LMFDB - Embedded newform 308.4.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [308,4,Mod(1,308)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(308, 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("308.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 308 = 2^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	308.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:	$18.1725882818$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	308.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-7.00000 q^{3} -1.00000 q^{5} +7.00000 q^{7} +22.0000 q^{9} +O(q^{10})$

$q-7.00000 q^{3} -1.00000 q^{5} +7.00000 q^{7} +22.0000 q^{9} +11.0000 q^{11} +12.0000 q^{13} +7.00000 q^{15} +2.00000 q^{17} +82.0000 q^{19} -49.0000 q^{21} +7.00000 q^{23} -124.000 q^{25} +35.0000 q^{27} -102.000 q^{29} -171.000 q^{31} -77.0000 q^{33} -7.00000 q^{35} -357.000 q^{37} -84.0000 q^{39} -114.000 q^{41} -344.000 q^{43} -22.0000 q^{45} +96.0000 q^{47} +49.0000 q^{49} -14.0000 q^{51} -430.000 q^{53} -11.0000 q^{55} -574.000 q^{57} -201.000 q^{59} -2.00000 q^{61} +154.000 q^{63} -12.0000 q^{65} +313.000 q^{67} -49.0000 q^{69} -579.000 q^{71} -438.000 q^{73} +868.000 q^{75} +77.0000 q^{77} +494.000 q^{79} -839.000 q^{81} +748.000 q^{83} -2.00000 q^{85} +714.000 q^{87} +457.000 q^{89} +84.0000 q^{91} +1197.00 q^{93} -82.0000 q^{95} -1037.00 q^{97} +242.000 q^{99} +O(q^{100})$

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
$3$	−7.00000	−1.34715	−0.673575	−	0.739119i	$-0.735242\pi$
$3$	−7.00000	−1.34715	−0.673575	+	0.739119i	$0.735242\pi$
$4$	0	0
$5$	−1.00000	−0.0894427	−0.0447214	−	0.998999i	$-0.514240\pi$
$5$	−1.00000	−0.0894427	−0.0447214	+	0.998999i	$0.514240\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	22.0000	0.814815
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	12.0000	0.256015	0.128008	−	0.991773i	$-0.459142\pi$
$13$	12.0000	0.256015	0.128008	+	0.991773i	$0.459142\pi$
$14$	0	0
$15$	7.00000	0.120493
$16$	0	0
$17$	2.00000	0.0285336	0.0142668	−	0.999898i	$-0.495459\pi$
$17$	2.00000	0.0285336	0.0142668	+	0.999898i	$0.495459\pi$
$18$	0	0
$19$	82.0000	0.990110	0.495055	−	0.868862i	$-0.335148\pi$
$19$	82.0000	0.990110	0.495055	+	0.868862i	$0.335148\pi$
$20$	0	0
$21$	−49.0000	−0.509175
$22$	0	0
$23$	7.00000	0.0634609	0.0317305	−	0.999496i	$-0.489898\pi$
$23$	7.00000	0.0634609	0.0317305	+	0.999496i	$0.489898\pi$
$24$	0	0
$25$	−124.000	−0.992000
$26$	0	0
$27$	35.0000	0.249472
$28$	0	0
$29$	−102.000	−0.653135	−0.326568	−	0.945174i	$-0.605892\pi$
$29$	−102.000	−0.653135	−0.326568	+	0.945174i	$0.605892\pi$
$30$	0	0
$31$	−171.000	−0.990726	−0.495363	−	0.868686i	$-0.664965\pi$
$31$	−171.000	−0.990726	−0.495363	+	0.868686i	$0.664965\pi$
$32$	0	0
$33$	−77.0000	−0.406181
$34$	0	0
$35$	−7.00000	−0.0338062
$36$	0	0
$37$	−357.000	−1.58623	−0.793114	−	0.609073i	$-0.791542\pi$
$37$	−357.000	−1.58623	−0.793114	+	0.609073i	$0.791542\pi$
$38$	0	0
$39$	−84.0000	−0.344891
$40$	0	0
$41$	−114.000	−0.434239	−0.217120	−	0.976145i	$-0.569666\pi$
$41$	−114.000	−0.434239	−0.217120	+	0.976145i	$0.569666\pi$
$42$	0	0
$43$	−344.000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−344.000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	0	0
$45$	−22.0000	−0.0728793
$46$	0	0
$47$	96.0000	0.297937	0.148969	−	0.988842i	$-0.452405\pi$
$47$	96.0000	0.297937	0.148969	+	0.988842i	$0.452405\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−14.0000	−0.0384391
$52$	0	0
$53$	−430.000	−1.11443	−0.557217	−	0.830367i	$-0.688131\pi$
$53$	−430.000	−1.11443	−0.557217	+	0.830367i	$0.688131\pi$
$54$	0	0
$55$	−11.0000	−0.0269680
$56$	0	0
$57$	−574.000	−1.33383
$58$	0	0
$59$	−201.000	−0.443525	−0.221762	−	0.975101i	$-0.571181\pi$
$59$	−201.000	−0.443525	−0.221762	+	0.975101i	$0.571181\pi$
$60$	0	0
$61$	−2.00000	−0.00419793	−0.00209897	−	0.999998i	$-0.500668\pi$
$61$	−2.00000	−0.00419793	−0.00209897	+	0.999998i	$0.500668\pi$
$62$	0	0
$63$	154.000	0.307971
$64$	0	0
$65$	−12.0000	−0.0228987
$66$	0	0
$67$	313.000	0.570732	0.285366	−	0.958419i	$-0.407885\pi$
$67$	313.000	0.570732	0.285366	+	0.958419i	$0.407885\pi$
$68$	0	0
$69$	−49.0000	−0.0854914
$70$	0	0
$71$	−579.000	−0.967812	−0.483906	−	0.875120i	$-0.660782\pi$
$71$	−579.000	−0.967812	−0.483906	+	0.875120i	$0.660782\pi$
$72$	0	0
$73$	−438.000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−438.000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	0	0
$75$	868.000	1.33637
$76$	0	0
$77$	77.0000	0.113961
$78$	0	0
$79$	494.000	0.703536	0.351768	−	0.936087i	$-0.385581\pi$
$79$	494.000	0.703536	0.351768	+	0.936087i	$0.385581\pi$
$80$	0	0
$81$	−839.000	−1.15089
$82$	0	0
$83$	748.000	0.989201	0.494600	−	0.869121i	$-0.335314\pi$
$83$	748.000	0.989201	0.494600	+	0.869121i	$0.335314\pi$
$84$	0	0
$85$	−2.00000	−0.00255212
$86$	0	0
$87$	714.000	0.879872
$88$	0	0
$89$	457.000	0.544291	0.272146	−	0.962256i	$-0.412267\pi$
$89$	457.000	0.544291	0.272146	+	0.962256i	$0.412267\pi$
$90$	0	0
$91$	84.0000	0.0967648
$92$	0	0
$93$	1197.00	1.33466
$94$	0	0
$95$	−82.0000	−0.0885581
$96$	0	0
$97$	−1037.00	−1.08548	−0.542739	−	0.839901i	$-0.682613\pi$
$97$	−1037.00	−1.08548	−0.542739	+	0.839901i	$0.682613\pi$
$98$	0	0
$99$	242.000	0.245676
$100$	0	0
$101$	68.0000	0.0669926	0.0334963	−	0.999439i	$-0.489336\pi$
$101$	68.0000	0.0669926	0.0334963	+	0.999439i	$0.489336\pi$
$102$	0	0
$103$	−468.000	−0.447703	−0.223852	−	0.974623i	$-0.571863\pi$
$103$	−468.000	−0.447703	−0.223852	+	0.974623i	$0.571863\pi$
$104$	0	0
$105$	49.0000	0.0455420
$106$	0	0
$107$	1582.00	1.42932	0.714662	−	0.699470i	$-0.246580\pi$
$107$	1582.00	1.42932	0.714662	+	0.699470i	$0.246580\pi$
$108$	0	0
$109$	516.000	0.453430	0.226715	−	0.973961i	$-0.427201\pi$
$109$	516.000	0.453430	0.226715	+	0.973961i	$0.427201\pi$
$110$	0	0
$111$	2499.00	2.13689
$112$	0	0
$113$	−1627.00	−1.35447	−0.677236	−	0.735766i	$-0.736822\pi$
$113$	−1627.00	−1.35447	−0.677236	+	0.735766i	$0.736822\pi$
$114$	0	0
$115$	−7.00000	−0.00567612
$116$	0	0
$117$	264.000	0.208605
$118$	0	0
$119$	14.0000	0.0107847
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	798.000	0.584986
$124$	0	0
$125$	249.000	0.178170
$126$	0	0
$127$	−2790.00	−1.94939	−0.974695	−	0.223540i	$-0.928239\pi$
$127$	−2790.00	−1.94939	−0.974695	+	0.223540i	$0.928239\pi$
$128$	0	0
$129$	2408.00	1.64351
$130$	0	0
$131$	730.000	0.486873	0.243437	−	0.969917i	$-0.421725\pi$
$131$	730.000	0.486873	0.243437	+	0.969917i	$0.421725\pi$
$132$	0	0
$133$	574.000	0.374226
$134$	0	0
$135$	−35.0000	−0.0223135
$136$	0	0
$137$	−1819.00	−1.13436	−0.567181	−	0.823593i	$-0.691966\pi$
$137$	−1819.00	−1.13436	−0.567181	+	0.823593i	$0.691966\pi$
$138$	0	0
$139$	−802.000	−0.489387	−0.244693	−	0.969601i	$-0.578687\pi$
$139$	−802.000	−0.489387	−0.244693	+	0.969601i	$0.578687\pi$
$140$	0	0
$141$	−672.000	−0.401366
$142$	0	0
$143$	132.000	0.0771916
$144$	0	0
$145$	102.000	0.0584182
$146$	0	0
$147$	−343.000	−0.192450
$148$	0	0
$149$	314.000	0.172644	0.0863218	−	0.996267i	$-0.472489\pi$
$149$	314.000	0.172644	0.0863218	+	0.996267i	$0.472489\pi$
$150$	0	0
$151$	766.000	0.412822	0.206411	−	0.978465i	$-0.433822\pi$
$151$	766.000	0.412822	0.206411	+	0.978465i	$0.433822\pi$
$152$	0	0
$153$	44.0000	0.0232496
$154$	0	0
$155$	171.000	0.0886132
$156$	0	0
$157$	−1681.00	−0.854512	−0.427256	−	0.904131i	$-0.640520\pi$
$157$	−1681.00	−0.854512	−0.427256	+	0.904131i	$0.640520\pi$
$158$	0	0
$159$	3010.00	1.50131
$160$	0	0
$161$	49.0000	0.0239860
$162$	0	0
$163$	724.000	0.347902	0.173951	−	0.984754i	$-0.444347\pi$
$163$	724.000	0.347902	0.173951	+	0.984754i	$0.444347\pi$
$164$	0	0
$165$	77.0000	0.0363300
$166$	0	0
$167$	−1242.00	−0.575502	−0.287751	−	0.957705i	$-0.592908\pi$
$167$	−1242.00	−0.575502	−0.287751	+	0.957705i	$0.592908\pi$
$168$	0	0
$169$	−2053.00	−0.934456
$170$	0	0
$171$	1804.00	0.806756
$172$	0	0
$173$	2432.00	1.06880	0.534398	−	0.845233i	$-0.320539\pi$
$173$	2432.00	1.06880	0.534398	+	0.845233i	$0.320539\pi$
$174$	0	0
$175$	−868.000	−0.374941
$176$	0	0
$177$	1407.00	0.597495
$178$	0	0
$179$	−2035.00	−0.849738	−0.424869	−	0.905255i	$-0.639680\pi$
$179$	−2035.00	−0.849738	−0.424869	+	0.905255i	$0.639680\pi$
$180$	0	0
$181$	1405.00	0.576977	0.288488	−	0.957483i	$-0.406847\pi$
$181$	1405.00	0.576977	0.288488	+	0.957483i	$0.406847\pi$
$182$	0	0
$183$	14.0000	0.00565524
$184$	0	0
$185$	357.000	0.141877
$186$	0	0
$187$	22.0000	0.00860320
$188$	0	0
$189$	245.000	0.0942917
$190$	0	0
$191$	−1671.00	−0.633033	−0.316517	−	0.948587i	$-0.602513\pi$
$191$	−1671.00	−0.633033	−0.316517	+	0.948587i	$0.602513\pi$
$192$	0	0
$193$	−2066.00	−0.770539	−0.385269	−	0.922804i	$-0.625891\pi$
$193$	−2066.00	−0.770539	−0.385269	+	0.922804i	$0.625891\pi$
$194$	0	0
$195$	84.0000	0.0308480
$196$	0	0
$197$	3442.00	1.24483	0.622417	−	0.782686i	$-0.286151\pi$
$197$	3442.00	1.24483	0.622417	+	0.782686i	$0.286151\pi$
$198$	0	0
$199$	3008.00	1.07151	0.535757	−	0.844372i	$-0.320026\pi$
$199$	3008.00	1.07151	0.535757	+	0.844372i	$0.320026\pi$
$200$	0	0
$201$	−2191.00	−0.768862
$202$	0	0
$203$	−714.000	−0.246862
$204$	0	0
$205$	114.000	0.0388395
$206$	0	0
$207$	154.000	0.0517089
$208$	0	0
$209$	902.000	0.298529
$210$	0	0
$211$	−778.000	−0.253838	−0.126919	−	0.991913i	$-0.540509\pi$
$211$	−778.000	−0.253838	−0.126919	+	0.991913i	$0.540509\pi$
$212$	0	0
$213$	4053.00	1.30379
$214$	0	0
$215$	344.000	0.109119
$216$	0	0
$217$	−1197.00	−0.374459
$218$	0	0
$219$	3066.00	0.946032
$220$	0	0
$221$	24.0000	0.00730504
$222$	0	0
$223$	−2739.00	−0.822498	−0.411249	−	0.911523i	$-0.634907\pi$
$223$	−2739.00	−0.822498	−0.411249	+	0.911523i	$0.634907\pi$
$224$	0	0
$225$	−2728.00	−0.808296
$226$	0	0
$227$	−732.000	−0.214029	−0.107014	−	0.994257i	$-0.534129\pi$
$227$	−732.000	−0.214029	−0.107014	+	0.994257i	$0.534129\pi$
$228$	0	0
$229$	−823.000	−0.237491	−0.118745	−	0.992925i	$-0.537887\pi$
$229$	−823.000	−0.237491	−0.118745	+	0.992925i	$0.537887\pi$
$230$	0	0
$231$	−539.000	−0.153522
$232$	0	0
$233$	1462.00	0.411068	0.205534	−	0.978650i	$-0.434107\pi$
$233$	1462.00	0.411068	0.205534	+	0.978650i	$0.434107\pi$
$234$	0	0
$235$	−96.0000	−0.0266483
$236$	0	0
$237$	−3458.00	−0.947769
$238$	0	0
$239$	1700.00	0.460100	0.230050	−	0.973179i	$-0.426111\pi$
$239$	1700.00	0.460100	0.230050	+	0.973179i	$0.426111\pi$
$240$	0	0
$241$	3460.00	0.924806	0.462403	−	0.886670i	$-0.346987\pi$
$241$	3460.00	0.924806	0.462403	+	0.886670i	$0.346987\pi$
$242$	0	0
$243$	4928.00	1.30095
$244$	0	0
$245$	−49.0000	−0.0127775
$246$	0	0
$247$	984.000	0.253483
$248$	0	0
$249$	−5236.00	−1.33260
$250$	0	0
$251$	−265.000	−0.0666400	−0.0333200	−	0.999445i	$-0.510608\pi$
$251$	−265.000	−0.0666400	−0.0333200	+	0.999445i	$0.510608\pi$
$252$	0	0
$253$	77.0000	0.0191342
$254$	0	0
$255$	14.0000	0.00343809
$256$	0	0
$257$	−6502.00	−1.57815	−0.789073	−	0.614299i	$-0.789439\pi$
$257$	−6502.00	−1.57815	−0.789073	+	0.614299i	$0.789439\pi$
$258$	0	0
$259$	−2499.00	−0.599538
$260$	0	0
$261$	−2244.00	−0.532184
$262$	0	0
$263$	−1398.00	−0.327773	−0.163887	−	0.986479i	$-0.552403\pi$
$263$	−1398.00	−0.327773	−0.163887	+	0.986479i	$0.552403\pi$
$264$	0	0
$265$	430.000	0.0996781
$266$	0	0
$267$	−3199.00	−0.733242
$268$	0	0
$269$	3686.00	0.835462	0.417731	−	0.908571i	$-0.362825\pi$
$269$	3686.00	0.835462	0.417731	+	0.908571i	$0.362825\pi$
$270$	0	0
$271$	−3808.00	−0.853578	−0.426789	−	0.904351i	$-0.640355\pi$
$271$	−3808.00	−0.853578	−0.426789	+	0.904351i	$0.640355\pi$
$272$	0	0
$273$	−588.000	−0.130357
$274$	0	0
$275$	−1364.00	−0.299099
$276$	0	0
$277$	−8872.00	−1.92443	−0.962214	−	0.272293i	$-0.912218\pi$
$277$	−8872.00	−1.92443	−0.962214	+	0.272293i	$0.912218\pi$
$278$	0	0
$279$	−3762.00	−0.807258
$280$	0	0
$281$	−2788.00	−0.591879	−0.295940	−	0.955207i	$-0.595633\pi$
$281$	−2788.00	−0.591879	−0.295940	+	0.955207i	$0.595633\pi$
$282$	0	0
$283$	160.000	0.0336078	0.0168039	−	0.999859i	$-0.494651\pi$
$283$	160.000	0.0336078	0.0168039	+	0.999859i	$0.494651\pi$
$284$	0	0
$285$	574.000	0.119301
$286$	0	0
$287$	−798.000	−0.164127
$288$	0	0
$289$	−4909.00	−0.999186
$290$	0	0
$291$	7259.00	1.46230
$292$	0	0
$293$	−1142.00	−0.227701	−0.113850	−	0.993498i	$-0.536318\pi$
$293$	−1142.00	−0.227701	−0.113850	+	0.993498i	$0.536318\pi$
$294$	0	0
$295$	201.000	0.0396701
$296$	0	0
$297$	385.000	0.0752187
$298$	0	0
$299$	84.0000	0.0162470
$300$	0	0
$301$	−2408.00	−0.461112
$302$	0	0
$303$	−476.000	−0.0902491
$304$	0	0
$305$	2.00000	0.000375474 0
$306$	0	0
$307$	−10220.0	−1.89996	−0.949978	−	0.312318i	$-0.898894\pi$
$307$	−10220.0	−1.89996	−0.949978	+	0.312318i	$0.898894\pi$
$308$	0	0
$309$	3276.00	0.603123
$310$	0	0
$311$	4864.00	0.886856	0.443428	−	0.896310i	$-0.353762\pi$
$311$	4864.00	0.886856	0.443428	+	0.896310i	$0.353762\pi$
$312$	0	0
$313$	−3703.00	−0.668709	−0.334355	−	0.942447i	$-0.608518\pi$
$313$	−3703.00	−0.668709	−0.334355	+	0.942447i	$0.608518\pi$
$314$	0	0
$315$	−154.000	−0.0275458
$316$	0	0
$317$	81.0000	0.0143515	0.00717573	−	0.999974i	$-0.497716\pi$
$317$	81.0000	0.0143515	0.00717573	+	0.999974i	$0.497716\pi$
$318$	0	0
$319$	−1122.00	−0.196928
$320$	0	0
$321$	−11074.0	−1.92552
$322$	0	0
$323$	164.000	0.0282514
$324$	0	0
$325$	−1488.00	−0.253967
$326$	0	0
$327$	−3612.00	−0.610838
$328$	0	0
$329$	672.000	0.112610
$330$	0	0
$331$	2675.00	0.444203	0.222102	−	0.975024i	$-0.428708\pi$
$331$	2675.00	0.444203	0.222102	+	0.975024i	$0.428708\pi$
$332$	0	0
$333$	−7854.00	−1.29248
$334$	0	0
$335$	−313.000	−0.0510478
$336$	0	0
$337$	1422.00	0.229855	0.114928	−	0.993374i	$-0.463336\pi$
$337$	1422.00	0.229855	0.114928	+	0.993374i	$0.463336\pi$
$338$	0	0
$339$	11389.0	1.82468
$340$	0	0
$341$	−1881.00	−0.298715
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	49.0000	0.00764658
$346$	0	0
$347$	9110.00	1.40937	0.704684	−	0.709522i	$-0.251089\pi$
$347$	9110.00	1.40937	0.704684	+	0.709522i	$0.251089\pi$
$348$	0	0
$349$	−114.000	−0.0174850	−0.00874252	−	0.999962i	$-0.502783\pi$
$349$	−114.000	−0.0174850	−0.00874252	+	0.999962i	$0.502783\pi$
$350$	0	0
$351$	420.000	0.0638688
$352$	0	0
$353$	12465.0	1.87945	0.939724	−	0.341934i	$-0.111082\pi$
$353$	12465.0	1.87945	0.939724	+	0.341934i	$0.111082\pi$
$354$	0	0
$355$	579.000	0.0865637
$356$	0	0
$357$	−98.0000	−0.0145286
$358$	0	0
$359$	7944.00	1.16788	0.583939	−	0.811797i	$-0.301511\pi$
$359$	7944.00	1.16788	0.583939	+	0.811797i	$0.301511\pi$
$360$	0	0
$361$	−135.000	−0.0196822
$362$	0	0
$363$	−847.000	−0.122468
$364$	0	0
$365$	438.000	0.0628109
$366$	0	0
$367$	−1591.00	−0.226293	−0.113146	−	0.993578i	$-0.536093\pi$
$367$	−1591.00	−0.226293	−0.113146	+	0.993578i	$0.536093\pi$
$368$	0	0
$369$	−2508.00	−0.353825
$370$	0	0
$371$	−3010.00	−0.421217
$372$	0	0
$373$	6892.00	0.956714	0.478357	−	0.878166i	$-0.341233\pi$
$373$	6892.00	0.956714	0.478357	+	0.878166i	$0.341233\pi$
$374$	0	0
$375$	−1743.00	−0.240022
$376$	0	0
$377$	−1224.00	−0.167213
$378$	0	0
$379$	3935.00	0.533318	0.266659	−	0.963791i	$-0.414080\pi$
$379$	3935.00	0.533318	0.266659	+	0.963791i	$0.414080\pi$
$380$	0	0
$381$	19530.0	2.62612
$382$	0	0
$383$	7325.00	0.977259	0.488629	−	0.872492i	$-0.337497\pi$
$383$	7325.00	0.977259	0.488629	+	0.872492i	$0.337497\pi$
$384$	0	0
$385$	−77.0000	−0.0101929
$386$	0	0
$387$	−7568.00	−0.994065
$388$	0	0
$389$	8937.00	1.16484	0.582421	−	0.812887i	$-0.302105\pi$
$389$	8937.00	1.16484	0.582421	+	0.812887i	$0.302105\pi$
$390$	0	0
$391$	14.0000	0.00181077
$392$	0	0
$393$	−5110.00	−0.655892
$394$	0	0
$395$	−494.000	−0.0629262
$396$	0	0
$397$	−7526.00	−0.951434	−0.475717	−	0.879599i	$-0.657811\pi$
$397$	−7526.00	−0.951434	−0.475717	+	0.879599i	$0.657811\pi$
$398$	0	0
$399$	−4018.00	−0.504139
$400$	0	0
$401$	6202.00	0.772352	0.386176	−	0.922425i	$-0.373796\pi$
$401$	6202.00	0.772352	0.386176	+	0.922425i	$0.373796\pi$
$402$	0	0
$403$	−2052.00	−0.253641
$404$	0	0
$405$	839.000	0.102939
$406$	0	0
$407$	−3927.00	−0.478266
$408$	0	0
$409$	1366.00	0.165145	0.0825726	−	0.996585i	$-0.473686\pi$
$409$	1366.00	0.165145	0.0825726	+	0.996585i	$0.473686\pi$
$410$	0	0
$411$	12733.0	1.52816
$412$	0	0
$413$	−1407.00	−0.167637
$414$	0	0
$415$	−748.000	−0.0884768
$416$	0	0
$417$	5614.00	0.659278
$418$	0	0
$419$	6384.00	0.744341	0.372170	−	0.928164i	$-0.378614\pi$
$419$	6384.00	0.744341	0.372170	+	0.928164i	$0.378614\pi$
$420$	0	0
$421$	5534.00	0.640643	0.320321	−	0.947309i	$-0.396209\pi$
$421$	5534.00	0.640643	0.320321	+	0.947309i	$0.396209\pi$
$422$	0	0
$423$	2112.00	0.242763
$424$	0	0
$425$	−248.000	−0.0283053
$426$	0	0
$427$	−14.0000	−0.00158667
$428$	0	0
$429$	−924.000	−0.103989
$430$	0	0
$431$	−10740.0	−1.20030	−0.600148	−	0.799889i	$-0.704892\pi$
$431$	−10740.0	−1.20030	−0.600148	+	0.799889i	$0.704892\pi$
$432$	0	0
$433$	16151.0	1.79253	0.896267	−	0.443514i	$-0.146268\pi$
$433$	16151.0	1.79253	0.896267	+	0.443514i	$0.146268\pi$
$434$	0	0
$435$	−714.000	−0.0786981
$436$	0	0
$437$	574.000	0.0628333
$438$	0	0
$439$	5914.00	0.642961	0.321480	−	0.946916i	$-0.395820\pi$
$439$	5914.00	0.642961	0.321480	+	0.946916i	$0.395820\pi$
$440$	0	0
$441$	1078.00	0.116402
$442$	0	0
$443$	7237.00	0.776163	0.388082	−	0.921625i	$-0.373138\pi$
$443$	7237.00	0.776163	0.388082	+	0.921625i	$0.373138\pi$
$444$	0	0
$445$	−457.000	−0.0486829
$446$	0	0
$447$	−2198.00	−0.232577
$448$	0	0
$449$	9839.00	1.03415	0.517073	−	0.855942i	$-0.327022\pi$
$449$	9839.00	1.03415	0.517073	+	0.855942i	$0.327022\pi$
$450$	0	0
$451$	−1254.00	−0.130928
$452$	0	0
$453$	−5362.00	−0.556134
$454$	0	0
$455$	−84.0000	−0.00865490
$456$	0	0
$457$	−6824.00	−0.698497	−0.349249	−	0.937030i	$-0.613563\pi$
$457$	−6824.00	−0.698497	−0.349249	+	0.937030i	$0.613563\pi$
$458$	0	0
$459$	70.0000	0.00711834
$460$	0	0
$461$	13922.0	1.40653	0.703267	−	0.710926i	$-0.251724\pi$
$461$	13922.0	1.40653	0.703267	+	0.710926i	$0.251724\pi$
$462$	0	0
$463$	−1551.00	−0.155683	−0.0778413	−	0.996966i	$-0.524803\pi$
$463$	−1551.00	−0.155683	−0.0778413	+	0.996966i	$0.524803\pi$
$464$	0	0
$465$	−1197.00	−0.119375
$466$	0	0
$467$	10719.0	1.06213	0.531067	−	0.847330i	$-0.321791\pi$
$467$	10719.0	1.06213	0.531067	+	0.847330i	$0.321791\pi$
$468$	0	0
$469$	2191.00	0.215716
$470$	0	0
$471$	11767.0	1.15116
$472$	0	0
$473$	−3784.00	−0.367840
$474$	0	0
$475$	−10168.0	−0.982189
$476$	0	0
$477$	−9460.00	−0.908058
$478$	0	0
$479$	−12252.0	−1.16870	−0.584351	−	0.811501i	$-0.698651\pi$
$479$	−12252.0	−1.16870	−0.584351	+	0.811501i	$0.698651\pi$
$480$	0	0
$481$	−4284.00	−0.406099
$482$	0	0
$483$	−343.000	−0.0323127
$484$	0	0
$485$	1037.00	0.0970881
$486$	0	0
$487$	−8521.00	−0.792861	−0.396431	−	0.918065i	$-0.629751\pi$
$487$	−8521.00	−0.792861	−0.396431	+	0.918065i	$0.629751\pi$
$488$	0	0
$489$	−5068.00	−0.468677
$490$	0	0
$491$	7946.00	0.730342	0.365171	−	0.930940i	$-0.381011\pi$
$491$	7946.00	0.730342	0.365171	+	0.930940i	$0.381011\pi$
$492$	0	0
$493$	−204.000	−0.0186363
$494$	0	0
$495$	−242.000	−0.0219739
$496$	0	0
$497$	−4053.00	−0.365799
$498$	0	0
$499$	−4660.00	−0.418057	−0.209028	−	0.977910i	$-0.567030\pi$
$499$	−4660.00	−0.418057	−0.209028	+	0.977910i	$0.567030\pi$
$500$	0	0
$501$	8694.00	0.775288
$502$	0	0
$503$	−18880.0	−1.67359	−0.836797	−	0.547514i	$-0.815574\pi$
$503$	−18880.0	−1.67359	−0.836797	+	0.547514i	$0.815574\pi$
$504$	0	0
$505$	−68.0000	−0.00599200
$506$	0	0
$507$	14371.0	1.25885
$508$	0	0
$509$	−1999.00	−0.174075	−0.0870374	−	0.996205i	$-0.527740\pi$
$509$	−1999.00	−0.174075	−0.0870374	+	0.996205i	$0.527740\pi$
$510$	0	0
$511$	−3066.00	−0.265424
$512$	0	0
$513$	2870.00	0.247005
$514$	0	0
$515$	468.000	0.0400438
$516$	0	0
$517$	1056.00	0.0898314
$518$	0	0
$519$	−17024.0	−1.43983
$520$	0	0
$521$	18063.0	1.51891	0.759457	−	0.650557i	$-0.225465\pi$
$521$	18063.0	1.51891	0.759457	+	0.650557i	$0.225465\pi$
$522$	0	0
$523$	7712.00	0.644784	0.322392	−	0.946606i	$-0.395513\pi$
$523$	7712.00	0.644784	0.322392	+	0.946606i	$0.395513\pi$
$524$	0	0
$525$	6076.00	0.505102
$526$	0	0
$527$	−342.000	−0.0282690
$528$	0	0
$529$	−12118.0	−0.995973
$530$	0	0
$531$	−4422.00	−0.361391
$532$	0	0
$533$	−1368.00	−0.111172
$534$	0	0
$535$	−1582.00	−0.127843
$536$	0	0
$537$	14245.0	1.14472
$538$	0	0
$539$	539.000	0.0430730
$540$	0	0
$541$	−704.000	−0.0559470	−0.0279735	−	0.999609i	$-0.508905\pi$
$541$	−704.000	−0.0559470	−0.0279735	+	0.999609i	$0.508905\pi$
$542$	0	0
$543$	−9835.00	−0.777275
$544$	0	0
$545$	−516.000	−0.0405560
$546$	0	0
$547$	−20680.0	−1.61648	−0.808239	−	0.588855i	$-0.799579\pi$
$547$	−20680.0	−1.61648	−0.808239	+	0.588855i	$0.799579\pi$
$548$	0	0
$549$	−44.0000	−0.00342054
$550$	0	0
$551$	−8364.00	−0.646676
$552$	0	0
$553$	3458.00	0.265912
$554$	0	0
$555$	−2499.00	−0.191129
$556$	0	0
$557$	−19266.0	−1.46558	−0.732789	−	0.680456i	$-0.761782\pi$
$557$	−19266.0	−1.46558	−0.732789	+	0.680456i	$0.761782\pi$
$558$	0	0
$559$	−4128.00	−0.312336
$560$	0	0
$561$	−154.000	−0.0115898
$562$	0	0
$563$	−18956.0	−1.41901	−0.709503	−	0.704703i	$-0.751080\pi$
$563$	−18956.0	−1.41901	−0.709503	+	0.704703i	$0.751080\pi$
$564$	0	0
$565$	1627.00	0.121148
$566$	0	0
$567$	−5873.00	−0.434996
$568$	0	0
$569$	13726.0	1.01129	0.505645	−	0.862742i	$-0.331255\pi$
$569$	13726.0	1.01129	0.505645	+	0.862742i	$0.331255\pi$
$570$	0	0
$571$	8620.00	0.631761	0.315881	−	0.948799i	$-0.397700\pi$
$571$	8620.00	0.631761	0.315881	+	0.948799i	$0.397700\pi$
$572$	0	0
$573$	11697.0	0.852791
$574$	0	0
$575$	−868.000	−0.0629532
$576$	0	0
$577$	−5761.00	−0.415656	−0.207828	−	0.978165i	$-0.566639\pi$
$577$	−5761.00	−0.415656	−0.207828	+	0.978165i	$0.566639\pi$
$578$	0	0
$579$	14462.0	1.03803
$580$	0	0
$581$	5236.00	0.373883
$582$	0	0
$583$	−4730.00	−0.336015
$584$	0	0
$585$	−264.000	−0.0186582
$586$	0	0
$587$	13188.0	0.927303	0.463652	−	0.886018i	$-0.346539\pi$
$587$	13188.0	0.927303	0.463652	+	0.886018i	$0.346539\pi$
$588$	0	0
$589$	−14022.0	−0.980928
$590$	0	0
$591$	−24094.0	−1.67698
$592$	0	0
$593$	26766.0	1.85354	0.926769	−	0.375632i	$-0.122574\pi$
$593$	26766.0	1.85354	0.926769	+	0.375632i	$0.122574\pi$
$594$	0	0
$595$	−14.0000	−0.000964612 0
$596$	0	0
$597$	−21056.0	−1.44349
$598$	0	0
$599$	20664.0	1.40953	0.704765	−	0.709441i	$-0.251053\pi$
$599$	20664.0	1.40953	0.704765	+	0.709441i	$0.251053\pi$
$600$	0	0
$601$	23656.0	1.60557	0.802786	−	0.596268i	$-0.203350\pi$
$601$	23656.0	1.60557	0.802786	+	0.596268i	$0.203350\pi$
$602$	0	0
$603$	6886.00	0.465041
$604$	0	0
$605$	−121.000	−0.00813116
$606$	0	0
$607$	−18674.0	−1.24869	−0.624345	−	0.781149i	$-0.714634\pi$
$607$	−18674.0	−1.24869	−0.624345	+	0.781149i	$0.714634\pi$
$608$	0	0
$609$	4998.00	0.332560
$610$	0	0
$611$	1152.00	0.0762765
$612$	0	0
$613$	7248.00	0.477559	0.238780	−	0.971074i	$-0.423253\pi$
$613$	7248.00	0.477559	0.238780	+	0.971074i	$0.423253\pi$
$614$	0	0
$615$	−798.000	−0.0523227
$616$	0	0
$617$	2066.00	0.134804	0.0674020	−	0.997726i	$-0.478529\pi$
$617$	2066.00	0.134804	0.0674020	+	0.997726i	$0.478529\pi$
$618$	0	0
$619$	1997.00	0.129671	0.0648354	−	0.997896i	$-0.479348\pi$
$619$	1997.00	0.129671	0.0648354	+	0.997896i	$0.479348\pi$
$620$	0	0
$621$	245.000	0.0158317
$622$	0	0
$623$	3199.00	0.205723
$624$	0	0
$625$	15251.0	0.976064
$626$	0	0
$627$	−6314.00	−0.402164
$628$	0	0
$629$	−714.000	−0.0452608
$630$	0	0
$631$	−5129.00	−0.323585	−0.161793	−	0.986825i	$-0.551728\pi$
$631$	−5129.00	−0.323585	−0.161793	+	0.986825i	$0.551728\pi$
$632$	0	0
$633$	5446.00	0.341957
$634$	0	0
$635$	2790.00	0.174359
$636$	0	0
$637$	588.000	0.0365736
$638$	0	0
$639$	−12738.0	−0.788588
$640$	0	0
$641$	−15129.0	−0.932230	−0.466115	−	0.884724i	$-0.654347\pi$
$641$	−15129.0	−0.932230	−0.466115	+	0.884724i	$0.654347\pi$
$642$	0	0
$643$	2807.00	0.172158	0.0860788	−	0.996288i	$-0.472566\pi$
$643$	2807.00	0.172158	0.0860788	+	0.996288i	$0.472566\pi$
$644$	0	0
$645$	−2408.00	−0.147000
$646$	0	0
$647$	−20137.0	−1.22360	−0.611798	−	0.791014i	$-0.709554\pi$
$647$	−20137.0	−1.22360	−0.611798	+	0.791014i	$0.709554\pi$
$648$	0	0
$649$	−2211.00	−0.133728
$650$	0	0
$651$	8379.00	0.504453
$652$	0	0
$653$	26247.0	1.57293	0.786466	−	0.617634i	$-0.211909\pi$
$653$	26247.0	1.57293	0.786466	+	0.617634i	$0.211909\pi$
$654$	0	0
$655$	−730.000	−0.0435473
$656$	0	0
$657$	−9636.00	−0.572201
$658$	0	0
$659$	−28074.0	−1.65950	−0.829748	−	0.558138i	$-0.811516\pi$
$659$	−28074.0	−1.65950	−0.829748	+	0.558138i	$0.811516\pi$
$660$	0	0
$661$	2355.00	0.138576	0.0692881	−	0.997597i	$-0.477927\pi$
$661$	2355.00	0.138576	0.0692881	+	0.997597i	$0.477927\pi$
$662$	0	0
$663$	−168.000	−0.00984099
$664$	0	0
$665$	−574.000	−0.0334718
$666$	0	0
$667$	−714.000	−0.0414486
$668$	0	0
$669$	19173.0	1.10803
$670$	0	0
$671$	−22.0000	−0.00126572
$672$	0	0
$673$	−19180.0	−1.09857	−0.549283	−	0.835637i	$-0.685099\pi$
$673$	−19180.0	−1.09857	−0.549283	+	0.835637i	$0.685099\pi$
$674$	0	0
$675$	−4340.00	−0.247477
$676$	0	0
$677$	−27482.0	−1.56015	−0.780073	−	0.625688i	$-0.784818\pi$
$677$	−27482.0	−1.56015	−0.780073	+	0.625688i	$0.784818\pi$
$678$	0	0
$679$	−7259.00	−0.410272
$680$	0	0
$681$	5124.00	0.288329
$682$	0	0
$683$	−24468.0	−1.37078	−0.685389	−	0.728177i	$-0.740368\pi$
$683$	−24468.0	−1.37078	−0.685389	+	0.728177i	$0.740368\pi$
$684$	0	0
$685$	1819.00	0.101460
$686$	0	0
$687$	5761.00	0.319936
$688$	0	0
$689$	−5160.00	−0.285313
$690$	0	0
$691$	147.000	0.00809283	0.00404641	−	0.999992i	$-0.498712\pi$
$691$	147.000	0.00809283	0.00404641	+	0.999992i	$0.498712\pi$
$692$	0	0
$693$	1694.00	0.0928568
$694$	0	0
$695$	802.000	0.0437721
$696$	0	0
$697$	−228.000	−0.0123904
$698$	0	0
$699$	−10234.0	−0.553770
$700$	0	0
$701$	5760.00	0.310346	0.155173	−	0.987887i	$-0.450407\pi$
$701$	5760.00	0.310346	0.155173	+	0.987887i	$0.450407\pi$
$702$	0	0
$703$	−29274.0	−1.57054
$704$	0	0
$705$	672.000	0.0358993
$706$	0	0
$707$	476.000	0.0253208
$708$	0	0
$709$	11775.0	0.623723	0.311861	−	0.950128i	$-0.399048\pi$
$709$	11775.0	0.623723	0.311861	+	0.950128i	$0.399048\pi$
$710$	0	0
$711$	10868.0	0.573252
$712$	0	0
$713$	−1197.00	−0.0628724
$714$	0	0
$715$	−132.000	−0.00690422
$716$	0	0
$717$	−11900.0	−0.619824
$718$	0	0
$719$	−21687.0	−1.12488	−0.562440	−	0.826838i	$-0.690137\pi$
$719$	−21687.0	−1.12488	−0.562440	+	0.826838i	$0.690137\pi$
$720$	0	0
$721$	−3276.00	−0.169216
$722$	0	0
$723$	−24220.0	−1.24585
$724$	0	0
$725$	12648.0	0.647910
$726$	0	0
$727$	33585.0	1.71334	0.856670	−	0.515864i	$-0.172529\pi$
$727$	33585.0	1.71334	0.856670	+	0.515864i	$0.172529\pi$
$728$	0	0
$729$	−11843.0	−0.601687
$730$	0	0
$731$	−688.000	−0.0348107
$732$	0	0
$733$	−32404.0	−1.63284	−0.816418	−	0.577461i	$-0.804044\pi$
$733$	−32404.0	−1.63284	−0.816418	+	0.577461i	$0.804044\pi$
$734$	0	0
$735$	343.000	0.0172133
$736$	0	0
$737$	3443.00	0.172082
$738$	0	0
$739$	6038.00	0.300557	0.150278	−	0.988644i	$-0.451983\pi$
$739$	6038.00	0.300557	0.150278	+	0.988644i	$0.451983\pi$
$740$	0	0
$741$	−6888.00	−0.341480
$742$	0	0
$743$	−3656.00	−0.180519	−0.0902595	−	0.995918i	$-0.528770\pi$
$743$	−3656.00	−0.180519	−0.0902595	+	0.995918i	$0.528770\pi$
$744$	0	0
$745$	−314.000	−0.0154417
$746$	0	0
$747$	16456.0	0.806015
$748$	0	0
$749$	11074.0	0.540234
$750$	0	0
$751$	−4915.00	−0.238816	−0.119408	−	0.992845i	$-0.538100\pi$
$751$	−4915.00	−0.238816	−0.119408	+	0.992845i	$0.538100\pi$
$752$	0	0
$753$	1855.00	0.0897742
$754$	0	0
$755$	−766.000	−0.0369240
$756$	0	0
$757$	−19186.0	−0.921172	−0.460586	−	0.887615i	$-0.652361\pi$
$757$	−19186.0	−0.921172	−0.460586	+	0.887615i	$0.652361\pi$
$758$	0	0
$759$	−539.000	−0.0257766
$760$	0	0
$761$	2448.00	0.116610	0.0583048	−	0.998299i	$-0.481430\pi$
$761$	2448.00	0.116610	0.0583048	+	0.998299i	$0.481430\pi$
$762$	0	0
$763$	3612.00	0.171380
$764$	0	0
$765$	−44.0000	−0.00207951
$766$	0	0
$767$	−2412.00	−0.113549
$768$	0	0
$769$	−8312.00	−0.389777	−0.194888	−	0.980825i	$-0.562434\pi$
$769$	−8312.00	−0.389777	−0.194888	+	0.980825i	$0.562434\pi$
$770$	0	0
$771$	45514.0	2.12600
$772$	0	0
$773$	−17598.0	−0.818831	−0.409415	−	0.912348i	$-0.634267\pi$
$773$	−17598.0	−0.818831	−0.409415	+	0.912348i	$0.634267\pi$
$774$	0	0
$775$	21204.0	0.982800
$776$	0	0
$777$	17493.0	0.807668
$778$	0	0
$779$	−9348.00	−0.429945
$780$	0	0
$781$	−6369.00	−0.291806
$782$	0	0
$783$	−3570.00	−0.162939
$784$	0	0
$785$	1681.00	0.0764299
$786$	0	0
$787$	−7982.00	−0.361534	−0.180767	−	0.983526i	$-0.557858\pi$
$787$	−7982.00	−0.361534	−0.180767	+	0.983526i	$0.557858\pi$
$788$	0	0
$789$	9786.00	0.441560
$790$	0	0
$791$	−11389.0	−0.511942
$792$	0	0
$793$	−24.0000	−0.00107474
$794$	0	0
$795$	−3010.00	−0.134281
$796$	0	0
$797$	−12489.0	−0.555060	−0.277530	−	0.960717i	$-0.589516\pi$
$797$	−12489.0	−0.555060	−0.277530	+	0.960717i	$0.589516\pi$
$798$	0	0
$799$	192.000	0.00850122
$800$	0	0
$801$	10054.0	0.443496
$802$	0	0
$803$	−4818.00	−0.211735
$804$	0	0
$805$	−49.0000	−0.00214537
$806$	0	0
$807$	−25802.0	−1.12549
$808$	0	0
$809$	31966.0	1.38920	0.694601	−	0.719395i	$-0.255581\pi$
$809$	31966.0	1.38920	0.694601	+	0.719395i	$0.255581\pi$
$810$	0	0
$811$	20786.0	0.899994	0.449997	−	0.893030i	$-0.351425\pi$
$811$	20786.0	0.899994	0.449997	+	0.893030i	$0.351425\pi$
$812$	0	0
$813$	26656.0	1.14990
$814$	0	0
$815$	−724.000	−0.0311173
$816$	0	0
$817$	−28208.0	−1.20792
$818$	0	0
$819$	1848.00	0.0788454
$820$	0	0
$821$	−33038.0	−1.40443	−0.702213	−	0.711967i	$-0.747805\pi$
$821$	−33038.0	−1.40443	−0.702213	+	0.711967i	$0.747805\pi$
$822$	0	0
$823$	−6877.00	−0.291272	−0.145636	−	0.989338i	$-0.546523\pi$
$823$	−6877.00	−0.291272	−0.145636	+	0.989338i	$0.546523\pi$
$824$	0	0
$825$	9548.00	0.402932
$826$	0	0
$827$	28964.0	1.21787	0.608934	−	0.793221i	$-0.291597\pi$
$827$	28964.0	1.21787	0.608934	+	0.793221i	$0.291597\pi$
$828$	0	0
$829$	−30365.0	−1.27216	−0.636080	−	0.771623i	$-0.719445\pi$
$829$	−30365.0	−1.27216	−0.636080	+	0.771623i	$0.719445\pi$
$830$	0	0
$831$	62104.0	2.59250
$832$	0	0
$833$	98.0000	0.00407623
$834$	0	0
$835$	1242.00	0.0514745
$836$	0	0
$837$	−5985.00	−0.247159
$838$	0	0
$839$	27393.0	1.12719	0.563594	−	0.826052i	$-0.309418\pi$
$839$	27393.0	1.12719	0.563594	+	0.826052i	$0.309418\pi$
$840$	0	0
$841$	−13985.0	−0.573414
$842$	0	0
$843$	19516.0	0.797351
$844$	0	0
$845$	2053.00	0.0835803
$846$	0	0
$847$	847.000	0.0343604
$848$	0	0
$849$	−1120.00	−0.0452748
$850$	0	0
$851$	−2499.00	−0.100663
$852$	0	0
$853$	−38994.0	−1.56522	−0.782608	−	0.622515i	$-0.786111\pi$
$853$	−38994.0	−1.56522	−0.782608	+	0.622515i	$0.786111\pi$
$854$	0	0
$855$	−1804.00	−0.0721585
$856$	0	0
$857$	2564.00	0.102199	0.0510995	−	0.998694i	$-0.483727\pi$
$857$	2564.00	0.102199	0.0510995	+	0.998694i	$0.483727\pi$
$858$	0	0
$859$	21643.0	0.859662	0.429831	−	0.902909i	$-0.358573\pi$
$859$	21643.0	0.859662	0.429831	+	0.902909i	$0.358573\pi$
$860$	0	0
$861$	5586.00	0.221104
$862$	0	0
$863$	10988.0	0.433414	0.216707	−	0.976237i	$-0.430468\pi$
$863$	10988.0	0.433414	0.216707	+	0.976237i	$0.430468\pi$
$864$	0	0
$865$	−2432.00	−0.0955959
$866$	0	0
$867$	34363.0	1.34605
$868$	0	0
$869$	5434.00	0.212124
$870$	0	0
$871$	3756.00	0.146116
$872$	0	0
$873$	−22814.0	−0.884464
$874$	0	0
$875$	1743.00	0.0673419
$876$	0	0
$877$	23082.0	0.888739	0.444369	−	0.895844i	$-0.353428\pi$
$877$	23082.0	0.888739	0.444369	+	0.895844i	$0.353428\pi$
$878$	0	0
$879$	7994.00	0.306747
$880$	0	0
$881$	29107.0	1.11310	0.556549	−	0.830815i	$-0.312125\pi$
$881$	29107.0	1.11310	0.556549	+	0.830815i	$0.312125\pi$
$882$	0	0
$883$	40852.0	1.55694	0.778471	−	0.627681i	$-0.215996\pi$
$883$	40852.0	1.55694	0.778471	+	0.627681i	$0.215996\pi$
$884$	0	0
$885$	−1407.00	−0.0534416
$886$	0	0
$887$	−35034.0	−1.32619	−0.663093	−	0.748537i	$-0.730757\pi$
$887$	−35034.0	−1.32619	−0.663093	+	0.748537i	$0.730757\pi$
$888$	0	0
$889$	−19530.0	−0.736800
$890$	0	0
$891$	−9229.00	−0.347007
$892$	0	0
$893$	7872.00	0.294990
$894$	0	0
$895$	2035.00	0.0760028
$896$	0	0
$897$	−588.000	−0.0218871
$898$	0	0
$899$	17442.0	0.647078
$900$	0	0
$901$	−860.000	−0.0317988
$902$	0	0
$903$	16856.0	0.621188
$904$	0	0
$905$	−1405.00	−0.0516064
$906$	0	0
$907$	50728.0	1.85711	0.928553	−	0.371199i	$-0.121053\pi$
$907$	50728.0	1.85711	0.928553	+	0.371199i	$0.121053\pi$
$908$	0	0
$909$	1496.00	0.0545866
$910$	0	0
$911$	20088.0	0.730565	0.365283	−	0.930897i	$-0.380972\pi$
$911$	20088.0	0.730565	0.365283	+	0.930897i	$0.380972\pi$
$912$	0	0
$913$	8228.00	0.298255
$914$	0	0
$915$	−14.0000	−0.000505820 0
$916$	0	0
$917$	5110.00	0.184021
$918$	0	0
$919$	11504.0	0.412929	0.206465	−	0.978454i	$-0.433804\pi$
$919$	11504.0	0.412929	0.206465	+	0.978454i	$0.433804\pi$
$920$	0	0
$921$	71540.0	2.55953
$922$	0	0
$923$	−6948.00	−0.247775
$924$	0	0
$925$	44268.0	1.57354
$926$	0	0
$927$	−10296.0	−0.364795
$928$	0	0
$929$	−7134.00	−0.251947	−0.125974	−	0.992034i	$-0.540205\pi$
$929$	−7134.00	−0.251947	−0.125974	+	0.992034i	$0.540205\pi$
$930$	0	0
$931$	4018.00	0.141444
$932$	0	0
$933$	−34048.0	−1.19473
$934$	0	0
$935$	−22.0000	−0.000769494 0
$936$	0	0
$937$	10036.0	0.349906	0.174953	−	0.984577i	$-0.444023\pi$
$937$	10036.0	0.349906	0.174953	+	0.984577i	$0.444023\pi$
$938$	0	0
$939$	25921.0	0.900852
$940$	0	0
$941$	−54438.0	−1.88590	−0.942948	−	0.332940i	$-0.891959\pi$
$941$	−54438.0	−1.88590	−0.942948	+	0.332940i	$0.891959\pi$
$942$	0	0
$943$	−798.000	−0.0275572
$944$	0	0
$945$	−245.000	−0.00843370
$946$	0	0
$947$	−15975.0	−0.548171	−0.274085	−	0.961705i	$-0.588375\pi$
$947$	−15975.0	−0.548171	−0.274085	+	0.961705i	$0.588375\pi$
$948$	0	0
$949$	−5256.00	−0.179786
$950$	0	0
$951$	−567.000	−0.0193336
$952$	0	0
$953$	33996.0	1.15555	0.577775	−	0.816196i	$-0.303921\pi$
$953$	33996.0	1.15555	0.577775	+	0.816196i	$0.303921\pi$
$954$	0	0
$955$	1671.00	0.0566202
$956$	0	0
$957$	7854.00	0.265291
$958$	0	0
$959$	−12733.0	−0.428749
$960$	0	0
$961$	−550.000	−0.0184620
$962$	0	0
$963$	34804.0	1.16463
$964$	0	0
$965$	2066.00	0.0689191
$966$	0	0
$967$	40966.0	1.36233	0.681167	−	0.732128i	$-0.261473\pi$
$967$	40966.0	1.36233	0.681167	+	0.732128i	$0.261473\pi$
$968$	0	0
$969$	−1148.00	−0.0380589
$970$	0	0
$971$	23793.0	0.786358	0.393179	−	0.919462i	$-0.371375\pi$
$971$	23793.0	0.786358	0.393179	+	0.919462i	$0.371375\pi$
$972$	0	0
$973$	−5614.00	−0.184971
$974$	0	0
$975$	10416.0	0.342132
$976$	0	0
$977$	−439.000	−0.0143755	−0.00718775	−	0.999974i	$-0.502288\pi$
$977$	−439.000	−0.0143755	−0.00718775	+	0.999974i	$0.502288\pi$
$978$	0	0
$979$	5027.00	0.164110
$980$	0	0
$981$	11352.0	0.369461
$982$	0	0
$983$	−7863.00	−0.255128	−0.127564	−	0.991830i	$-0.540716\pi$
$983$	−7863.00	−0.255128	−0.127564	+	0.991830i	$0.540716\pi$
$984$	0	0
$985$	−3442.00	−0.111341
$986$	0	0
$987$	−4704.00	−0.151702
$988$	0	0
$989$	−2408.00	−0.0774216
$990$	0	0
$991$	−46360.0	−1.48605	−0.743024	−	0.669265i	$-0.766609\pi$
$991$	−46360.0	−1.48605	−0.743024	+	0.669265i	$0.766609\pi$
$992$	0	0
$993$	−18725.0	−0.598409
$994$	0	0
$995$	−3008.00	−0.0958392
$996$	0	0
$997$	5860.00	0.186147	0.0930733	−	0.995659i	$-0.470331\pi$
$997$	5860.00	0.186147	0.0930733	+	0.995659i	$0.470331\pi$
$998$	0	0
$999$	−12495.0	−0.395720

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	308.4.a.a.1.1	✓	1
4.3	odd	2		1232.4.a.h.1.1		1
7.6	odd	2		2156.4.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
308.4.a.a.1.1	✓	1	1.1	even	1	trivial
1232.4.a.h.1.1		1	4.3	odd	2
2156.4.a.c.1.1		1	7.6	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:	\( N \)	\(=\)	\( 308 = 2^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	308.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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