LMFDB - Embedded newform 675.3.c.r.26.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [675,3,Mod(26,675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(675, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("675.26");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	675.c (of order $2$, degree $1$, 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:	$18.3924178443$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.60217600.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 16x^{4} + 64x^{2} + 4 $ x^6 + 16x^4 + 64x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			26.3
Root			$-0.252000i$ of defining polynomial
Character	$\chi$	$=$	675.26
Dual form			675.3.c.r.26.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.25200i q^{2} +2.43250 q^{4} +7.62099 q^{7} -8.05349i q^{8} -15.8015i q^{11} +10.7005 q^{13} -9.54148i q^{14} -0.352988 q^{16} +1.55749i q^{17} -23.9980 q^{19} -19.7835 q^{22} +33.5655i q^{23} -13.3970i q^{26} +18.5380 q^{28} -22.0615i q^{29} +27.0515 q^{31} -31.7720i q^{32} +1.94997 q^{34} +28.7800 q^{37} +30.0455i q^{38} -48.2815i q^{41} +9.02199 q^{43} -38.4370i q^{44} +42.0240 q^{46} -48.8230i q^{47} +9.07951 q^{49} +26.0289 q^{52} +59.6165i q^{53} -61.3755i q^{56} -27.6210 q^{58} +99.8889i q^{59} -89.2119 q^{61} -33.8685i q^{62} -41.1905 q^{64} +1.85951 q^{67} +3.78858i q^{68} -126.109i q^{71} +101.003 q^{73} -36.0326i q^{74} -58.3750 q^{76} -120.423i q^{77} +108.641 q^{79} -60.4484 q^{82} +3.54753i q^{83} -11.2955i q^{86} -127.257 q^{88} -108.553i q^{89} +81.5484 q^{91} +81.6479i q^{92} -61.1264 q^{94} -88.8919 q^{97} -11.3675i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 14 q^{4} - 16 q^{7} + 22 q^{13} + 46 q^{16} - 16 q^{19} + 86 q^{22} + 212 q^{28} - 56 q^{31} - 80 q^{34} + 150 q^{37} - 92 q^{43} + 234 q^{46} + 74 q^{49} - 354 q^{52} - 104 q^{58} - 46 q^{61} - 342 q^{64}+ \cdots + 162 q^{97}+O(q^{100}) $ 6 * q - 14 * q^4 - 16 * q^7 + 22 * q^13 + 46 * q^16 - 16 * q^19 + 86 * q^22 + 212 * q^28 - 56 * q^31 - 80 * q^34 + 150 * q^37 - 92 * q^43 + 234 * q^46 + 74 * q^49 - 354 * q^52 - 104 * q^58 - 46 * q^61 - 342 * q^64 + 8 * q^67 + 424 * q^73 - 156 * q^76 + 316 * q^79 + 196 * q^82 - 858 * q^88 + 44 * q^91 + 526 * q^94 + 162 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/675\mathbb{Z}\right)^\times$.

$n$	$326$	$352$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 1.25200i	− 0.626000i	−0.949753	−	0.313000i	$-0.898666\pi$
$2$	− 1.25200i	− 0.626000i	0.949753	−	0.313000i	$-0.101334\pi$
$3$	0	0
$3$	0	0
$4$	2.43250	0.608124
$5$	0	0
$5$	0	0
$6$	0	0
$7$	7.62099	1.08871	0.544357	−	0.838854i	$-0.316774\pi$
$7$	7.62099	1.08871	0.544357	+	0.838854i	$0.316774\pi$
$8$	− 8.05349i	− 1.00669i
$9$	0	0
$10$	0	0
$11$	− 15.8015i	− 1.43650i	−0.695786	−	0.718249i	$-0.744944\pi$
$11$	− 15.8015i	− 1.43650i	0.695786	−	0.718249i	$-0.255056\pi$
$12$	0	0
$13$	10.7005	0.823115	0.411558	−	0.911384i	$-0.364985\pi$
$13$	10.7005	0.823115	0.411558	+	0.911384i	$0.364985\pi$
$14$	− 9.54148i	− 0.681535i
$15$	0	0
$16$	−0.352988	−0.0220617
$17$	1.55749i	0.0916169i	0.998950	+	0.0458084i	$0.0145864\pi$
$17$	1.55749i	0.0916169i	−0.998950	+	0.0458084i	$0.985414\pi$
$18$	0	0
$19$	−23.9980	−1.26305	−0.631526	−	0.775355i	$-0.717571\pi$
$19$	−23.9980	−1.26305	−0.631526	+	0.775355i	$0.717571\pi$
$20$	0	0
$21$	0	0
$22$	−19.7835	−0.899248
$23$	33.5655i	1.45937i	0.683784	+	0.729685i	$0.260333\pi$
$23$	33.5655i	1.45937i	−0.683784	+	0.729685i	$0.739667\pi$
$24$	0	0
$25$	0	0
$26$	− 13.3970i	− 0.515270i
$27$	0	0
$28$	18.5380	0.662072
$29$	− 22.0615i	− 0.760741i	−0.924834	−	0.380370i	$-0.875796\pi$
$29$	− 22.0615i	− 0.760741i	0.924834	−	0.380370i	$-0.124204\pi$
$30$	0	0
$31$	27.0515	0.872628	0.436314	−	0.899794i	$-0.356284\pi$
$31$	27.0515	0.872628	0.436314	+	0.899794i	$0.356284\pi$
$32$	− 31.7720i	− 0.992875i
$33$	0	0
$34$	1.94997	0.0573522
$35$	0	0
$36$	0	0
$37$	28.7800	0.777838	0.388919	−	0.921272i	$-0.372849\pi$
$37$	28.7800	0.777838	0.388919	+	0.921272i	$0.372849\pi$
$38$	30.0455i	0.790671i
$39$	0	0
$40$	0	0
$41$	− 48.2815i	− 1.17760i	−0.808280	−	0.588799i	$-0.799601\pi$
$41$	− 48.2815i	− 1.17760i	0.808280	−	0.588799i	$-0.200399\pi$
$42$	0	0
$43$	9.02199	0.209814	0.104907	−	0.994482i	$-0.466546\pi$
$43$	9.02199	0.209814	0.104907	+	0.994482i	$0.466546\pi$
$44$	− 38.4370i	− 0.873569i
$45$	0	0
$46$	42.0240	0.913565
$47$	− 48.8230i	− 1.03879i	−0.854535	−	0.519393i	$-0.826158\pi$
$47$	− 48.8230i	− 1.03879i	0.854535	−	0.519393i	$-0.173842\pi$
$48$	0	0
$49$	9.07951	0.185296
$50$	0	0
$51$	0	0
$52$	26.0289	0.500556
$53$	59.6165i	1.12484i	0.826852	+	0.562419i	$0.190129\pi$
$53$	59.6165i	1.12484i	−0.826852	+	0.562419i	$0.809871\pi$
$54$	0	0
$55$	0	0
$56$	− 61.3755i	− 1.09599i
$57$	0	0
$58$	−27.6210	−0.476224
$59$	99.8889i	1.69303i	0.532363	+	0.846516i	$0.321304\pi$
$59$	99.8889i	1.69303i	−0.532363	+	0.846516i	$0.678696\pi$
$60$	0	0
$61$	−89.2119	−1.46249	−0.731245	−	0.682115i	$-0.761061\pi$
$61$	−89.2119	−1.46249	−0.731245	+	0.682115i	$0.761061\pi$
$62$	− 33.8685i	− 0.546265i
$63$	0	0
$64$	−41.1905	−0.643602
$65$	0	0
$66$	0	0
$67$	1.85951	0.0277539	0.0138770	−	0.999904i	$-0.495583\pi$
$67$	1.85951	0.0277539	0.0138770	+	0.999904i	$0.495583\pi$
$68$	3.78858i	0.0557144i
$69$	0	0
$70$	0	0
$71$	− 126.109i	− 1.77618i	−0.459668	−	0.888091i	$-0.652031\pi$
$71$	− 126.109i	− 1.77618i	0.459668	−	0.888091i	$-0.347969\pi$
$72$	0	0
$73$	101.003	1.38361	0.691805	−	0.722085i	$-0.256816\pi$
$73$	101.003	1.38361	0.691805	+	0.722085i	$0.256816\pi$
$74$	− 36.0326i	− 0.486927i
$75$	0	0
$76$	−58.3750	−0.768092
$77$	− 120.423i	− 1.56393i
$78$	0	0
$79$	108.641	1.37520	0.687601	−	0.726089i	$-0.258664\pi$
$79$	108.641	1.37520	0.687601	+	0.726089i	$0.258664\pi$
$80$	0	0
$81$	0	0
$82$	−60.4484	−0.737176
$83$	3.54753i	0.0427414i	0.999772	+	0.0213707i	$0.00680302\pi$
$83$	3.54753i	0.0427414i	−0.999772	+	0.0213707i	$0.993197\pi$
$84$	0	0
$85$	0	0
$86$	− 11.2955i	− 0.131343i
$87$	0	0
$88$	−127.257	−1.44610
$89$	− 108.553i	− 1.21970i	−0.792516	−	0.609851i	$-0.791229\pi$
$89$	− 108.553i	− 1.21970i	0.792516	−	0.609851i	$-0.208771\pi$
$90$	0	0
$91$	81.5484	0.896136
$92$	81.6479i	0.887477i
$93$	0	0
$94$	−61.1264	−0.650281
$95$	0	0
$96$	0	0
$97$	−88.8919	−0.916411	−0.458206	−	0.888846i	$-0.651508\pi$
$97$	−88.8919	−0.916411	−0.458206	+	0.888846i	$0.651508\pi$
$98$	− 11.3675i	− 0.115995i
$99$	0	0
$100$	0	0
$101$	30.4230i	0.301218i	0.988593	+	0.150609i	$0.0481234\pi$
$101$	30.4230i	0.301218i	−0.988593	+	0.150609i	$0.951877\pi$
$102$	0	0
$103$	−184.773	−1.79392	−0.896958	−	0.442115i	$-0.854228\pi$
$103$	−184.773	−1.79392	−0.896958	+	0.442115i	$0.854228\pi$
$104$	− 86.1763i	− 0.828618i
$105$	0	0
$106$	74.6398	0.704149
$107$	− 26.7615i	− 0.250107i	−0.992150	−	0.125054i	$-0.960090\pi$
$107$	− 26.7615i	− 0.250107i	0.992150	−	0.125054i	$-0.0399103\pi$
$108$	0	0
$109$	6.01248	0.0551604	0.0275802	−	0.999620i	$-0.491220\pi$
$109$	6.01248	0.0551604	0.0275802	+	0.999620i	$0.491220\pi$
$110$	0	0
$111$	0	0
$112$	−2.69012	−0.0240189
$113$	194.730i	1.72328i	0.507522	+	0.861639i	$0.330562\pi$
$113$	194.730i	1.72328i	−0.507522	+	0.861639i	$0.669438\pi$
$114$	0	0
$115$	0	0
$116$	− 53.6645i	− 0.462625i
$117$	0	0
$118$	125.061	1.05984
$119$	11.8696i	0.0997445i
$120$	0	0
$121$	−128.687	−1.06353
$122$	111.693i	0.915519i
$123$	0	0
$124$	65.8026	0.530666
$125$	0	0
$126$	0	0
$127$	203.372	1.60136	0.800679	−	0.599094i	$-0.204472\pi$
$127$	203.372	1.60136	0.800679	+	0.599094i	$0.204472\pi$
$128$	− 75.5175i	− 0.589980i
$129$	0	0
$130$	0	0
$131$	137.404i	1.04889i	0.851445	+	0.524444i	$0.175727\pi$
$131$	137.404i	1.04889i	−0.851445	+	0.524444i	$0.824273\pi$
$132$	0	0
$133$	−182.888	−1.37510
$134$	− 2.32811i	− 0.0173740i
$135$	0	0
$136$	12.5432	0.0922294
$137$	− 27.3030i	− 0.199292i	−0.995023	−	0.0996458i	$-0.968229\pi$
$137$	− 27.3030i	− 0.199292i	0.995023	−	0.0996458i	$-0.0317710\pi$
$138$	0	0
$139$	140.553	1.01118	0.505588	−	0.862775i	$-0.331276\pi$
$139$	140.553	1.01118	0.505588	+	0.862775i	$0.331276\pi$
$140$	0	0
$141$	0	0
$142$	−157.888	−1.11189
$143$	− 169.084i	− 1.18240i
$144$	0	0
$145$	0	0
$146$	− 126.456i	− 0.866140i
$147$	0	0
$148$	70.0072	0.473022
$149$	131.719i	0.884017i	0.897011	+	0.442008i	$0.145734\pi$
$149$	131.719i	0.884017i	−0.897011	+	0.442008i	$0.854266\pi$
$150$	0	0
$151$	−232.712	−1.54114	−0.770571	−	0.637354i	$-0.780029\pi$
$151$	−232.712	−1.54114	−0.770571	+	0.637354i	$0.780029\pi$
$152$	193.267i	1.27150i
$153$	0	0
$154$	−150.770	−0.979024
$155$	0	0
$156$	0	0
$157$	−7.63107	−0.0486056	−0.0243028	−	0.999705i	$-0.507737\pi$
$157$	−7.63107	−0.0486056	−0.0243028	+	0.999705i	$0.507737\pi$
$158$	− 136.019i	− 0.860877i
$159$	0	0
$160$	0	0
$161$	255.802i	1.58883i
$162$	0	0
$163$	309.047	1.89600	0.947998	−	0.318275i	$-0.103104\pi$
$163$	309.047	1.89600	0.947998	+	0.318275i	$0.103104\pi$
$164$	− 117.444i	− 0.716125i
$165$	0	0
$166$	4.44151	0.0267561
$167$	− 74.2799i	− 0.444790i	−0.974957	−	0.222395i	$-0.928613\pi$
$167$	− 74.2799i	− 0.444790i	0.974957	−	0.222395i	$-0.0713874\pi$
$168$	0	0
$169$	−54.4993	−0.322481
$170$	0	0
$171$	0	0
$172$	21.9459	0.127593
$173$	− 7.61407i	− 0.0440120i	−0.999758	−	0.0220060i	$-0.992995\pi$
$173$	− 7.61407i	− 0.0440120i	0.999758	−	0.0220060i	$-0.00700529\pi$
$174$	0	0
$175$	0	0
$176$	5.57773i	0.0316917i
$177$	0	0
$178$	−135.909	−0.763533
$179$	301.982i	1.68705i	0.537088	+	0.843526i	$0.319524\pi$
$179$	301.982i	1.68705i	−0.537088	+	0.843526i	$0.680476\pi$
$180$	0	0
$181$	224.339	1.23944	0.619721	−	0.784822i	$-0.287246\pi$
$181$	224.339	1.23944	0.619721	+	0.784822i	$0.287246\pi$
$182$	− 102.099i	− 0.560982i
$183$	0	0
$184$	270.319	1.46913
$185$	0	0
$186$	0	0
$187$	24.6106	0.131608
$188$	− 118.762i	− 0.631711i
$189$	0	0
$190$	0	0
$191$	295.040i	1.54471i	0.635190	+	0.772356i	$0.280922\pi$
$191$	295.040i	1.54471i	−0.635190	+	0.772356i	$0.719078\pi$
$192$	0	0
$193$	−8.23700	−0.0426787	−0.0213394	−	0.999772i	$-0.506793\pi$
$193$	−8.23700	−0.0426787	−0.0213394	+	0.999772i	$0.506793\pi$
$194$	111.293i	0.573674i
$195$	0	0
$196$	22.0859	0.112683
$197$	21.9989i	0.111669i	0.998440	+	0.0558347i	$0.0177820\pi$
$197$	21.9989i	0.111669i	−0.998440	+	0.0558347i	$0.982218\pi$
$198$	0	0
$199$	−340.073	−1.70891	−0.854455	−	0.519525i	$-0.826109\pi$
$199$	−340.073	−1.70891	−0.854455	+	0.519525i	$0.826109\pi$
$200$	0	0
$201$	0	0
$202$	38.0896	0.188562
$203$	− 168.130i	− 0.828229i
$204$	0	0
$205$	0	0
$206$	231.336i	1.12299i
$207$	0	0
$208$	−3.77714	−0.0181594
$209$	379.204i	1.81437i
$210$	0	0
$211$	167.918	0.795820	0.397910	−	0.917424i	$-0.369736\pi$
$211$	167.918	0.795820	0.397910	+	0.917424i	$0.369736\pi$
$212$	145.017i	0.684041i
$213$	0	0
$214$	−33.5054	−0.156567
$215$	0	0
$216$	0	0
$217$	206.159	0.950042
$218$	− 7.52763i	− 0.0345304i
$219$	0	0
$220$	0	0
$221$	16.6659i	0.0754113i
$222$	0	0
$223$	−44.4215	−0.199199	−0.0995997	−	0.995028i	$-0.531756\pi$
$223$	−44.4215	−0.199199	−0.0995997	+	0.995028i	$0.531756\pi$
$224$	− 242.134i	− 1.08096i
$225$	0	0
$226$	243.803	1.07877
$227$	367.970i	1.62101i	0.585729	+	0.810507i	$0.300808\pi$
$227$	367.970i	1.62101i	−0.585729	+	0.810507i	$0.699192\pi$
$228$	0	0
$229$	−91.5505	−0.399784	−0.199892	−	0.979818i	$-0.564059\pi$
$229$	−91.5505	−0.399784	−0.199892	+	0.979818i	$0.564059\pi$
$230$	0	0
$231$	0	0
$232$	−177.672	−0.765827
$233$	313.399i	1.34506i	0.740069	+	0.672531i	$0.234793\pi$
$233$	313.399i	1.34506i	−0.740069	+	0.672531i	$0.765207\pi$
$234$	0	0
$235$	0	0
$236$	242.979i	1.02957i
$237$	0	0
$238$	14.8607	0.0624401
$239$	− 19.4749i	− 0.0814849i	−0.999170	−	0.0407425i	$-0.987028\pi$
$239$	− 19.4749i	− 0.0814849i	0.999170	−	0.0407425i	$-0.0129723\pi$
$240$	0	0
$241$	67.1869	0.278784	0.139392	−	0.990237i	$-0.455485\pi$
$241$	67.1869	0.278784	0.139392	+	0.990237i	$0.455485\pi$
$242$	161.116i	0.665769i
$243$	0	0
$244$	−217.008	−0.889375
$245$	0	0
$246$	0	0
$247$	−256.790	−1.03964
$248$	− 217.859i	− 0.878462i
$249$	0	0
$250$	0	0
$251$	180.227i	0.718034i	0.933331	+	0.359017i	$0.116888\pi$
$251$	180.227i	0.718034i	−0.933331	+	0.359017i	$0.883112\pi$
$252$	0	0
$253$	530.385	2.09638
$254$	− 254.622i	− 1.00245i
$255$	0	0
$256$	−259.310	−1.01293
$257$	104.434i	0.406359i	0.979141	+	0.203180i	$0.0651275\pi$
$257$	104.434i	0.406359i	−0.979141	+	0.203180i	$0.934872\pi$
$258$	0	0
$259$	219.332	0.846842
$260$	0	0
$261$	0	0
$262$	172.030	0.656605
$263$	− 5.89260i	− 0.0224053i	−0.999937	−	0.0112027i	$-0.996434\pi$
$263$	− 5.89260i	− 0.0224053i	0.999937	−	0.0112027i	$-0.00356599\pi$
$264$	0	0
$265$	0	0
$266$	228.976i	0.860813i
$267$	0	0
$268$	4.52326	0.0168778
$269$	271.843i	1.01057i	0.862953	+	0.505284i	$0.168612\pi$
$269$	271.843i	1.01057i	−0.862953	+	0.505284i	$0.831388\pi$
$270$	0	0
$271$	−198.634	−0.732966	−0.366483	−	0.930425i	$-0.619438\pi$
$271$	−198.634	−0.732966	−0.366483	+	0.930425i	$0.619438\pi$
$272$	− 0.549774i	− 0.00202123i
$273$	0	0
$274$	−34.1833	−0.124757
$275$	0	0
$276$	0	0
$277$	142.471	0.514335	0.257167	−	0.966367i	$-0.417211\pi$
$277$	142.471	0.514335	0.257167	+	0.966367i	$0.417211\pi$
$278$	− 175.973i	− 0.632996i
$279$	0	0
$280$	0	0
$281$	67.3771i	0.239776i	0.992787	+	0.119888i	$0.0382536\pi$
$281$	67.3771i	0.239776i	−0.992787	+	0.119888i	$0.961746\pi$
$282$	0	0
$283$	82.0477	0.289921	0.144961	−	0.989437i	$-0.453694\pi$
$283$	82.0477	0.289921	0.144961	+	0.989437i	$0.453694\pi$
$284$	− 306.759i	− 1.08014i
$285$	0	0
$286$	−211.693	−0.740185
$287$	− 367.953i	− 1.28207i
$288$	0	0
$289$	286.574	0.991606
$290$	0	0
$291$	0	0
$292$	245.690	0.841406
$293$	− 145.755i	− 0.497456i	−0.968573	−	0.248728i	$-0.919987\pi$
$293$	− 145.755i	− 0.497456i	0.968573	−	0.248728i	$-0.0800126\pi$
$294$	0	0
$295$	0	0
$296$	− 231.779i	− 0.783038i
$297$	0	0
$298$	164.912	0.553395
$299$	359.167i	1.20123i
$300$	0	0
$301$	68.7565	0.228427
$302$	291.356i	0.964755i
$303$	0	0
$304$	8.47099	0.0278651
$305$	0	0
$306$	0	0
$307$	−125.568	−0.409018	−0.204509	−	0.978865i	$-0.565560\pi$
$307$	−125.568	−0.409018	−0.204509	+	0.978865i	$0.565560\pi$
$308$	− 292.928i	− 0.951066i
$309$	0	0
$310$	0	0
$311$	107.587i	0.345940i	0.984927	+	0.172970i	$0.0553364\pi$
$311$	107.587i	0.345940i	−0.984927	+	0.172970i	$0.944664\pi$
$312$	0	0
$313$	−341.034	−1.08957	−0.544783	−	0.838577i	$-0.683388\pi$
$313$	−341.034	−1.08957	−0.544783	+	0.838577i	$0.683388\pi$
$314$	9.55411i	0.0304271i
$315$	0	0
$316$	264.269	0.836293
$317$	− 27.2585i	− 0.0859889i	−0.999075	−	0.0429944i	$-0.986310\pi$
$317$	− 27.2585i	− 0.0859889i	0.999075	−	0.0429944i	$-0.0136898\pi$
$318$	0	0
$319$	−348.604	−1.09280
$320$	0	0
$321$	0	0
$322$	320.265	0.994610
$323$	− 37.3765i	− 0.115717i
$324$	0	0
$325$	0	0
$326$	− 386.928i	− 1.18689i
$327$	0	0
$328$	−388.834	−1.18547
$329$	− 372.079i	− 1.13094i
$330$	0	0
$331$	116.289	0.351328	0.175664	−	0.984450i	$-0.443793\pi$
$331$	116.289	0.351328	0.175664	+	0.984450i	$0.443793\pi$
$332$	8.62936i	0.0259920i
$333$	0	0
$334$	−92.9985	−0.278439
$335$	0	0
$336$	0	0
$337$	−253.128	−0.751123	−0.375562	−	0.926797i	$-0.622550\pi$
$337$	−253.128	−0.751123	−0.375562	+	0.926797i	$0.622550\pi$
$338$	68.2332i	0.201873i
$339$	0	0
$340$	0	0
$341$	− 427.453i	− 1.25353i
$342$	0	0
$343$	−304.234	−0.886979
$344$	− 72.6585i	− 0.211216i
$345$	0	0
$346$	−9.53282	−0.0275515
$347$	194.883i	0.561623i	0.959763	+	0.280811i	$0.0906035\pi$
$347$	194.883i	0.561623i	−0.959763	+	0.280811i	$0.909396\pi$
$348$	0	0
$349$	384.734	1.10239	0.551195	−	0.834377i	$-0.314172\pi$
$349$	384.734	1.10239	0.551195	+	0.834377i	$0.314172\pi$
$350$	0	0
$351$	0	0
$352$	−502.045	−1.42626
$353$	36.6392i	0.103794i	0.998652	+	0.0518969i	$0.0165267\pi$
$353$	36.6392i	0.103794i	−0.998652	+	0.0518969i	$0.983473\pi$
$354$	0	0
$355$	0	0
$356$	− 264.056i	− 0.741729i
$357$	0	0
$358$	378.082	1.05609
$359$	219.662i	0.611870i	0.952052	+	0.305935i	$0.0989691\pi$
$359$	219.662i	0.611870i	−0.952052	+	0.305935i	$0.901031\pi$
$360$	0	0
$361$	214.903	0.595300
$362$	− 280.872i	− 0.775890i
$363$	0	0
$364$	198.366	0.544962
$365$	0	0
$366$	0	0
$367$	−117.245	−0.319469	−0.159734	−	0.987160i	$-0.551064\pi$
$367$	−117.245	−0.319469	−0.159734	+	0.987160i	$0.551064\pi$
$368$	− 11.8482i	− 0.0321962i
$369$	0	0
$370$	0	0
$371$	454.336i	1.22463i
$372$	0	0
$373$	−517.173	−1.38652	−0.693261	−	0.720686i	$-0.743827\pi$
$373$	−517.173	−1.38652	−0.693261	+	0.720686i	$0.743827\pi$
$374$	− 30.8125i	− 0.0823863i
$375$	0	0
$376$	−393.195	−1.04573
$377$	− 236.069i	− 0.626177i
$378$	0	0
$379$	22.2216	0.0586323	0.0293161	−	0.999570i	$-0.490667\pi$
$379$	22.2216	0.0586323	0.0293161	+	0.999570i	$0.490667\pi$
$380$	0	0
$381$	0	0
$382$	369.390	0.966990
$383$	418.624i	1.09301i	0.837455	+	0.546507i	$0.184043\pi$
$383$	418.624i	1.09301i	−0.837455	+	0.546507i	$0.815957\pi$
$384$	0	0
$385$	0	0
$386$	10.3127i	0.0267169i
$387$	0	0
$388$	−216.229	−0.557291
$389$	− 5.33849i	− 0.0137236i	−0.999976	−	0.00686182i	$-0.997816\pi$
$389$	− 5.33849i	− 0.0137236i	0.999976	−	0.00686182i	$-0.00218420\pi$
$390$	0	0
$391$	−52.2778	−0.133703
$392$	− 73.1217i	− 0.186535i
$393$	0	0
$394$	27.5426	0.0699050
$395$	0	0
$396$	0	0
$397$	−7.32331	−0.0184466	−0.00922332	−	0.999957i	$-0.502936\pi$
$397$	−7.32331	−0.0184466	−0.00922332	+	0.999957i	$0.502936\pi$
$398$	425.772i	1.06978i
$399$	0	0
$400$	0	0
$401$	− 711.217i	− 1.77361i	−0.462145	−	0.886804i	$-0.652920\pi$
$401$	− 711.217i	− 1.77361i	0.462145	−	0.886804i	$-0.347080\pi$
$402$	0	0
$403$	289.464	0.718273
$404$	74.0038i	0.183178i
$405$	0	0
$406$	−210.499	−0.518471
$407$	− 454.767i	− 1.11736i
$408$	0	0
$409$	−473.476	−1.15764	−0.578821	−	0.815454i	$-0.696487\pi$
$409$	−473.476	−1.15764	−0.578821	+	0.815454i	$0.696487\pi$
$410$	0	0
$411$	0	0
$412$	−449.460	−1.09092
$413$	761.253i	1.84323i
$414$	0	0
$415$	0	0
$416$	− 339.976i	− 0.817251i
$417$	0	0
$418$	474.763	1.13580
$419$	− 113.330i	− 0.270478i	−0.990813	−	0.135239i	$-0.956820\pi$
$419$	− 113.330i	− 0.270478i	0.990813	−	0.135239i	$-0.0431803\pi$
$420$	0	0
$421$	−235.223	−0.558725	−0.279363	−	0.960186i	$-0.590123\pi$
$421$	−235.223	−0.558725	−0.279363	+	0.960186i	$0.590123\pi$
$422$	− 210.233i	− 0.498183i
$423$	0	0
$424$	480.120	1.13236
$425$	0	0
$426$	0	0
$427$	−679.883	−1.59223
$428$	− 65.0972i	− 0.152096i
$429$	0	0
$430$	0	0
$431$	491.160i	1.13958i	0.821790	+	0.569791i	$0.192976\pi$
$431$	491.160i	1.13958i	−0.821790	+	0.569791i	$0.807024\pi$
$432$	0	0
$433$	−456.794	−1.05495	−0.527475	−	0.849570i	$-0.676861\pi$
$433$	−456.794	−1.05495	−0.527475	+	0.849570i	$0.676861\pi$
$434$	− 258.111i	− 0.594726i
$435$	0	0
$436$	14.6253	0.0335443
$437$	− 805.504i	− 1.84326i
$438$	0	0
$439$	−179.662	−0.409253	−0.204626	−	0.978840i	$-0.565598\pi$
$439$	−179.662	−0.409253	−0.204626	+	0.978840i	$0.565598\pi$
$440$	0	0
$441$	0	0
$442$	20.8657	0.0472075
$443$	392.984i	0.887098i	0.896250	+	0.443549i	$0.146281\pi$
$443$	392.984i	0.887098i	−0.896250	+	0.443549i	$0.853719\pi$
$444$	0	0
$445$	0	0
$446$	55.6157i	0.124699i
$447$	0	0
$448$	−313.913	−0.700698
$449$	− 515.004i	− 1.14700i	−0.819204	−	0.573502i	$-0.805585\pi$
$449$	− 515.004i	− 1.14700i	0.819204	−	0.573502i	$-0.194415\pi$
$450$	0	0
$451$	−762.919	−1.69162
$452$	473.681i	1.04797i
$453$	0	0
$454$	460.699	1.01476
$455$	0	0
$456$	0	0
$457$	593.405	1.29848	0.649240	−	0.760584i	$-0.275087\pi$
$457$	593.405	1.29848	0.649240	+	0.760584i	$0.275087\pi$
$458$	114.621i	0.250265i
$459$	0	0
$460$	0	0
$461$	− 819.771i	− 1.77824i	−0.457670	−	0.889122i	$-0.651316\pi$
$461$	− 819.771i	− 1.77824i	0.457670	−	0.889122i	$-0.348684\pi$
$462$	0	0
$463$	−24.6308	−0.0531982	−0.0265991	−	0.999646i	$-0.508468\pi$
$463$	−24.6308	−0.0531982	−0.0265991	+	0.999646i	$0.508468\pi$
$464$	7.78744i	0.0167833i
$465$	0	0
$466$	392.376	0.842009
$467$	355.316i	0.760848i	0.924812	+	0.380424i	$0.124222\pi$
$467$	355.316i	0.760848i	−0.924812	+	0.380424i	$0.875778\pi$
$468$	0	0
$469$	14.1713	0.0302161
$470$	0	0
$471$	0	0
$472$	804.454	1.70435
$473$	− 142.561i	− 0.301397i
$474$	0	0
$475$	0	0
$476$	28.8727i	0.0606570i
$477$	0	0
$478$	−24.3826	−0.0510096
$479$	− 192.628i	− 0.402147i	−0.979576	−	0.201073i	$-0.935557\pi$
$479$	− 192.628i	− 0.402147i	0.979576	−	0.201073i	$-0.0644429\pi$
$480$	0	0
$481$	307.960	0.640250
$482$	− 84.1180i	− 0.174519i
$483$	0	0
$484$	−313.030	−0.646757
$485$	0	0
$486$	0	0
$487$	597.148	1.22618	0.613088	−	0.790015i	$-0.289927\pi$
$487$	597.148	1.22618	0.613088	+	0.790015i	$0.289927\pi$
$488$	718.467i	1.47227i
$489$	0	0
$490$	0	0
$491$	− 571.283i	− 1.16351i	−0.813364	−	0.581755i	$-0.802366\pi$
$491$	− 571.283i	− 1.16351i	0.813364	−	0.581755i	$-0.197634\pi$
$492$	0	0
$493$	34.3605	0.0696967
$494$	321.502i	0.650813i
$495$	0	0
$496$	−9.54884	−0.0192517
$497$	− 961.075i	− 1.93375i
$498$	0	0
$499$	451.031	0.903869	0.451934	−	0.892051i	$-0.350734\pi$
$499$	451.031	0.903869	0.451934	+	0.892051i	$0.350734\pi$
$500$	0	0
$501$	0	0
$502$	225.644	0.449490
$503$	− 244.289i	− 0.485664i	−0.970068	−	0.242832i	$-0.921924\pi$
$503$	− 244.289i	− 0.485664i	0.970068	−	0.242832i	$-0.0780764\pi$
$504$	0	0
$505$	0	0
$506$	− 664.042i	− 1.31234i
$507$	0	0
$508$	494.702	0.973823
$509$	288.068i	0.565950i	0.959127	+	0.282975i	$0.0913213\pi$
$509$	288.068i	0.565950i	−0.959127	+	0.282975i	$0.908679\pi$
$510$	0	0
$511$	769.746	1.50635
$512$	22.5863i	0.0441138i
$513$	0	0
$514$	130.752	0.254381
$515$	0	0
$516$	0	0
$517$	−771.475	−1.49222
$518$	− 274.604i	− 0.530123i
$519$	0	0
$520$	0	0
$521$	− 845.882i	− 1.62357i	−0.583953	−	0.811787i	$-0.698495\pi$
$521$	− 845.882i	− 1.62357i	0.583953	−	0.811787i	$-0.301505\pi$
$522$	0	0
$523$	−551.696	−1.05487	−0.527434	−	0.849596i	$-0.676846\pi$
$523$	−551.696	−1.05487	−0.527434	+	0.849596i	$0.676846\pi$
$524$	334.236i	0.637854i
$525$	0	0
$526$	−7.37754	−0.0140257
$527$	42.1323i	0.0799475i
$528$	0	0
$529$	−597.642	−1.12976
$530$	0	0
$531$	0	0
$532$	−444.875	−0.836232
$533$	− 516.636i	− 0.969298i
$534$	0	0
$535$	0	0
$536$	− 14.9756i	− 0.0279395i
$537$	0	0
$538$	340.348	0.632616
$539$	− 143.470i	− 0.266178i
$540$	0	0
$541$	−424.879	−0.785358	−0.392679	−	0.919676i	$-0.628452\pi$
$541$	−424.879	−0.785358	−0.392679	+	0.919676i	$0.628452\pi$
$542$	248.690i	0.458837i
$543$	0	0
$544$	49.4845	0.0909641
$545$	0	0
$546$	0	0
$547$	−202.160	−0.369579	−0.184789	−	0.982778i	$-0.559160\pi$
$547$	−202.160	−0.369579	−0.184789	+	0.982778i	$0.559160\pi$
$548$	− 66.4143i	− 0.121194i
$549$	0	0
$550$	0	0
$551$	529.431i	0.960855i
$552$	0	0
$553$	827.952	1.49720
$554$	− 178.373i	− 0.321974i
$555$	0	0
$556$	341.895	0.614920
$557$	− 638.115i	− 1.14563i	−0.819685	−	0.572814i	$-0.805852\pi$
$557$	− 638.115i	− 1.14563i	0.819685	−	0.572814i	$-0.194148\pi$
$558$	0	0
$559$	96.5398	0.172701
$560$	0	0
$561$	0	0
$562$	84.3561	0.150100
$563$	472.368i	0.839019i	0.907751	+	0.419509i	$0.137798\pi$
$563$	472.368i	0.839019i	−0.907751	+	0.419509i	$0.862202\pi$
$564$	0	0
$565$	0	0
$566$	− 102.724i	− 0.181491i
$567$	0	0
$568$	−1015.62	−1.78806
$569$	− 655.843i	− 1.15262i	−0.817230	−	0.576312i	$-0.804491\pi$
$569$	− 655.843i	− 1.15262i	0.817230	−	0.576312i	$-0.195509\pi$
$570$	0	0
$571$	491.508	0.860785	0.430392	−	0.902642i	$-0.358375\pi$
$571$	491.508	0.860785	0.430392	+	0.902642i	$0.358375\pi$
$572$	− 411.295i	− 0.719048i
$573$	0	0
$574$	−460.677	−0.802573
$575$	0	0
$576$	0	0
$577$	−789.855	−1.36890	−0.684450	−	0.729060i	$-0.739957\pi$
$577$	−789.855	−1.36890	−0.684450	+	0.729060i	$0.739957\pi$
$578$	− 358.791i	− 0.620746i
$579$	0	0
$580$	0	0
$581$	27.0357i	0.0465331i
$582$	0	0
$583$	942.028	1.61583
$584$	− 813.430i	− 1.39286i
$585$	0	0
$586$	−182.485	−0.311408
$587$	1079.13i	1.83839i	0.393808	+	0.919193i	$0.371158\pi$
$587$	1079.13i	1.83839i	−0.393808	+	0.919193i	$0.628842\pi$
$588$	0	0
$589$	−649.181	−1.10217
$590$	0	0
$591$	0	0
$592$	−10.1590	−0.0171605
$593$	− 448.894i	− 0.756989i	−0.925604	−	0.378494i	$-0.876442\pi$
$593$	− 448.894i	− 0.756989i	0.925604	−	0.378494i	$-0.123558\pi$
$594$	0	0
$595$	0	0
$596$	320.405i	0.537592i
$597$	0	0
$598$	449.678	0.751970
$599$	− 229.646i	− 0.383382i	−0.981455	−	0.191691i	$-0.938603\pi$
$599$	− 229.646i	− 0.383382i	0.981455	−	0.191691i	$-0.0613972\pi$
$600$	0	0
$601$	172.020	0.286223	0.143112	−	0.989707i	$-0.454289\pi$
$601$	172.020	0.286223	0.143112	+	0.989707i	$0.454289\pi$
$602$	− 86.0832i	− 0.142995i
$603$	0	0
$604$	−566.072	−0.937205
$605$	0	0
$606$	0	0
$607$	703.607	1.15915	0.579577	−	0.814917i	$-0.303218\pi$
$607$	703.607	1.15915	0.579577	+	0.814917i	$0.303218\pi$
$608$	762.464i	1.25405i
$609$	0	0
$610$	0	0
$611$	− 522.430i	− 0.855041i
$612$	0	0
$613$	338.822	0.552728	0.276364	−	0.961053i	$-0.410870\pi$
$613$	338.822	0.552728	0.276364	+	0.961053i	$0.410870\pi$
$614$	157.212i	0.256045i
$615$	0	0
$616$	−969.825	−1.57439
$617$	699.196i	1.13322i	0.823987	+	0.566609i	$0.191745\pi$
$617$	699.196i	1.13322i	−0.823987	+	0.566609i	$0.808255\pi$
$618$	0	0
$619$	−210.885	−0.340686	−0.170343	−	0.985385i	$-0.554488\pi$
$619$	−210.885	−0.340686	−0.170343	+	0.985385i	$0.554488\pi$
$620$	0	0
$621$	0	0
$622$	134.699	0.216559
$623$	− 827.284i	− 1.32790i
$624$	0	0
$625$	0	0
$626$	426.975i	0.682069i
$627$	0	0
$628$	−18.5625	−0.0295582
$629$	44.8245i	0.0712631i
$630$	0	0
$631$	26.6344	0.0422099	0.0211049	−	0.999777i	$-0.493282\pi$
$631$	26.6344	0.0422099	0.0211049	+	0.999777i	$0.493282\pi$
$632$	− 874.939i	− 1.38440i
$633$	0	0
$634$	−34.1276	−0.0538291
$635$	0	0
$636$	0	0
$637$	97.1553	0.152520
$638$	436.453i	0.684095i
$639$	0	0
$640$	0	0
$641$	244.672i	0.381704i	0.981619	+	0.190852i	$0.0611250\pi$
$641$	244.672i	0.381704i	−0.981619	+	0.190852i	$0.938875\pi$
$642$	0	0
$643$	−267.024	−0.415279	−0.207639	−	0.978205i	$-0.566578\pi$
$643$	−267.024	−0.415279	−0.207639	+	0.978205i	$0.566578\pi$
$644$	622.238i	0.966208i
$645$	0	0
$646$	−46.7955	−0.0724388
$647$	974.715i	1.50651i	0.657726	+	0.753257i	$0.271519\pi$
$647$	974.715i	1.50651i	−0.657726	+	0.753257i	$0.728481\pi$
$648$	0	0
$649$	1578.39	2.43204
$650$	0	0
$651$	0	0
$652$	751.756	1.15300
$653$	752.254i	1.15200i	0.817451	+	0.575998i	$0.195386\pi$
$653$	752.254i	1.15200i	−0.817451	+	0.575998i	$0.804614\pi$
$654$	0	0
$655$	0	0
$656$	17.0428i	0.0259798i
$657$	0	0
$658$	−465.844	−0.707969
$659$	− 93.3999i	− 0.141730i	−0.997486	−	0.0708649i	$-0.977424\pi$
$659$	− 93.3999i	− 0.141730i	0.997486	−	0.0708649i	$-0.0225759\pi$
$660$	0	0
$661$	1221.56	1.84805	0.924025	−	0.382333i	$-0.124879\pi$
$661$	1221.56	1.84805	0.924025	+	0.382333i	$0.124879\pi$
$662$	− 145.594i	− 0.219931i
$663$	0	0
$664$	28.5700	0.0430271
$665$	0	0
$666$	0	0
$667$	740.505	1.11020
$668$	− 180.686i	− 0.270487i
$669$	0	0
$670$	0	0
$671$	1409.68i	2.10087i
$672$	0	0
$673$	926.253	1.37630	0.688152	−	0.725566i	$-0.258422\pi$
$673$	926.253	1.37630	0.688152	+	0.725566i	$0.258422\pi$
$674$	316.917i	0.470203i
$675$	0	0
$676$	−132.569	−0.196109
$677$	672.436i	0.993258i	0.867963	+	0.496629i	$0.165429\pi$
$677$	672.436i	0.993258i	−0.867963	+	0.496629i	$0.834571\pi$
$678$	0	0
$679$	−677.444	−0.997709
$680$	0	0
$681$	0	0
$682$	−535.172	−0.784709
$683$	− 64.4780i	− 0.0944041i	−0.998885	−	0.0472020i	$-0.984970\pi$
$683$	− 64.4780i	− 0.0944041i	0.998885	−	0.0472020i	$-0.0150305\pi$
$684$	0	0
$685$	0	0
$686$	380.901i	0.555249i
$687$	0	0
$688$	−3.18465	−0.00462885
$689$	637.926i	0.925872i
$690$	0	0
$691$	605.634	0.876460	0.438230	−	0.898863i	$-0.355605\pi$
$691$	605.634	0.876460	0.438230	+	0.898863i	$0.355605\pi$
$692$	− 18.5212i	− 0.0267647i
$693$	0	0
$694$	243.994	0.351576
$695$	0	0
$696$	0	0
$697$	75.1978	0.107888
$698$	− 481.687i	− 0.690096i
$699$	0	0
$700$	0	0
$701$	63.2631i	0.0902469i	0.998981	+	0.0451234i	$0.0143681\pi$
$701$	63.2631i	0.0902469i	−0.998981	+	0.0451234i	$0.985632\pi$
$702$	0	0
$703$	−690.662	−0.982450
$704$	650.871i	0.924533i
$705$	0	0
$706$	45.8723	0.0649750
$707$	231.853i	0.327940i
$708$	0	0
$709$	−427.518	−0.602988	−0.301494	−	0.953468i	$-0.597485\pi$
$709$	−427.518	−0.602988	−0.301494	+	0.953468i	$0.597485\pi$
$710$	0	0
$711$	0	0
$712$	−874.233	−1.22786
$713$	907.996i	1.27349i
$714$	0	0
$715$	0	0
$716$	734.570i	1.02594i
$717$	0	0
$718$	275.016	0.383031
$719$	− 396.331i	− 0.551226i	−0.961269	−	0.275613i	$-0.911119\pi$
$719$	− 396.331i	− 0.551226i	0.961269	−	0.275613i	$-0.0888808\pi$
$720$	0	0
$721$	−1408.16	−1.95306
$722$	− 269.059i	− 0.372658i
$723$	0	0
$724$	545.703	0.753734
$725$	0	0
$726$	0	0
$727$	625.939	0.860989	0.430495	−	0.902593i	$-0.358339\pi$
$727$	625.939	0.860989	0.430495	+	0.902593i	$0.358339\pi$
$728$	− 656.749i	− 0.902128i
$729$	0	0
$730$	0	0
$731$	14.0516i	0.0192225i
$732$	0	0
$733$	−187.721	−0.256099	−0.128050	−	0.991768i	$-0.540872\pi$
$733$	−187.721	−0.256099	−0.128050	+	0.991768i	$0.540872\pi$
$734$	146.791i	0.199987i
$735$	0	0
$736$	1066.44	1.44897
$737$	− 29.3831i	− 0.0398685i
$738$	0	0
$739$	1271.61	1.72072	0.860361	−	0.509686i	$-0.170238\pi$
$739$	1271.61	1.72072	0.860361	+	0.509686i	$0.170238\pi$
$740$	0	0
$741$	0	0
$742$	568.829	0.766616
$743$	− 1411.00i	− 1.89906i	−0.313684	−	0.949528i	$-0.601563\pi$
$743$	− 1411.00i	− 1.89906i	0.313684	−	0.949528i	$-0.398437\pi$
$744$	0	0
$745$	0	0
$746$	647.501i	0.867964i
$747$	0	0
$748$	59.8652	0.0800337
$749$	− 203.949i	− 0.272295i
$750$	0	0
$751$	−210.475	−0.280260	−0.140130	−	0.990133i	$-0.544752\pi$
$751$	−210.475	−0.280260	−0.140130	+	0.990133i	$0.544752\pi$
$752$	17.2339i	0.0229174i
$753$	0	0
$754$	−295.558	−0.391987
$755$	0	0
$756$	0	0
$757$	6.79805	0.00898024	0.00449012	−	0.999990i	$-0.498571\pi$
$757$	6.79805	0.00898024	0.00449012	+	0.999990i	$0.498571\pi$
$758$	− 27.8215i	− 0.0367038i
$759$	0	0
$760$	0	0
$761$	− 652.208i	− 0.857040i	−0.903532	−	0.428520i	$-0.859035\pi$
$761$	− 652.208i	− 0.857040i	0.903532	−	0.428520i	$-0.140965\pi$
$762$	0	0
$763$	45.8210	0.0600538
$764$	717.683i	0.939376i
$765$	0	0
$766$	524.118	0.684227
$767$	1068.86i	1.39356i
$768$	0	0
$769$	−920.972	−1.19762	−0.598811	−	0.800890i	$-0.704360\pi$
$769$	−920.972	−1.19762	−0.598811	+	0.800890i	$0.704360\pi$
$770$	0	0
$771$	0	0
$772$	−20.0365	−0.0259540
$773$	− 990.700i	− 1.28163i	−0.767696	−	0.640815i	$-0.778597\pi$
$773$	− 990.700i	− 1.28163i	0.767696	−	0.640815i	$-0.221403\pi$
$774$	0	0
$775$	0	0
$776$	715.890i	0.922538i
$777$	0	0
$778$	−6.68380	−0.00859100
$779$	1158.66i	1.48737i
$780$	0	0
$781$	−1992.71	−2.55148
$782$	65.4518i	0.0836980i
$783$	0	0
$784$	−3.20495	−0.00408795
$785$	0	0
$786$	0	0
$787$	−1328.97	−1.68866	−0.844328	−	0.535827i	$-0.820000\pi$
$787$	−1328.97	−1.68866	−0.844328	+	0.535827i	$0.820000\pi$
$788$	53.5121i	0.0679088i
$789$	0	0
$790$	0	0
$791$	1484.04i	1.87616i
$792$	0	0
$793$	−954.612	−1.20380
$794$	9.16879i	0.0115476i
$795$	0	0
$796$	−827.226	−1.03923
$797$	− 1276.68i	− 1.60186i	−0.598758	−	0.800930i	$-0.704339\pi$
$797$	− 1276.68i	− 1.60186i	0.598758	−	0.800930i	$-0.295661\pi$
$798$	0	0
$799$	76.0411	0.0951704
$800$	0	0
$801$	0	0
$802$	−890.444	−1.11028
$803$	− 1596.00i	− 1.98755i
$804$	0	0
$805$	0	0
$806$	− 362.409i	− 0.449639i
$807$	0	0
$808$	245.011	0.303232
$809$	− 765.367i	− 0.946065i	−0.881045	−	0.473033i	$-0.843159\pi$
$809$	− 765.367i	− 0.946065i	0.881045	−	0.473033i	$-0.156841\pi$
$810$	0	0
$811$	−89.9037	−0.110855	−0.0554277	−	0.998463i	$-0.517652\pi$
$811$	−89.9037	−0.110855	−0.0554277	+	0.998463i	$0.517652\pi$
$812$	− 408.976i	− 0.503665i
$813$	0	0
$814$	−569.368	−0.699470
$815$	0	0
$816$	0	0
$817$	−216.510	−0.265006
$818$	592.792i	0.724685i
$819$	0	0
$820$	0	0
$821$	− 459.145i	− 0.559251i	−0.960109	−	0.279626i	$-0.909790\pi$
$821$	− 459.145i	− 0.559251i	0.960109	−	0.279626i	$-0.0902104\pi$
$822$	0	0
$823$	−504.672	−0.613210	−0.306605	−	0.951837i	$-0.599193\pi$
$823$	−504.672	−0.613210	−0.306605	+	0.951837i	$0.599193\pi$
$824$	1488.07i	1.80591i
$825$	0	0
$826$	953.089	1.15386
$827$	1399.26i	1.69197i	0.533204	+	0.845986i	$0.320988\pi$
$827$	1399.26i	1.69197i	−0.533204	+	0.845986i	$0.679012\pi$
$828$	0	0
$829$	−19.9383	−0.0240510	−0.0120255	−	0.999928i	$-0.503828\pi$
$829$	−19.9383	−0.0240510	−0.0120255	+	0.999928i	$0.503828\pi$
$830$	0	0
$831$	0	0
$832$	−440.759	−0.529758
$833$	14.1412i	0.0169762i
$834$	0	0
$835$	0	0
$836$	922.411i	1.10336i
$837$	0	0
$838$	−141.890	−0.169320
$839$	277.472i	0.330717i	0.986233	+	0.165359i	$0.0528782\pi$
$839$	277.472i	0.330717i	−0.986233	+	0.165359i	$0.947122\pi$
$840$	0	0
$841$	354.291	0.421273
$842$	294.500i	0.349762i
$843$	0	0
$844$	408.460	0.483957
$845$	0	0
$846$	0	0
$847$	−980.722	−1.15788
$848$	− 21.0439i	− 0.0248159i
$849$	0	0
$850$	0	0
$851$	966.015i	1.13515i
$852$	0	0
$853$	−534.191	−0.626250	−0.313125	−	0.949712i	$-0.601376\pi$
$853$	−534.191	−0.626250	−0.313125	+	0.949712i	$0.601376\pi$
$854$	851.214i	0.996738i
$855$	0	0
$856$	−215.523	−0.251779
$857$	744.422i	0.868637i	0.900759	+	0.434318i	$0.143011\pi$
$857$	744.422i	0.868637i	−0.900759	+	0.434318i	$0.856989\pi$
$858$	0	0
$859$	−292.861	−0.340932	−0.170466	−	0.985364i	$-0.554527\pi$
$859$	−292.861	−0.340932	−0.170466	+	0.985364i	$0.554527\pi$
$860$	0	0
$861$	0	0
$862$	614.932	0.713378
$863$	− 599.294i	− 0.694431i	−0.937785	−	0.347215i	$-0.887127\pi$
$863$	− 599.294i	− 0.694431i	0.937785	−	0.347215i	$-0.112873\pi$
$864$	0	0
$865$	0	0
$866$	571.906i	0.660399i
$867$	0	0
$868$	501.481	0.577743
$869$	− 1716.69i	− 1.97548i
$870$	0	0
$871$	19.8977	0.0228447
$872$	− 48.4214i	− 0.0555291i
$873$	0	0
$874$	−1008.49	−1.15388
$875$	0	0
$876$	0	0
$877$	1279.44	1.45889	0.729443	−	0.684041i	$-0.239779\pi$
$877$	1279.44	1.45889	0.729443	+	0.684041i	$0.239779\pi$
$878$	224.937i	0.256192i
$879$	0	0
$880$	0	0
$881$	− 600.782i	− 0.681932i	−0.940076	−	0.340966i	$-0.889246\pi$
$881$	− 600.782i	− 0.681932i	0.940076	−	0.340966i	$-0.110754\pi$
$882$	0	0
$883$	−968.646	−1.09699	−0.548497	−	0.836153i	$-0.684800\pi$
$883$	−968.646	−1.09699	−0.548497	+	0.836153i	$0.684800\pi$
$884$	40.5397i	0.0458594i
$885$	0	0
$886$	492.017	0.555324
$887$	396.806i	0.447357i	0.974663	+	0.223679i	$0.0718066\pi$
$887$	396.806i	0.447357i	−0.974663	+	0.223679i	$0.928193\pi$
$888$	0	0
$889$	1549.90	1.74342
$890$	0	0
$891$	0	0
$892$	−108.055	−0.121138
$893$	1171.65i	1.31204i
$894$	0	0
$895$	0	0
$896$	− 575.518i	− 0.642319i
$897$	0	0
$898$	−644.786	−0.718024
$899$	− 596.796i	− 0.663844i
$900$	0	0
$901$	−92.8518	−0.103054
$902$	955.175i	1.05895i
$903$	0	0
$904$	1568.26	1.73480
$905$	0	0
$906$	0	0
$907$	1396.36	1.53954	0.769768	−	0.638324i	$-0.220372\pi$
$907$	1396.36	1.53954	0.769768	+	0.638324i	$0.220372\pi$
$908$	895.086i	0.985777i
$909$	0	0
$910$	0	0
$911$	268.377i	0.294596i	0.989092	+	0.147298i	$0.0470576\pi$
$911$	268.377i	0.294596i	−0.989092	+	0.147298i	$0.952942\pi$
$912$	0	0
$913$	56.0563	0.0613979
$914$	− 742.943i	− 0.812848i
$915$	0	0
$916$	−222.696	−0.243118
$917$	1047.16i	1.14194i
$918$	0	0
$919$	494.547	0.538136	0.269068	−	0.963121i	$-0.413284\pi$
$919$	494.547	0.538136	0.269068	+	0.963121i	$0.413284\pi$
$920$	0	0
$921$	0	0
$922$	−1026.35	−1.11318
$923$	− 1349.43i	− 1.46200i
$924$	0	0
$925$	0	0
$926$	30.8377i	0.0333021i
$927$	0	0
$928$	−700.938	−0.755321
$929$	− 590.099i	− 0.635198i	−0.948225	−	0.317599i	$-0.897123\pi$
$929$	− 590.099i	− 0.635198i	0.948225	−	0.317599i	$-0.102877\pi$
$930$	0	0
$931$	−217.890	−0.234039
$932$	762.342i	0.817964i
$933$	0	0
$934$	444.856	0.476291
$935$	0	0
$936$	0	0
$937$	1768.81	1.88774	0.943869	−	0.330320i	$-0.107157\pi$
$937$	1768.81	1.88774	0.943869	+	0.330320i	$0.107157\pi$
$938$	− 17.7425i	− 0.0189153i
$939$	0	0
$940$	0	0
$941$	678.388i	0.720922i	0.932774	+	0.360461i	$0.117381\pi$
$941$	678.388i	0.720922i	−0.932774	+	0.360461i	$0.882619\pi$
$942$	0	0
$943$	1620.59	1.71855
$944$	− 35.2596i	− 0.0373512i
$945$	0	0
$946$	−178.486	−0.188675
$947$	− 1642.89i	− 1.73484i	−0.497577	−	0.867420i	$-0.665777\pi$
$947$	− 1642.89i	− 1.73484i	0.497577	−	0.867420i	$-0.334223\pi$
$948$	0	0
$949$	1080.79	1.13887
$950$	0	0
$951$	0	0
$952$	95.5916	0.100411
$953$	− 1296.78i	− 1.36074i	−0.732869	−	0.680369i	$-0.761819\pi$
$953$	− 1296.78i	− 1.36074i	0.732869	−	0.680369i	$-0.238181\pi$
$954$	0	0
$955$	0	0
$956$	− 47.3726i	− 0.0495529i
$957$	0	0
$958$	−241.171	−0.251744
$959$	− 208.076i	− 0.216971i
$960$	0	0
$961$	−229.218	−0.238520
$962$	− 385.567i	− 0.400797i
$963$	0	0
$964$	163.432	0.169535
$965$	0	0
$966$	0	0
$967$	−1429.01	−1.47777	−0.738886	−	0.673830i	$-0.764648\pi$
$967$	−1429.01	−1.47777	−0.738886	+	0.673830i	$0.764648\pi$
$968$	1036.38i	1.07064i
$969$	0	0
$970$	0	0
$971$	1195.60i	1.23131i	0.788018	+	0.615653i	$0.211108\pi$
$971$	1195.60i	1.23131i	−0.788018	+	0.615653i	$0.788892\pi$
$972$	0	0
$973$	1071.16	1.10088
$974$	− 747.629i	− 0.767586i
$975$	0	0
$976$	31.4907	0.0322651
$977$	1046.06i	1.07069i	0.844635	+	0.535343i	$0.179817\pi$
$977$	1046.06i	1.07069i	−0.844635	+	0.535343i	$0.820183\pi$
$978$	0	0
$979$	−1715.30	−1.75210
$980$	0	0
$981$	0	0
$982$	−715.247	−0.728357
$983$	− 549.332i	− 0.558832i	−0.960170	−	0.279416i	$-0.909859\pi$
$983$	− 549.332i	− 0.558832i	0.960170	−	0.279416i	$-0.0901409\pi$
$984$	0	0
$985$	0	0
$986$	− 43.0193i	− 0.0436302i
$987$	0	0
$988$	−624.641	−0.632228
$989$	302.827i	0.306196i
$990$	0	0
$991$	−731.635	−0.738280	−0.369140	−	0.929374i	$-0.620348\pi$
$991$	−731.635	−0.738280	−0.369140	+	0.929374i	$0.620348\pi$
$992$	− 859.479i	− 0.866411i
$993$	0	0
$994$	−1203.27	−1.21053
$995$	0	0
$996$	0	0
$997$	−187.559	−0.188123	−0.0940615	−	0.995566i	$-0.529985\pi$
$997$	−187.559	−0.188123	−0.0940615	+	0.995566i	$0.529985\pi$
$998$	− 564.690i	− 0.565822i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.3.c.r.26.3	✓	6
3.2	odd	2	inner	675.3.c.r.26.4	yes	6
5.2	odd	4		675.3.d.k.674.4		6
5.3	odd	4		675.3.d.j.674.3		6
5.4	even	2		675.3.c.s.26.4	yes	6
15.2	even	4		675.3.d.j.674.4		6
15.8	even	4		675.3.d.k.674.3		6
15.14	odd	2		675.3.c.s.26.3	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
675.3.c.r.26.3	✓	6	1.1	even	1	trivial
675.3.c.r.26.4	yes	6	3.2	odd	2	inner
675.3.c.s.26.3	yes	6	15.14	odd	2
675.3.c.s.26.4	yes	6	5.4	even	2
675.3.d.j.674.3		6	5.3	odd	4
675.3.d.j.674.4		6	15.2	even	4
675.3.d.k.674.3		6	15.8	even	4
675.3.d.k.674.4		6	5.2	odd	4

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:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	675.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.25200i q^{2} +2.43250 q^{4} +7.62099 q^{7} -8.05349i q^{8} -15.8015i q^{11} +10.7005 q^{13} -9.54148i q^{14} -0.352988 q^{16} +1.55749i q^{17} -23.9980 q^{19} -19.7835 q^{22} +33.5655i q^{23} -13.3970i q^{26} +18.5380 q^{28} -22.0615i q^{29} +27.0515 q^{31} -31.7720i q^{32} +1.94997 q^{34} +28.7800 q^{37} +30.0455i q^{38} -48.2815i q^{41} +9.02199 q^{43} -38.4370i q^{44} +42.0240 q^{46} -48.8230i q^{47} +9.07951 q^{49} +26.0289 q^{52} +59.6165i q^{53} -61.3755i q^{56} -27.6210 q^{58} +99.8889i q^{59} -89.2119 q^{61} -33.8685i q^{62} -41.1905 q^{64} +1.85951 q^{67} +3.78858i q^{68} -126.109i q^{71} +101.003 q^{73} -36.0326i q^{74} -58.3750 q^{76} -120.423i q^{77} +108.641 q^{79} -60.4484 q^{82} +3.54753i q^{83} -11.2955i q^{86} -127.257 q^{88} -108.553i q^{89} +81.5484 q^{91} +81.6479i q^{92} -61.1264 q^{94} -88.8919 q^{97} -11.3675i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 14 q^{4} - 16 q^{7} + 22 q^{13} + 46 q^{16} - 16 q^{19} + 86 q^{22} + 212 q^{28} - 56 q^{31} - 80 q^{34} + 150 q^{37} - 92 q^{43} + 234 q^{46} + 74 q^{49} - 354 q^{52} - 104 q^{58} - 46 q^{61} - 342 q^{64}+ \cdots + 162 q^{97}+O(q^{100}) \) 6 * q - 14 * q^4 - 16 * q^7 + 22 * q^13 + 46 * q^16 - 16 * q^19 + 86 * q^22 + 212 * q^28 - 56 * q^31 - 80 * q^34 + 150 * q^37 - 92 * q^43 + 234 * q^46 + 74 * q^49 - 354 * q^52 - 104 * q^58 - 46 * q^61 - 342 * q^64 + 8 * q^67 + 424 * q^73 - 156 * q^76 + 316 * q^79 + 196 * q^82 - 858 * q^88 + 44 * q^91 + 526 * q^94 + 162 * q^97

Introduction

L-functions

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Database

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