.703.17		36
8.3	odd	2		352.4.e.a.351.20	yes	36
8.5	even	2		352.4.e.a.351.31	yes	36
11.10	odd	2	inner	704.4.e.g.703.18		36
44.43	even	2	inner	704.4.e.g.703.5		36
88.21	odd	2		352.4.e.a.351.19	✓	36
88.43	even	2		352.4.e.a.351.32	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
352.4.e.a.351.19	✓	36	88.21	odd	2
352.4.e.a.351.20	yes	36	8.3	odd	2
352.4.e.a.351.31	yes	36	8.5	even	2
352.4.e.a.351.32	yes	36	88.43	even	2
704.4.e.g.703.5		36	44.43	even	2	inner
704.4.e.g.703.6		36	1.1	even	1	trivial
704.4.e.g.703.17		36	4.3	odd	2	inner
704.4.e.g.703.18		36	11.10	odd	2	inner

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:	\( N \)	\(=\)	\( 704 = 2^{6} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	704.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.43252i q^{3} -17.0450 q^{5} -13.3052 q^{7} +21.0829 q^{9} +(30.7486 + 19.6347i) q^{11} +66.3043i q^{13} -41.4623i q^{15} -10.3897i q^{17} -86.7262 q^{19} -32.3650i q^{21} +134.362i q^{23} +165.533 q^{25} +116.962i q^{27} +69.9754i q^{29} -79.2776i q^{31} +(-47.7618 + 74.7966i) q^{33} +226.787 q^{35} +111.615 q^{37} -161.286 q^{39} -261.845i q^{41} -128.924 q^{43} -359.358 q^{45} -422.628i q^{47} -165.973 q^{49} +25.2731 q^{51} -727.354 q^{53} +(-524.111 - 334.674i) q^{55} -210.963i q^{57} +573.045i q^{59} -889.645i q^{61} -280.511 q^{63} -1130.16i q^{65} +137.512i q^{67} -326.838 q^{69} -35.6256i q^{71} -845.012i q^{73} +402.662i q^{75} +(-409.115 - 261.243i) q^{77} +603.045 q^{79} +284.724 q^{81} +1071.60 q^{83} +177.093i q^{85} -170.216 q^{87} -532.570 q^{89} -882.189i q^{91} +192.844 q^{93} +1478.25 q^{95} -371.900 q^{97} +(648.269 + 413.956i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 364 q^{9} + 1236 q^{25} + 592 q^{33} - 984 q^{45} + 2372 q^{49} - 376 q^{53} + 2824 q^{69} + 2592 q^{77} + 6756 q^{81} + 512 q^{89} + 872 q^{93} + 3904 q^{97}+O(q^{100}) \) 36 * q - 364 * q^9 + 1236 * q^25 + 592 * q^33 - 984 * q^45 + 2372 * q^49 - 376 * q^53 + 2824 * q^69 + 2592 * q^77 + 6756 * q^81 + 512 * q^89 + 872 * q^93 + 3904 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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