LMFDB - Embedded newform 1350.4.c.u.649.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1350,4,Mod(649,1350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1350, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1350.649");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1350 = 2 \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1350.c (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:	$79.6525785077$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{401})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 201x^{2} + 10000 $ x^4 + 201*x^2 + 10000
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 270)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.2
Root			$-10.5125i$ of defining polynomial
Character	$\chi$	$=$	1350.649
Dual form			1350.4.c.u.649.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{2} -4.00000 q^{4} +30.5375i q^{7} +8.00000i q^{8} +O(q^{10})$	$q-2.00000i q^{2} -4.00000 q^{4} +30.5375i q^{7} +8.00000i q^{8} +13.5375 q^{11} +28.0750i q^{13} +61.0750 q^{14} +16.0000 q^{16} -55.5375i q^{17} -27.4625 q^{19} -27.0750i q^{22} -139.687i q^{23} +56.1499 q^{26} -122.150i q^{28} +178.762 q^{29} +297.687 q^{31} -32.0000i q^{32} -111.075 q^{34} +159.612i q^{37} +54.9250i q^{38} +140.775 q^{41} +5.68738i q^{43} -54.1499 q^{44} -279.375 q^{46} +301.987i q^{47} -589.537 q^{49} -112.300i q^{52} -122.925i q^{53} -244.300 q^{56} -357.525i q^{58} -864.300 q^{59} -47.6124 q^{61} -595.375i q^{62} -64.0000 q^{64} +402.388i q^{67} +222.150i q^{68} +927.225 q^{71} +1019.61i q^{73} +319.225 q^{74} +109.850 q^{76} +413.400i q^{77} -812.225 q^{79} -281.550i q^{82} +1381.05i q^{83} +11.3748 q^{86} +108.300i q^{88} -1423.35 q^{89} -857.338 q^{91} +558.750i q^{92} +603.974 q^{94} -124.863i q^{97} +1179.07i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 16 q^{4}+O(q^{10}) $ 4 * q - 16 * q^4	$ 4 q - 16 q^{4} - 66 q^{11} + 4 q^{14} + 64 q^{16} - 230 q^{19} - 256 q^{26} - 126 q^{29} + 590 q^{31} - 204 q^{34} + 1284 q^{41} + 264 q^{44} + 84 q^{46} - 2238 q^{49} - 16 q^{56} - 2496 q^{59} + 170 q^{61} - 256 q^{64} + 2988 q^{71} + 556 q^{74} + 920 q^{76} - 2528 q^{79} - 1156 q^{86} - 1368 q^{89} - 7154 q^{91} - 708 q^{94}+O(q^{100}) $ 4 * q - 16 * q^4 - 66 * q^11 + 4 * q^14 + 64 * q^16 - 230 * q^19 - 256 * q^26 - 126 * q^29 + 590 * q^31 - 204 * q^34 + 1284 * q^41 + 264 * q^44 + 84 * q^46 - 2238 * q^49 - 16 * q^56 - 2496 * q^59 + 170 * q^61 - 256 * q^64 + 2988 * q^71 + 556 * q^74 + 920 * q^76 - 2528 * q^79 - 1156 * q^86 - 1368 * q^89 - 7154 * q^91 - 708 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times$.

$n$	$1001$	$1027$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 2.00000i	− 0.707107i
$2$	− 2.00000i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−4.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	30.5375i	1.64887i	0.565958	+	0.824434i	$0.308507\pi$
$7$	30.5375i	1.64887i	−0.565958	+	0.824434i	$0.691493\pi$
$8$	8.00000i	0.353553i
$9$	0	0
$10$	0	0
$11$	13.5375	0.371064	0.185532	−	0.982638i	$-0.440599\pi$
$11$	13.5375	0.371064	0.185532	+	0.982638i	$0.440599\pi$
$12$	0	0
$13$	28.0750i	0.598969i	0.954101	+	0.299484i	$0.0968146\pi$
$13$	28.0750i	0.598969i	−0.954101	+	0.299484i	$0.903185\pi$
$14$	61.0750	1.16593
$15$	0	0
$16$	16.0000	0.250000
$17$	− 55.5375i	− 0.792342i	−0.918177	−	0.396171i	$-0.870339\pi$
$17$	− 55.5375i	− 0.792342i	0.918177	−	0.396171i	$-0.129661\pi$
$18$	0	0
$19$	−27.4625	−0.331597	−0.165798	−	0.986160i	$-0.553020\pi$
$19$	−27.4625	−0.331597	−0.165798	+	0.986160i	$0.553020\pi$
$20$	0	0
$21$	0	0
$22$	− 27.0750i	− 0.262382i
$23$	− 139.687i	− 1.26638i	−0.773995	−	0.633192i	$-0.781744\pi$
$23$	− 139.687i	− 1.26638i	0.773995	−	0.633192i	$-0.218256\pi$
$24$	0	0
$25$	0	0
$26$	56.1499	0.423535
$27$	0	0
$28$	− 122.150i	− 0.824434i
$29$	178.762	1.14467	0.572333	−	0.820021i	$-0.306038\pi$
$29$	178.762	1.14467	0.572333	+	0.820021i	$0.306038\pi$
$30$	0	0
$31$	297.687	1.72472	0.862359	−	0.506298i	$-0.168986\pi$
$31$	297.687	1.72472	0.862359	+	0.506298i	$0.168986\pi$
$32$	− 32.0000i	− 0.176777i
$33$	0	0
$34$	−111.075	−0.560271
$35$	0	0
$36$	0	0
$37$	159.612i	0.709192i	0.935020	+	0.354596i	$0.115382\pi$
$37$	159.612i	0.709192i	−0.935020	+	0.354596i	$0.884618\pi$
$38$	54.9250i	0.234474i
$39$	0	0
$40$	0	0
$41$	140.775	0.536229	0.268114	−	0.963387i	$-0.413599\pi$
$41$	140.775	0.536229	0.268114	+	0.963387i	$0.413599\pi$
$42$	0	0
$43$	5.68738i	0.0201702i	0.999949	+	0.0100851i	$0.00321024\pi$
$43$	5.68738i	0.0201702i	−0.999949	+	0.0100851i	$0.996790\pi$
$44$	−54.1499	−0.185532
$45$	0	0
$46$	−279.375	−0.895469
$47$	301.987i	0.937220i	0.883405	+	0.468610i	$0.155245\pi$
$47$	301.987i	0.937220i	−0.883405	+	0.468610i	$0.844755\pi$
$48$	0	0
$49$	−589.537	−1.71877
$50$	0	0
$51$	0	0
$52$	− 112.300i	− 0.299484i
$53$	− 122.925i	− 0.318586i	−0.987231	−	0.159293i	$-0.949079\pi$
$53$	− 122.925i	− 0.318586i	0.987231	−	0.159293i	$-0.0509214\pi$
$54$	0	0
$55$	0	0
$56$	−244.300	−0.582963
$57$	0	0
$58$	− 357.525i	− 0.809402i
$59$	−864.300	−1.90716	−0.953578	−	0.301145i	$-0.902631\pi$
$59$	−864.300	−1.90716	−0.953578	+	0.301145i	$0.902631\pi$
$60$	0	0
$61$	−47.6124	−0.0999368	−0.0499684	−	0.998751i	$-0.515912\pi$
$61$	−47.6124	−0.0999368	−0.0499684	+	0.998751i	$0.515912\pi$
$62$	− 595.375i	− 1.21956i
$63$	0	0
$64$	−64.0000	−0.125000
$65$	0	0
$66$	0	0
$67$	402.388i	0.733723i	0.930275	+	0.366862i	$0.119568\pi$
$67$	402.388i	0.733723i	−0.930275	+	0.366862i	$0.880432\pi$
$68$	222.150i	0.396171i
$69$	0	0
$70$	0	0
$71$	927.225	1.54988	0.774939	−	0.632036i	$-0.217781\pi$
$71$	927.225	1.54988	0.774939	+	0.632036i	$0.217781\pi$
$72$	0	0
$73$	1019.61i	1.63475i	0.576107	+	0.817374i	$0.304571\pi$
$73$	1019.61i	1.63475i	−0.576107	+	0.817374i	$0.695429\pi$
$74$	319.225	0.501475
$75$	0	0
$76$	109.850	0.165798
$77$	413.400i	0.611836i
$78$	0	0
$79$	−812.225	−1.15674	−0.578370	−	0.815775i	$-0.696311\pi$
$79$	−812.225	−1.15674	−0.578370	+	0.815775i	$0.696311\pi$
$80$	0	0
$81$	0	0
$82$	− 281.550i	− 0.379171i
$83$	1381.05i	1.82638i	0.407530	+	0.913192i	$0.366390\pi$
$83$	1381.05i	1.82638i	−0.407530	+	0.913192i	$0.633610\pi$
$84$	0	0
$85$	0	0
$86$	11.3748	0.0142625
$87$	0	0
$88$	108.300i	0.131191i
$89$	−1423.35	−1.69522	−0.847611	−	0.530619i	$-0.821960\pi$
$89$	−1423.35	−1.69522	−0.847611	+	0.530619i	$0.821960\pi$
$90$	0	0
$91$	−857.338	−0.987621
$92$	558.750i	0.633192i
$93$	0	0
$94$	603.974	0.662715
$95$	0	0
$96$	0	0
$97$	− 124.863i	− 0.130700i	−0.997862	−	0.0653500i	$-0.979184\pi$
$97$	− 124.863i	− 0.130700i	0.997862	−	0.0653500i	$-0.0208164\pi$
$98$	1179.07i	1.21535i
$99$	0	0
$100$	0	0
$101$	571.537	0.563070	0.281535	−	0.959551i	$-0.409156\pi$
$101$	571.537	0.563070	0.281535	+	0.959551i	$0.409156\pi$
$102$	0	0
$103$	1200.51i	1.14845i	0.818699	+	0.574223i	$0.194696\pi$
$103$	1200.51i	1.14845i	−0.818699	+	0.574223i	$0.805304\pi$
$104$	−224.600	−0.211767
$105$	0	0
$106$	−245.850	−0.225274
$107$	− 763.499i	− 0.689815i	−0.938637	−	0.344908i	$-0.887910\pi$
$107$	− 763.499i	− 0.689815i	0.938637	−	0.344908i	$-0.112090\pi$
$108$	0	0
$109$	1531.52	1.34581	0.672906	−	0.739728i	$-0.265046\pi$
$109$	1531.52	1.34581	0.672906	+	0.739728i	$0.265046\pi$
$110$	0	0
$111$	0	0
$112$	488.600i	0.412217i
$113$	1390.76i	1.15780i	0.815397	+	0.578902i	$0.196519\pi$
$113$	1390.76i	1.15780i	−0.815397	+	0.578902i	$0.803481\pi$
$114$	0	0
$115$	0	0
$116$	−715.049	−0.572333
$117$	0	0
$118$	1728.60i	1.34856i
$119$	1695.97	1.30647
$120$	0	0
$121$	−1147.74	−0.862312
$122$	95.2249i	0.0706660i
$123$	0	0
$124$	−1190.75	−0.862359
$125$	0	0
$126$	0	0
$127$	− 2439.02i	− 1.70416i	−0.523411	−	0.852080i	$-0.675341\pi$
$127$	− 2439.02i	− 1.70416i	0.523411	−	0.852080i	$-0.324659\pi$
$128$	128.000i	0.0883883i
$129$	0	0
$130$	0	0
$131$	85.6874	0.0571492	0.0285746	−	0.999592i	$-0.490903\pi$
$131$	85.6874	0.0571492	0.0285746	+	0.999592i	$0.490903\pi$
$132$	0	0
$133$	− 838.636i	− 0.546759i
$134$	804.775	0.518821
$135$	0	0
$136$	444.300	0.280135
$137$	2515.80i	1.56890i	0.620193	+	0.784450i	$0.287054\pi$
$137$	2515.80i	1.56890i	−0.620193	+	0.784450i	$0.712946\pi$
$138$	0	0
$139$	−3078.31	−1.87841	−0.939205	−	0.343358i	$-0.888436\pi$
$139$	−3078.31	−1.87841	−0.939205	+	0.343358i	$0.888436\pi$
$140$	0	0
$141$	0	0
$142$	− 1854.45i	− 1.09593i
$143$	380.064i	0.222256i
$144$	0	0
$145$	0	0
$146$	2039.22	1.15594
$147$	0	0
$148$	− 638.450i	− 0.354596i
$149$	224.737	0.123565	0.0617824	−	0.998090i	$-0.480322\pi$
$149$	224.737	0.123565	0.0617824	+	0.998090i	$0.480322\pi$
$150$	0	0
$151$	−898.225	−0.484083	−0.242041	−	0.970266i	$-0.577817\pi$
$151$	−898.225	−0.484083	−0.242041	+	0.970266i	$0.577817\pi$
$152$	− 219.700i	− 0.117237i
$153$	0	0
$154$	826.801	0.432633
$155$	0	0
$156$	0	0
$157$	1813.21i	0.921720i	0.887473	+	0.460860i	$0.152459\pi$
$157$	1813.21i	0.921720i	−0.887473	+	0.460860i	$0.847541\pi$
$158$	1624.45i	0.817938i
$159$	0	0
$160$	0	0
$161$	4265.70	2.08810
$162$	0	0
$163$	218.576i	0.105032i	0.998620	+	0.0525159i	$0.0167240\pi$
$163$	218.576i	0.105032i	−0.998620	+	0.0525159i	$0.983276\pi$
$164$	−563.101	−0.268114
$165$	0	0
$166$	2762.10	1.29145
$167$	1891.65i	0.876528i	0.898846	+	0.438264i	$0.144407\pi$
$167$	1891.65i	0.876528i	−0.898846	+	0.438264i	$0.855593\pi$
$168$	0	0
$169$	1408.80	0.641237
$170$	0	0
$171$	0	0
$172$	− 22.7495i	− 0.0100851i
$173$	331.726i	0.145784i	0.997340	+	0.0728920i	$0.0232229\pi$
$173$	331.726i	0.145784i	−0.997340	+	0.0728920i	$0.976777\pi$
$174$	0	0
$175$	0	0
$176$	216.600	0.0927660
$177$	0	0
$178$	2846.70i	1.19870i
$179$	3222.52	1.34560	0.672801	−	0.739824i	$-0.265091\pi$
$179$	3222.52	1.34560	0.672801	+	0.739824i	$0.265091\pi$
$180$	0	0
$181$	−3566.46	−1.46460	−0.732301	−	0.680981i	$-0.761554\pi$
$181$	−3566.46	−1.46460	−0.732301	+	0.680981i	$0.761554\pi$
$182$	1714.68i	0.698353i
$183$	0	0
$184$	1117.50	0.447734
$185$	0	0
$186$	0	0
$187$	− 751.837i	− 0.294010i
$188$	− 1207.95i	− 0.468610i
$189$	0	0
$190$	0	0
$191$	1185.52	0.449118	0.224559	−	0.974460i	$-0.427906\pi$
$191$	1185.52	0.449118	0.224559	+	0.974460i	$0.427906\pi$
$192$	0	0
$193$	− 1401.16i	− 0.522580i	−0.965260	−	0.261290i	$-0.915852\pi$
$193$	− 1401.16i	− 0.522580i	0.965260	−	0.261290i	$-0.0841479\pi$
$194$	−249.726	−0.0924189
$195$	0	0
$196$	2358.15	0.859384
$197$	3463.05i	1.25245i	0.779644	+	0.626223i	$0.215400\pi$
$197$	3463.05i	1.25245i	−0.779644	+	0.626223i	$0.784600\pi$
$198$	0	0
$199$	207.625	0.0739606	0.0369803	−	0.999316i	$-0.488226\pi$
$199$	207.625	0.0739606	0.0369803	+	0.999316i	$0.488226\pi$
$200$	0	0
$201$	0	0
$202$	− 1143.07i	− 0.398151i
$203$	5458.95i	1.88741i
$204$	0	0
$205$	0	0
$206$	2401.02	0.812074
$207$	0	0
$208$	449.199i	0.149742i
$209$	−371.773	−0.123044
$210$	0	0
$211$	−1240.04	−0.404586	−0.202293	−	0.979325i	$-0.564839\pi$
$211$	−1240.04	−0.404586	−0.202293	+	0.979325i	$0.564839\pi$
$212$	491.700i	0.159293i
$213$	0	0
$214$	−1527.00	−0.487773
$215$	0	0
$216$	0	0
$217$	9090.62i	2.84383i
$218$	− 3063.05i	− 0.951632i
$219$	0	0
$220$	0	0
$221$	1559.21	0.474588
$222$	0	0
$223$	− 5867.00i	− 1.76181i	−0.473295	−	0.880904i	$-0.656935\pi$
$223$	− 5867.00i	− 1.76181i	0.473295	−	0.880904i	$-0.343065\pi$
$224$	977.199	0.291482
$225$	0	0
$226$	2781.52	0.818691
$227$	− 6159.07i	− 1.80085i	−0.435016	−	0.900423i	$-0.643257\pi$
$227$	− 6159.07i	− 1.80085i	0.435016	−	0.900423i	$-0.356743\pi$
$228$	0	0
$229$	319.824	0.0922908	0.0461454	−	0.998935i	$-0.485306\pi$
$229$	319.824	0.0922908	0.0461454	+	0.998935i	$0.485306\pi$
$230$	0	0
$231$	0	0
$232$	1430.10i	0.404701i
$233$	5812.42i	1.63427i	0.576448	+	0.817134i	$0.304438\pi$
$233$	5812.42i	1.63427i	−0.576448	+	0.817134i	$0.695562\pi$
$234$	0	0
$235$	0	0
$236$	3457.20	0.953578
$237$	0	0
$238$	− 3391.95i	− 0.923813i
$239$	−3632.40	−0.983098	−0.491549	−	0.870850i	$-0.663569\pi$
$239$	−3632.40	−0.983098	−0.491549	+	0.870850i	$0.663569\pi$
$240$	0	0
$241$	2582.45	0.690249	0.345125	−	0.938557i	$-0.387837\pi$
$241$	2582.45	0.690249	0.345125	+	0.938557i	$0.387837\pi$
$242$	2295.47i	0.609746i
$243$	0	0
$244$	190.450	0.0499684
$245$	0	0
$246$	0	0
$247$	− 771.009i	− 0.198616i
$248$	2381.50i	0.609780i
$249$	0	0
$250$	0	0
$251$	−251.362	−0.0632105	−0.0316052	−	0.999500i	$-0.510062\pi$
$251$	−251.362	−0.0632105	−0.0316052	+	0.999500i	$0.510062\pi$
$252$	0	0
$253$	− 1891.01i	− 0.469909i
$254$	−4878.05	−1.20502
$255$	0	0
$256$	256.000	0.0625000
$257$	− 1725.34i	− 0.418769i	−0.977833	−	0.209384i	$-0.932854\pi$
$257$	− 1725.34i	− 0.418769i	0.977833	−	0.209384i	$-0.0671460\pi$
$258$	0	0
$259$	−4874.16	−1.16937
$260$	0	0
$261$	0	0
$262$	− 171.375i	− 0.0404106i
$263$	4534.50i	1.06315i	0.847010	+	0.531577i	$0.178400\pi$
$263$	4534.50i	1.06315i	−0.847010	+	0.531577i	$0.821600\pi$
$264$	0	0
$265$	0	0
$266$	−1677.27	−0.386617
$267$	0	0
$268$	− 1609.55i	− 0.366862i
$269$	−4179.79	−0.947383	−0.473692	−	0.880691i	$-0.657079\pi$
$269$	−4179.79	−0.947383	−0.473692	+	0.880691i	$0.657079\pi$
$270$	0	0
$271$	2411.41	0.540527	0.270263	−	0.962786i	$-0.412889\pi$
$271$	2411.41	0.540527	0.270263	+	0.962786i	$0.412889\pi$
$272$	− 888.600i	− 0.198086i
$273$	0	0
$274$	5031.60	1.10938
$275$	0	0
$276$	0	0
$277$	− 1626.85i	− 0.352880i	−0.984311	−	0.176440i	$-0.943542\pi$
$277$	− 1626.85i	− 0.352880i	0.984311	−	0.176440i	$-0.0564582\pi$
$278$	6156.62i	1.32824i
$279$	0	0
$280$	0	0
$281$	906.227	0.192388	0.0961939	−	0.995363i	$-0.469333\pi$
$281$	906.227	0.192388	0.0961939	+	0.995363i	$0.469333\pi$
$282$	0	0
$283$	1643.95i	0.345310i	0.984982	+	0.172655i	$0.0552345\pi$
$283$	1643.95i	0.345310i	−0.984982	+	0.172655i	$0.944765\pi$
$284$	−3708.90	−0.774939
$285$	0	0
$286$	760.128	0.157158
$287$	4298.92i	0.884171i
$288$	0	0
$289$	1828.59	0.372194
$290$	0	0
$291$	0	0
$292$	− 4078.45i	− 0.817374i
$293$	3737.10i	0.745132i	0.928006	+	0.372566i	$0.121522\pi$
$293$	3737.10i	0.745132i	−0.928006	+	0.372566i	$0.878478\pi$
$294$	0	0
$295$	0	0
$296$	−1276.90	−0.250737
$297$	0	0
$298$	− 449.473i	− 0.0873735i
$299$	3921.72	0.758524
$300$	0	0
$301$	−173.678	−0.0332580
$302$	1796.45i	0.342298i
$303$	0	0
$304$	−439.400	−0.0828991
$305$	0	0
$306$	0	0
$307$	4719.01i	0.877291i	0.898660	+	0.438646i	$0.144542\pi$
$307$	4719.01i	0.877291i	−0.898660	+	0.438646i	$0.855458\pi$
$308$	− 1653.60i	− 0.305918i
$309$	0	0
$310$	0	0
$311$	−5386.65	−0.982150	−0.491075	−	0.871117i	$-0.663396\pi$
$311$	−5386.65	−0.982150	−0.491075	+	0.871117i	$0.663396\pi$
$312$	0	0
$313$	2706.96i	0.488838i	0.969670	+	0.244419i	$0.0785972\pi$
$313$	2706.96i	0.488838i	−0.969670	+	0.244419i	$0.921403\pi$
$314$	3626.42	0.651755
$315$	0	0
$316$	3248.90	0.578370
$317$	− 6358.35i	− 1.12656i	−0.826265	−	0.563281i	$-0.809539\pi$
$317$	− 6358.35i	− 1.12656i	0.826265	−	0.563281i	$-0.190461\pi$
$318$	0	0
$319$	2419.99	0.424744
$320$	0	0
$321$	0	0
$322$	− 8531.40i	− 1.47651i
$323$	1525.20i	0.262738i
$324$	0	0
$325$	0	0
$326$	437.152	0.0742687
$327$	0	0
$328$	1126.20i	0.189586i
$329$	−9221.93	−1.54535
$330$	0	0
$331$	−4071.74	−0.676142	−0.338071	−	0.941121i	$-0.609774\pi$
$331$	−4071.74	−0.676142	−0.338071	+	0.941121i	$0.609774\pi$
$332$	− 5524.20i	− 0.913192i
$333$	0	0
$334$	3783.30	0.619799
$335$	0	0
$336$	0	0
$337$	6373.36i	1.03020i	0.857129	+	0.515102i	$0.172246\pi$
$337$	6373.36i	1.03020i	−0.857129	+	0.515102i	$0.827754\pi$
$338$	− 2817.59i	− 0.453423i
$339$	0	0
$340$	0	0
$341$	4029.94	0.639980
$342$	0	0
$343$	− 7528.63i	− 1.18515i
$344$	−45.4991	−0.00713124
$345$	0	0
$346$	663.452	0.103085
$347$	860.775i	0.133167i	0.997781	+	0.0665833i	$0.0212098\pi$
$347$	860.775i	0.133167i	−0.997781	+	0.0665833i	$0.978790\pi$
$348$	0	0
$349$	2926.71	0.448892	0.224446	−	0.974487i	$-0.427943\pi$
$349$	2926.71	0.448892	0.224446	+	0.974487i	$0.427943\pi$
$350$	0	0
$351$	0	0
$352$	− 433.199i	− 0.0655954i
$353$	− 7893.64i	− 1.19019i	−0.803657	−	0.595093i	$-0.797115\pi$
$353$	− 7893.64i	− 1.19019i	0.803657	−	0.595093i	$-0.202885\pi$
$354$	0	0
$355$	0	0
$356$	5693.40	0.847611
$357$	0	0
$358$	− 6445.05i	− 0.951484i
$359$	6685.57	0.982872	0.491436	−	0.870914i	$-0.336472\pi$
$359$	6685.57	0.982872	0.491436	+	0.870914i	$0.336472\pi$
$360$	0	0
$361$	−6104.81	−0.890044
$362$	7132.92i	1.03563i
$363$	0	0
$364$	3429.35	0.493810
$365$	0	0
$366$	0	0
$367$	− 3952.26i	− 0.562143i	−0.959687	−	0.281071i	$-0.909310\pi$
$367$	− 3952.26i	− 0.562143i	0.959687	−	0.281071i	$-0.0906898\pi$
$368$	− 2235.00i	− 0.316596i
$369$	0	0
$370$	0	0
$371$	3753.82	0.525306
$372$	0	0
$373$	9672.17i	1.34264i	0.741166	+	0.671322i	$0.234273\pi$
$373$	9672.17i	1.34264i	−0.741166	+	0.671322i	$0.765727\pi$
$374$	−1503.67	−0.207896
$375$	0	0
$376$	−2415.90	−0.331357
$377$	5018.74i	0.685619i
$378$	0	0
$379$	4049.53	0.548840	0.274420	−	0.961610i	$-0.411514\pi$
$379$	4049.53	0.548840	0.274420	+	0.961610i	$0.411514\pi$
$380$	0	0
$381$	0	0
$382$	− 2371.05i	− 0.317574i
$383$	9277.68i	1.23777i	0.785480	+	0.618887i	$0.212416\pi$
$383$	9277.68i	1.23777i	−0.785480	+	0.618887i	$0.787584\pi$
$384$	0	0
$385$	0	0
$386$	−2802.33	−0.369520
$387$	0	0
$388$	499.452i	0.0653500i
$389$	13899.5	1.81165	0.905825	−	0.423653i	$-0.139252\pi$
$389$	13899.5	1.81165	0.905825	+	0.423653i	$0.139252\pi$
$390$	0	0
$391$	−7757.88	−1.00341
$392$	− 4716.30i	− 0.607676i
$393$	0	0
$394$	6926.10	0.885614
$395$	0	0
$396$	0	0
$397$	6239.04i	0.788736i	0.918953	+	0.394368i	$0.129036\pi$
$397$	6239.04i	0.788736i	−0.918953	+	0.394368i	$0.870964\pi$
$398$	− 415.250i	− 0.0522981i
$399$	0	0
$400$	0	0
$401$	10106.0	1.25853	0.629265	−	0.777191i	$-0.283356\pi$
$401$	10106.0	1.25853	0.629265	+	0.777191i	$0.283356\pi$
$402$	0	0
$403$	8357.56i	1.03305i
$404$	−2286.15	−0.281535
$405$	0	0
$406$	10917.9	1.33460
$407$	2160.75i	0.263156i
$408$	0	0
$409$	15427.2	1.86510	0.932551	−	0.361038i	$-0.117578\pi$
$409$	15427.2	1.86510	0.932551	+	0.361038i	$0.117578\pi$
$410$	0	0
$411$	0	0
$412$	− 4802.05i	− 0.574223i
$413$	− 26393.5i	− 3.14465i
$414$	0	0
$415$	0	0
$416$	898.399	0.105884
$417$	0	0
$418$	743.547i	0.0870049i
$419$	−7245.19	−0.844751	−0.422376	−	0.906421i	$-0.638804\pi$
$419$	−7245.19	−0.844751	−0.422376	+	0.906421i	$0.638804\pi$
$420$	0	0
$421$	952.281	0.110241	0.0551204	−	0.998480i	$-0.482446\pi$
$421$	952.281	0.110241	0.0551204	+	0.998480i	$0.482446\pi$
$422$	2480.07i	0.286085i
$423$	0	0
$424$	983.400	0.112637
$425$	0	0
$426$	0	0
$427$	− 1453.96i	− 0.164783i
$428$	3054.00i	0.344908i
$429$	0	0
$430$	0	0
$431$	−6293.02	−0.703305	−0.351652	−	0.936131i	$-0.614380\pi$
$431$	−6293.02	−0.703305	−0.351652	+	0.936131i	$0.614380\pi$
$432$	0	0
$433$	− 3761.61i	− 0.417485i	−0.977971	−	0.208743i	$-0.933063\pi$
$433$	− 3761.61i	− 0.417485i	0.977971	−	0.208743i	$-0.0669371\pi$
$434$	18181.2	2.01089
$435$	0	0
$436$	−6126.10	−0.672906
$437$	3836.17i	0.419929i
$438$	0	0
$439$	−14644.1	−1.59208	−0.796042	−	0.605242i	$-0.793076\pi$
$439$	−14644.1	−1.59208	−0.796042	+	0.605242i	$0.793076\pi$
$440$	0	0
$441$	0	0
$442$	− 3118.42i	− 0.335584i
$443$	− 16912.8i	− 1.81389i	−0.421254	−	0.906943i	$-0.638410\pi$
$443$	− 16912.8i	− 1.81389i	0.421254	−	0.906943i	$-0.361590\pi$
$444$	0	0
$445$	0	0
$446$	−11734.0	−1.24579
$447$	0	0
$448$	− 1954.40i	− 0.206109i
$449$	−3957.82	−0.415994	−0.207997	−	0.978129i	$-0.566694\pi$
$449$	−3957.82	−0.415994	−0.207997	+	0.978129i	$0.566694\pi$
$450$	0	0
$451$	1905.74	0.198975
$452$	− 5563.05i	− 0.578902i
$453$	0	0
$454$	−12318.1	−1.27339
$455$	0	0
$456$	0	0
$457$	− 2778.33i	− 0.284386i	−0.989839	−	0.142193i	$-0.954585\pi$
$457$	− 2778.33i	− 0.284386i	0.989839	−	0.142193i	$-0.0454154\pi$
$458$	− 639.649i	− 0.0652595i
$459$	0	0
$460$	0	0
$461$	−3117.07	−0.314916	−0.157458	−	0.987526i	$-0.550330\pi$
$461$	−3117.07	−0.314916	−0.157458	+	0.987526i	$0.550330\pi$
$462$	0	0
$463$	1171.27i	0.117567i	0.998271	+	0.0587835i	$0.0187221\pi$
$463$	1171.27i	0.117567i	−0.998271	+	0.0587835i	$0.981278\pi$
$464$	2860.20	0.286167
$465$	0	0
$466$	11624.8	1.15560
$467$	− 13712.8i	− 1.35879i	−0.733773	−	0.679395i	$-0.762242\pi$
$467$	− 13712.8i	− 1.35879i	0.733773	−	0.679395i	$-0.237758\pi$
$468$	0	0
$469$	−12287.9	−1.20981
$470$	0	0
$471$	0	0
$472$	− 6914.40i	− 0.674282i
$473$	76.9928i	0.00748443i
$474$	0	0
$475$	0	0
$476$	−6783.90	−0.653234
$477$	0	0
$478$	7264.80i	0.695155i
$479$	13500.2	1.28777	0.643884	−	0.765123i	$-0.277322\pi$
$479$	13500.2	1.28777	0.643884	+	0.765123i	$0.277322\pi$
$480$	0	0
$481$	−4481.11	−0.424784
$482$	− 5164.90i	− 0.488080i
$483$	0	0
$484$	4590.95	0.431156
$485$	0	0
$486$	0	0
$487$	5522.12i	0.513821i	0.966435	+	0.256911i	$0.0827046\pi$
$487$	5522.12i	0.513821i	−0.966435	+	0.256911i	$0.917295\pi$
$488$	− 380.899i	− 0.0353330i
$489$	0	0
$490$	0	0
$491$	−4908.83	−0.451186	−0.225593	−	0.974222i	$-0.572432\pi$
$491$	−4908.83	−0.451186	−0.225593	+	0.974222i	$0.572432\pi$
$492$	0	0
$493$	− 9928.01i	− 0.906968i
$494$	−1542.02	−0.140443
$495$	0	0
$496$	4763.00	0.431179
$497$	28315.1i	2.55555i
$498$	0	0
$499$	−11374.9	−1.02046	−0.510229	−	0.860039i	$-0.670439\pi$
$499$	−11374.9	−1.02046	−0.510229	+	0.860039i	$0.670439\pi$
$500$	0	0
$501$	0	0
$502$	502.724i	0.0446965i
$503$	11799.9i	1.04598i	0.852338	+	0.522991i	$0.175184\pi$
$503$	11799.9i	1.04598i	−0.852338	+	0.522991i	$0.824816\pi$
$504$	0	0
$505$	0	0
$506$	−3782.03	−0.332276
$507$	0	0
$508$	9756.09i	0.852080i
$509$	−14238.9	−1.23994	−0.619970	−	0.784626i	$-0.712855\pi$
$509$	−14238.9	−1.23994	−0.619970	+	0.784626i	$0.712855\pi$
$510$	0	0
$511$	−31136.4	−2.69548
$512$	− 512.000i	− 0.0441942i
$513$	0	0
$514$	−3450.67	−0.296114
$515$	0	0
$516$	0	0
$517$	4088.14i	0.347769i
$518$	9748.32i	0.826866i
$519$	0	0
$520$	0	0
$521$	−10333.1	−0.868910	−0.434455	−	0.900694i	$-0.643059\pi$
$521$	−10333.1	−0.868910	−0.434455	+	0.900694i	$0.643059\pi$
$522$	0	0
$523$	8396.57i	0.702020i	0.936372	+	0.351010i	$0.114162\pi$
$523$	8396.57i	0.702020i	−0.936372	+	0.351010i	$0.885838\pi$
$524$	−342.750	−0.0285746
$525$	0	0
$526$	9069.00	0.751763
$527$	− 16532.8i	− 1.36657i
$528$	0	0
$529$	−7345.56	−0.603729
$530$	0	0
$531$	0	0
$532$	3354.54i	0.273380i
$533$	3952.26i	0.321184i
$534$	0	0
$535$	0	0
$536$	−3219.10	−0.259410
$537$	0	0
$538$	8359.57i	0.669901i
$539$	−7980.85	−0.637773
$540$	0	0
$541$	−13853.5	−1.10094	−0.550471	−	0.834854i	$-0.685552\pi$
$541$	−13853.5	−1.10094	−0.550471	+	0.834854i	$0.685552\pi$
$542$	− 4822.82i	− 0.382210i
$543$	0	0
$544$	−1777.20	−0.140068
$545$	0	0
$546$	0	0
$547$	22969.3i	1.79542i	0.440583	+	0.897712i	$0.354772\pi$
$547$	22969.3i	1.79542i	−0.440583	+	0.897712i	$0.645228\pi$
$548$	− 10063.2i	− 0.784450i
$549$	0	0
$550$	0	0
$551$	−4909.26	−0.379568
$552$	0	0
$553$	− 24803.3i	− 1.90731i
$554$	−3253.70	−0.249524
$555$	0	0
$556$	12313.2	0.939205
$557$	− 5636.25i	− 0.428753i	−0.976751	−	0.214377i	$-0.931228\pi$
$557$	− 5636.25i	− 0.428753i	0.976751	−	0.214377i	$-0.0687720\pi$
$558$	0	0
$559$	−159.673	−0.0120813
$560$	0	0
$561$	0	0
$562$	− 1812.45i	− 0.136039i
$563$	− 2856.37i	− 0.213822i	−0.994269	−	0.106911i	$-0.965904\pi$
$563$	− 2856.37i	− 0.213822i	0.994269	−	0.106911i	$-0.0340959\pi$
$564$	0	0
$565$	0	0
$566$	3287.90	0.244171
$567$	0	0
$568$	7417.80i	0.547965i
$569$	8555.25	0.630324	0.315162	−	0.949038i	$-0.397941\pi$
$569$	8555.25	0.630324	0.315162	+	0.949038i	$0.397941\pi$
$570$	0	0
$571$	17812.3	1.30547	0.652733	−	0.757588i	$-0.273623\pi$
$571$	17812.3	1.30547	0.652733	+	0.757588i	$0.273623\pi$
$572$	− 1520.26i	− 0.111128i
$573$	0	0
$574$	8597.84	0.625203
$575$	0	0
$576$	0	0
$577$	− 7464.06i	− 0.538532i	−0.963066	−	0.269266i	$-0.913219\pi$
$577$	− 7464.06i	− 0.538532i	0.963066	−	0.269266i	$-0.0867811\pi$
$578$	− 3657.18i	− 0.263181i
$579$	0	0
$580$	0	0
$581$	−42173.8	−3.01147
$582$	0	0
$583$	− 1664.09i	− 0.118216i
$584$	−8156.90	−0.577971
$585$	0	0
$586$	7474.20	0.526888
$587$	11631.9i	0.817887i	0.912560	+	0.408943i	$0.134103\pi$
$587$	11631.9i	0.817887i	−0.912560	+	0.408943i	$0.865897\pi$
$588$	0	0
$589$	−8175.25	−0.571910
$590$	0	0
$591$	0	0
$592$	2553.80i	0.177298i
$593$	− 20156.1i	− 1.39581i	−0.716192	−	0.697903i	$-0.754116\pi$
$593$	− 20156.1i	− 1.39581i	0.716192	−	0.697903i	$-0.245884\pi$
$594$	0	0
$595$	0	0
$596$	−898.947	−0.0617824
$597$	0	0
$598$	− 7843.43i	− 0.536358i
$599$	−20273.1	−1.38287	−0.691433	−	0.722441i	$-0.743020\pi$
$599$	−20273.1	−1.38287	−0.691433	+	0.722441i	$0.743020\pi$
$600$	0	0
$601$	18371.2	1.24688	0.623441	−	0.781870i	$-0.285734\pi$
$601$	18371.2	1.24688	0.623441	+	0.781870i	$0.285734\pi$
$602$	347.357i	0.0235169i
$603$	0	0
$604$	3592.90	0.242041
$605$	0	0
$606$	0	0
$607$	15917.0i	1.06434i	0.846638	+	0.532169i	$0.178623\pi$
$607$	15917.0i	1.06434i	−0.846638	+	0.532169i	$0.821377\pi$
$608$	878.801i	0.0586185i
$609$	0	0
$610$	0	0
$611$	−8478.28	−0.561366
$612$	0	0
$613$	− 13349.4i	− 0.879574i	−0.898102	−	0.439787i	$-0.855054\pi$
$613$	− 13349.4i	− 0.879574i	0.898102	−	0.439787i	$-0.144946\pi$
$614$	9438.03	0.620339
$615$	0	0
$616$	−3307.20	−0.216317
$617$	− 25466.1i	− 1.66163i	−0.556547	−	0.830816i	$-0.687874\pi$
$617$	− 25466.1i	− 1.66163i	0.556547	−	0.830816i	$-0.312126\pi$
$618$	0	0
$619$	−9642.68	−0.626126	−0.313063	−	0.949732i	$-0.601355\pi$
$619$	−9642.68	−0.626126	−0.313063	+	0.949732i	$0.601355\pi$
$620$	0	0
$621$	0	0
$622$	10773.3i	0.694485i
$623$	− 43465.5i	− 2.79520i
$624$	0	0
$625$	0	0
$626$	5413.92	0.345661
$627$	0	0
$628$	− 7252.85i	− 0.460860i
$629$	8864.47	0.561923
$630$	0	0
$631$	−24904.8	−1.57123	−0.785613	−	0.618718i	$-0.787652\pi$
$631$	−24904.8	−1.57123	−0.785613	+	0.618718i	$0.787652\pi$
$632$	− 6497.80i	− 0.408969i
$633$	0	0
$634$	−12716.7	−0.796600
$635$	0	0
$636$	0	0
$637$	− 16551.2i	− 1.02949i
$638$	− 4839.98i	− 0.300340i
$639$	0	0
$640$	0	0
$641$	22240.9	1.37045	0.685227	−	0.728329i	$-0.259703\pi$
$641$	22240.9	1.37045	0.685227	+	0.728329i	$0.259703\pi$
$642$	0	0
$643$	25250.1i	1.54862i	0.632804	+	0.774312i	$0.281904\pi$
$643$	25250.1i	1.54862i	−0.632804	+	0.774312i	$0.718096\pi$
$644$	−17062.8	−1.04405
$645$	0	0
$646$	3050.40	0.185784
$647$	− 6646.58i	− 0.403870i	−0.979399	−	0.201935i	$-0.935277\pi$
$647$	− 6646.58i	− 0.403870i	0.979399	−	0.201935i	$-0.0647230\pi$
$648$	0	0
$649$	−11700.4	−0.707677
$650$	0	0
$651$	0	0
$652$	− 874.304i	− 0.0525159i
$653$	27372.3i	1.64037i	0.572100	+	0.820184i	$0.306129\pi$
$653$	27372.3i	1.64037i	−0.572100	+	0.820184i	$0.693871\pi$
$654$	0	0
$655$	0	0
$656$	2252.40	0.134057
$657$	0	0
$658$	18443.9i	1.09273i
$659$	705.594	0.0417087	0.0208544	−	0.999783i	$-0.493361\pi$
$659$	705.594	0.0417087	0.0208544	+	0.999783i	$0.493361\pi$
$660$	0	0
$661$	15174.1	0.892895	0.446448	−	0.894810i	$-0.352689\pi$
$661$	15174.1	0.892895	0.446448	+	0.894810i	$0.352689\pi$
$662$	8143.47i	0.478104i
$663$	0	0
$664$	−11048.4	−0.645724
$665$	0	0
$666$	0	0
$667$	− 24970.8i	− 1.44959i
$668$	− 7566.60i	− 0.438264i
$669$	0	0
$670$	0	0
$671$	−644.552	−0.0370830
$672$	0	0
$673$	17105.5i	0.979747i	0.871793	+	0.489874i	$0.162957\pi$
$673$	17105.5i	0.979747i	−0.871793	+	0.489874i	$0.837043\pi$
$674$	12746.7	0.728464
$675$	0	0
$676$	−5635.19	−0.320618
$677$	− 27269.5i	− 1.54808i	−0.633134	−	0.774042i	$-0.718232\pi$
$677$	− 27269.5i	− 1.54808i	0.633134	−	0.774042i	$-0.281768\pi$
$678$	0	0
$679$	3813.00	0.215507
$680$	0	0
$681$	0	0
$682$	− 8059.87i	− 0.452534i
$683$	− 12070.7i	− 0.676242i	−0.941103	−	0.338121i	$-0.890209\pi$
$683$	− 12070.7i	− 0.676242i	0.941103	−	0.338121i	$-0.109791\pi$
$684$	0	0
$685$	0	0
$686$	−15057.3	−0.838031
$687$	0	0
$688$	90.9981i	0.00504255i
$689$	3451.11	0.190823
$690$	0	0
$691$	−880.230	−0.0484595	−0.0242298	−	0.999706i	$-0.507713\pi$
$691$	−880.230	−0.0484595	−0.0242298	+	0.999706i	$0.507713\pi$
$692$	− 1326.90i	− 0.0728920i
$693$	0	0
$694$	1721.55	0.0941630
$695$	0	0
$696$	0	0
$697$	− 7818.30i	− 0.424877i
$698$	− 5853.43i	− 0.317415i
$699$	0	0
$700$	0	0
$701$	9375.93	0.505170	0.252585	−	0.967575i	$-0.418719\pi$
$701$	9375.93	0.505170	0.252585	+	0.967575i	$0.418719\pi$
$702$	0	0
$703$	− 4383.36i	− 0.235166i
$704$	−866.399	−0.0463830
$705$	0	0
$706$	−15787.3	−0.841589
$707$	17453.3i	0.928429i
$708$	0	0
$709$	−794.475	−0.0420834	−0.0210417	−	0.999779i	$-0.506698\pi$
$709$	−794.475	−0.0420834	−0.0210417	+	0.999779i	$0.506698\pi$
$710$	0	0
$711$	0	0
$712$	− 11386.8i	− 0.599351i
$713$	− 41583.2i	− 2.18415i
$714$	0	0
$715$	0	0
$716$	−12890.1	−0.672801
$717$	0	0
$718$	− 13371.1i	− 0.694995i
$719$	21154.5	1.09726	0.548630	−	0.836065i	$-0.315150\pi$
$719$	21154.5	1.09726	0.548630	+	0.836065i	$0.315150\pi$
$720$	0	0
$721$	−36660.6	−1.89364
$722$	12209.6i	0.629356i
$723$	0	0
$724$	14265.8	0.732301
$725$	0	0
$726$	0	0
$727$	25133.1i	1.28217i	0.767471	+	0.641084i	$0.221515\pi$
$727$	25133.1i	1.28217i	−0.767471	+	0.641084i	$0.778485\pi$
$728$	− 6858.71i	− 0.349177i
$729$	0	0
$730$	0	0
$731$	315.863	0.0159817
$732$	0	0
$733$	1476.70i	0.0744108i	0.999308	+	0.0372054i	$0.0118456\pi$
$733$	1476.70i	0.0744108i	−0.999308	+	0.0372054i	$0.988154\pi$
$734$	−7904.53	−0.397495
$735$	0	0
$736$	−4470.00	−0.223867
$737$	5447.31i	0.272258i
$738$	0	0
$739$	12589.0	0.626649	0.313325	−	0.949646i	$-0.398557\pi$
$739$	12589.0	0.626649	0.313325	+	0.949646i	$0.398557\pi$
$740$	0	0
$741$	0	0
$742$	− 7507.64i	− 0.371448i
$743$	− 31901.9i	− 1.57519i	−0.616194	−	0.787595i	$-0.711326\pi$
$743$	− 31901.9i	− 1.57519i	0.616194	−	0.787595i	$-0.288674\pi$
$744$	0	0
$745$	0	0
$746$	19344.3	0.949393
$747$	0	0
$748$	3007.35i	0.147005i
$749$	23315.3	1.13741
$750$	0	0
$751$	−1446.78	−0.0702980	−0.0351490	−	0.999382i	$-0.511191\pi$
$751$	−1446.78	−0.0702980	−0.0351490	+	0.999382i	$0.511191\pi$
$752$	4831.80i	0.234305i
$753$	0	0
$754$	10037.5	0.484806
$755$	0	0
$756$	0	0
$757$	− 14461.0i	− 0.694312i	−0.937807	−	0.347156i	$-0.887147\pi$
$757$	− 14461.0i	− 0.694312i	0.937807	−	0.347156i	$-0.112853\pi$
$758$	− 8099.06i	− 0.388089i
$759$	0	0
$760$	0	0
$761$	−24463.5	−1.16531	−0.582655	−	0.812719i	$-0.697986\pi$
$761$	−24463.5	−1.16531	−0.582655	+	0.812719i	$0.697986\pi$
$762$	0	0
$763$	46768.9i	2.21907i
$764$	−4742.10	−0.224559
$765$	0	0
$766$	18555.4	0.875238
$767$	− 24265.2i	− 1.14233i
$768$	0	0
$769$	−18749.7	−0.879237	−0.439618	−	0.898185i	$-0.644886\pi$
$769$	−18749.7	−0.879237	−0.439618	+	0.898185i	$0.644886\pi$
$770$	0	0
$771$	0	0
$772$	5604.65i	0.261290i
$773$	29141.4i	1.35594i	0.735089	+	0.677971i	$0.237140\pi$
$773$	29141.4i	1.35594i	−0.735089	+	0.677971i	$0.762860\pi$
$774$	0	0
$775$	0	0
$776$	998.903	0.0462095
$777$	0	0
$778$	− 27799.0i	− 1.28103i
$779$	−3866.04	−0.177812
$780$	0	0
$781$	12552.3	0.575104
$782$	15515.8i	0.709518i
$783$	0	0
$784$	−9432.60	−0.429692
$785$	0	0
$786$	0	0
$787$	27495.2i	1.24536i	0.782477	+	0.622679i	$0.213956\pi$
$787$	27495.2i	1.24536i	−0.782477	+	0.622679i	$0.786044\pi$
$788$	− 13852.2i	− 0.626223i
$789$	0	0
$790$	0	0
$791$	−42470.4	−1.90907
$792$	0	0
$793$	− 1336.72i	− 0.0598590i
$794$	12478.1	0.557721
$795$	0	0
$796$	−830.501	−0.0369803
$797$	− 14174.6i	− 0.629973i	−0.949096	−	0.314987i	$-0.898000\pi$
$797$	− 14174.6i	− 0.629973i	0.949096	−	0.314987i	$-0.102000\pi$
$798$	0	0
$799$	16771.6	0.742599
$800$	0	0
$801$	0	0
$802$	− 20212.0i	− 0.889915i
$803$	13803.0i	0.606596i
$804$	0	0
$805$	0	0
$806$	16715.1	0.730478
$807$	0	0
$808$	4572.30i	0.199075i
$809$	7470.89	0.324675	0.162338	−	0.986735i	$-0.448097\pi$
$809$	7470.89	0.324675	0.162338	+	0.986735i	$0.448097\pi$
$810$	0	0
$811$	44105.0	1.90966	0.954832	−	0.297146i	$-0.0960349\pi$
$811$	44105.0	1.90966	0.954832	+	0.297146i	$0.0960349\pi$
$812$	− 21835.8i	− 0.943703i
$813$	0	0
$814$	4321.50	0.186079
$815$	0	0
$816$	0	0
$817$	− 156.190i	− 0.00668836i
$818$	− 30854.4i	− 1.31883i
$819$	0	0
$820$	0	0
$821$	−11137.6	−0.473452	−0.236726	−	0.971576i	$-0.576074\pi$
$821$	−11137.6	−0.473452	−0.236726	+	0.971576i	$0.576074\pi$
$822$	0	0
$823$	− 2369.19i	− 0.100346i	−0.998741	−	0.0501730i	$-0.984023\pi$
$823$	− 2369.19i	− 0.100346i	0.998741	−	0.0501730i	$-0.0159773\pi$
$824$	−9604.09	−0.406037
$825$	0	0
$826$	−52787.1	−2.22360
$827$	− 9602.48i	− 0.403762i	−0.979410	−	0.201881i	$-0.935295\pi$
$827$	− 9602.48i	− 0.403762i	0.979410	−	0.201881i	$-0.0647054\pi$
$828$	0	0
$829$	43349.7	1.81616	0.908081	−	0.418794i	$-0.137547\pi$
$829$	43349.7	1.81616	0.908081	+	0.418794i	$0.137547\pi$
$830$	0	0
$831$	0	0
$832$	− 1796.80i	− 0.0748711i
$833$	32741.4i	1.36185i
$834$	0	0
$835$	0	0
$836$	1487.09	0.0615218
$837$	0	0
$838$	14490.4i	0.597329i
$839$	−2827.86	−0.116363	−0.0581816	−	0.998306i	$-0.518530\pi$
$839$	−2827.86	−0.116363	−0.0581816	+	0.998306i	$0.518530\pi$
$840$	0	0
$841$	7566.97	0.310262
$842$	− 1904.56i	− 0.0779520i
$843$	0	0
$844$	4960.15	0.202293
$845$	0	0
$846$	0	0
$847$	− 35049.0i	− 1.42184i
$848$	− 1966.80i	− 0.0796465i
$849$	0	0
$850$	0	0
$851$	22295.8	0.898110
$852$	0	0
$853$	− 24726.8i	− 0.992532i	−0.868171	−	0.496266i	$-0.834704\pi$
$853$	− 24726.8i	− 0.992532i	0.868171	−	0.496266i	$-0.165296\pi$
$854$	−2907.93	−0.116519
$855$	0	0
$856$	6107.99	0.243887
$857$	− 14785.5i	− 0.589338i	−0.955599	−	0.294669i	$-0.904791\pi$
$857$	− 14785.5i	− 0.589338i	0.955599	−	0.294669i	$-0.0952094\pi$
$858$	0	0
$859$	10329.4	0.410286	0.205143	−	0.978732i	$-0.434234\pi$
$859$	10329.4	0.410286	0.205143	+	0.978732i	$0.434234\pi$
$860$	0	0
$861$	0	0
$862$	12586.0i	0.497311i
$863$	21051.9i	0.830375i	0.909736	+	0.415188i	$0.136284\pi$
$863$	21051.9i	0.830375i	−0.909736	+	0.415188i	$0.863716\pi$
$864$	0	0
$865$	0	0
$866$	−7523.21	−0.295207
$867$	0	0
$868$	− 36362.5i	− 1.42192i
$869$	−10995.5	−0.429224
$870$	0	0
$871$	−11297.0	−0.439477
$872$	12252.2i	0.475816i
$873$	0	0
$874$	7672.34	0.296934
$875$	0	0
$876$	0	0
$877$	2246.67i	0.0865048i	0.999064	+	0.0432524i	$0.0137720\pi$
$877$	2246.67i	0.0865048i	−0.999064	+	0.0432524i	$0.986228\pi$
$878$	29288.2i	1.12577i
$879$	0	0
$880$	0	0
$881$	20270.2	0.775167	0.387583	−	0.921835i	$-0.373310\pi$
$881$	20270.2	0.775167	0.387583	+	0.921835i	$0.373310\pi$
$882$	0	0
$883$	18547.6i	0.706882i	0.935457	+	0.353441i	$0.114989\pi$
$883$	18547.6i	0.706882i	−0.935457	+	0.353441i	$0.885011\pi$
$884$	−6236.85	−0.237294
$885$	0	0
$886$	−33825.6	−1.28261
$887$	39211.8i	1.48433i	0.670215	+	0.742167i	$0.266202\pi$
$887$	39211.8i	1.48433i	−0.670215	+	0.742167i	$0.733798\pi$
$888$	0	0
$889$	74481.6	2.80994
$890$	0	0
$891$	0	0
$892$	23468.0i	0.880904i
$893$	− 8293.33i	− 0.310779i
$894$	0	0
$895$	0	0
$896$	−3908.80	−0.145741
$897$	0	0
$898$	7915.65i	0.294152i
$899$	53215.3	1.97423
$900$	0	0
$901$	−6826.95	−0.252429
$902$	− 3811.48i	− 0.140697i
$903$	0	0
$904$	−11126.1	−0.409346
$905$	0	0
$906$	0	0
$907$	28405.1i	1.03989i	0.854201	+	0.519943i	$0.174047\pi$
$907$	28405.1i	1.03989i	−0.854201	+	0.519943i	$0.825953\pi$
$908$	24636.3i	0.900423i
$909$	0	0
$910$	0	0
$911$	25264.2	0.918814	0.459407	−	0.888226i	$-0.348062\pi$
$911$	25264.2	0.918814	0.459407	+	0.888226i	$0.348062\pi$
$912$	0	0
$913$	18695.9i	0.677705i
$914$	−5556.65	−0.201092
$915$	0	0
$916$	−1279.30	−0.0461454
$917$	2616.68i	0.0942315i
$918$	0	0
$919$	−37988.0	−1.36355	−0.681777	−	0.731560i	$-0.738793\pi$
$919$	−37988.0	−1.36355	−0.681777	+	0.731560i	$0.738793\pi$
$920$	0	0
$921$	0	0
$922$	6234.14i	0.222680i
$923$	26031.8i	0.928328i
$924$	0	0
$925$	0	0
$926$	2342.54	0.0831324
$927$	0	0
$928$	− 5720.39i	− 0.202350i
$929$	−43992.2	−1.55365	−0.776823	−	0.629719i	$-0.783170\pi$
$929$	−43992.2	−1.55365	−0.776823	+	0.629719i	$0.783170\pi$
$930$	0	0
$931$	16190.2	0.569938
$932$	− 23249.7i	− 0.817134i
$933$	0	0
$934$	−27425.7	−0.960809
$935$	0	0
$936$	0	0
$937$	4507.35i	0.157149i	0.996908	+	0.0785745i	$0.0250369\pi$
$937$	4507.35i	0.157149i	−0.996908	+	0.0785745i	$0.974963\pi$
$938$	24575.8i	0.855467i
$939$	0	0
$940$	0	0
$941$	−40420.2	−1.40028	−0.700138	−	0.714008i	$-0.746878\pi$
$941$	−40420.2	−1.40028	−0.700138	+	0.714008i	$0.746878\pi$
$942$	0	0
$943$	− 19664.5i	− 0.679072i
$944$	−13828.8	−0.476789
$945$	0	0
$946$	153.986	0.00529229
$947$	35199.4i	1.20784i	0.797044	+	0.603921i	$0.206396\pi$
$947$	35199.4i	1.20784i	−0.797044	+	0.603921i	$0.793604\pi$
$948$	0	0
$949$	−28625.6	−0.979163
$950$	0	0
$951$	0	0
$952$	13567.8i	0.461906i
$953$	− 18416.3i	− 0.625983i	−0.949756	−	0.312992i	$-0.898669\pi$
$953$	− 18416.3i	− 0.625983i	0.949756	−	0.312992i	$-0.101331\pi$
$954$	0	0
$955$	0	0
$956$	14529.6	0.491549
$957$	0	0
$958$	− 27000.5i	− 0.910590i
$959$	−76826.1	−2.58691
$960$	0	0
$961$	58826.8	1.97465
$962$	8962.22i	0.300368i
$963$	0	0
$964$	−10329.8	−0.345125
$965$	0	0
$966$	0	0
$967$	− 50169.3i	− 1.66839i	−0.551468	−	0.834196i	$-0.685932\pi$
$967$	− 50169.3i	− 1.66839i	0.551468	−	0.834196i	$-0.314068\pi$
$968$	− 9181.89i	− 0.304873i
$969$	0	0
$970$	0	0
$971$	24543.7	0.811169	0.405585	−	0.914057i	$-0.367068\pi$
$971$	24543.7	0.811169	0.405585	+	0.914057i	$0.367068\pi$
$972$	0	0
$973$	− 94003.8i	− 3.09725i
$974$	11044.2	0.363327
$975$	0	0
$976$	−761.799	−0.0249842
$977$	17798.0i	0.582813i	0.956599	+	0.291406i	$0.0941232\pi$
$977$	17798.0i	0.582813i	−0.956599	+	0.291406i	$0.905877\pi$
$978$	0	0
$979$	−19268.6	−0.629035
$980$	0	0
$981$	0	0
$982$	9817.65i	0.319037i
$983$	9299.82i	0.301748i	0.988553	+	0.150874i	$0.0482087\pi$
$983$	9299.82i	0.301748i	−0.988553	+	0.150874i	$0.951791\pi$
$984$	0	0
$985$	0	0
$986$	−19856.0	−0.641323
$987$	0	0
$988$	3084.04i	0.0993080i
$989$	794.456	0.0255432
$990$	0	0
$991$	−17970.7	−0.576044	−0.288022	−	0.957624i	$-0.592998\pi$
$991$	−17970.7	−0.576044	−0.288022	+	0.957624i	$0.592998\pi$
$992$	− 9526.00i	− 0.304890i
$993$	0	0
$994$	56630.2	1.80704
$995$	0	0
$996$	0	0
$997$	2105.88i	0.0668945i	0.999440	+	0.0334473i	$0.0106486\pi$
$997$	2105.88i	0.0668945i	−0.999440	+	0.0334473i	$0.989351\pi$
$998$	22749.7i	0.721572i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1350.4.c.u.649.2		4
3.2	odd	2		1350.4.c.bb.649.4		4
5.2	odd	4		1350.4.a.bm.1.1		2
5.3	odd	4		270.4.a.m.1.2	✓	2
5.4	even	2	inner	1350.4.c.u.649.3		4
15.2	even	4		1350.4.a.bf.1.1		2
15.8	even	4		270.4.a.n.1.2	yes	2
15.14	odd	2		1350.4.c.bb.649.1		4
20.3	even	4		2160.4.a.w.1.1		2
45.13	odd	12		810.4.e.bd.541.1		4
45.23	even	12		810.4.e.z.541.1		4
45.38	even	12		810.4.e.z.271.1		4
45.43	odd	12		810.4.e.bd.271.1		4
60.23	odd	4		2160.4.a.bb.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.4.a.m.1.2	✓	2	5.3	odd	4
270.4.a.n.1.2	yes	2	15.8	even	4
810.4.e.z.271.1		4	45.38	even	12
810.4.e.z.541.1		4	45.23	even	12
810.4.e.bd.271.1		4	45.43	odd	12
810.4.e.bd.541.1		4	45.13	odd	12
1350.4.a.bf.1.1		2	15.2	even	4
1350.4.a.bm.1.1		2	5.2	odd	4
1350.4.c.u.649.2		4	1.1	even	1	trivial
1350.4.c.u.649.3		4	5.4	even	2	inner
1350.4.c.bb.649.1		4	15.14	odd	2
1350.4.c.bb.649.4		4	3.2	odd	2
2160.4.a.w.1.1		2	20.3	even	4
2160.4.a.bb.1.1		2	60.23	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:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1350.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} -4.00000 q^{4} +30.5375i q^{7} +8.00000i q^{8} +O(q^{10})\)	\(q-2.00000i q^{2} -4.00000 q^{4} +30.5375i q^{7} +8.00000i q^{8} +13.5375 q^{11} +28.0750i q^{13} +61.0750 q^{14} +16.0000 q^{16} -55.5375i q^{17} -27.4625 q^{19} -27.0750i q^{22} -139.687i q^{23} +56.1499 q^{26} -122.150i q^{28} +178.762 q^{29} +297.687 q^{31} -32.0000i q^{32} -111.075 q^{34} +159.612i q^{37} +54.9250i q^{38} +140.775 q^{41} +5.68738i q^{43} -54.1499 q^{44} -279.375 q^{46} +301.987i q^{47} -589.537 q^{49} -112.300i q^{52} -122.925i q^{53} -244.300 q^{56} -357.525i q^{58} -864.300 q^{59} -47.6124 q^{61} -595.375i q^{62} -64.0000 q^{64} +402.388i q^{67} +222.150i q^{68} +927.225 q^{71} +1019.61i q^{73} +319.225 q^{74} +109.850 q^{76} +413.400i q^{77} -812.225 q^{79} -281.550i q^{82} +1381.05i q^{83} +11.3748 q^{86} +108.300i q^{88} -1423.35 q^{89} -857.338 q^{91} +558.750i q^{92} +603.974 q^{94} -124.863i q^{97} +1179.07i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 16 q^{4}+O(q^{10}) \) 4 * q - 16 * q^4	\( 4 q - 16 q^{4} - 66 q^{11} + 4 q^{14} + 64 q^{16} - 230 q^{19} - 256 q^{26} - 126 q^{29} + 590 q^{31} - 204 q^{34} + 1284 q^{41} + 264 q^{44} + 84 q^{46} - 2238 q^{49} - 16 q^{56} - 2496 q^{59} + 170 q^{61} - 256 q^{64} + 2988 q^{71} + 556 q^{74} + 920 q^{76} - 2528 q^{79} - 1156 q^{86} - 1368 q^{89} - 7154 q^{91} - 708 q^{94}+O(q^{100}) \) 4 * q - 16 * q^4 - 66 * q^11 + 4 * q^14 + 64 * q^16 - 230 * q^19 - 256 * q^26 - 126 * q^29 + 590 * q^31 - 204 * q^34 + 1284 * q^41 + 264 * q^44 + 84 * q^46 - 2238 * q^49 - 16 * q^56 - 2496 * q^59 + 170 * q^61 - 256 * q^64 + 2988 * q^71 + 556 * q^74 + 920 * q^76 - 2528 * q^79 - 1156 * q^86 - 1368 * q^89 - 7154 * q^91 - 708 * q^94

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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