LMFDB - Embedded newform 1350.4.c.bb.649.1

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			649.1
Root			$10.5125i$ of defining polynomial
Character	$\chi$	$=$	1350.649
Dual form			1350.4.c.bb.649.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{2} -4.00000 q^{4} -30.5375i q^{7} +8.00000i q^{8} +O(q^{10})$	$q-2.00000i q^{2} -4.00000 q^{4} -30.5375i q^{7} +8.00000i q^{8} -13.5375 q^{11} -28.0750i q^{13} -61.0750 q^{14} +16.0000 q^{16} -55.5375i q^{17} -27.4625 q^{19} +27.0750i q^{22} -139.687i q^{23} -56.1499 q^{26} +122.150i q^{28} -178.762 q^{29} +297.687 q^{31} -32.0000i q^{32} -111.075 q^{34} -159.612i q^{37} +54.9250i q^{38} -140.775 q^{41} -5.68738i q^{43} +54.1499 q^{44} -279.375 q^{46} +301.987i q^{47} -589.537 q^{49} +112.300i q^{52} -122.925i q^{53} +244.300 q^{56} +357.525i q^{58} +864.300 q^{59} -47.6124 q^{61} -595.375i q^{62} -64.0000 q^{64} -402.388i q^{67} +222.150i q^{68} -927.225 q^{71} -1019.61i q^{73} -319.225 q^{74} +109.850 q^{76} +413.400i q^{77} -812.225 q^{79} +281.550i q^{82} +1381.05i q^{83} -11.3748 q^{86} -108.300i q^{88} +1423.35 q^{89} -857.338 q^{91} +558.750i q^{92} +603.974 q^{94} +124.863i q^{97} +1179.07i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.691493\pi$
$7$	− 30.5375i	− 1.64887i	0.565958	−	0.824434i	$-0.308507\pi$
$8$	8.00000i	0.353553i
$9$	0	0
$10$	0	0
$11$	−13.5375	−0.371064	−0.185532	−	0.982638i	$-0.559401\pi$
$11$	−13.5375	−0.371064	−0.185532	+	0.982638i	$0.559401\pi$
$12$	0	0
$13$	− 28.0750i	− 0.598969i	−0.954101	−	0.299484i	$-0.903185\pi$
$13$	− 28.0750i	− 0.598969i	0.954101	−	0.299484i	$-0.0968146\pi$
$14$	−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	$-0.693962\pi$
$29$	−178.762	−1.14467	−0.572333	+	0.820021i	$0.693962\pi$
$30$	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	$-0.884618\pi$
$37$	− 159.612i	− 0.709192i	0.935020	−	0.354596i	$-0.115382\pi$
$38$	54.9250i	0.234474i
$39$	0	0
$40$	0	0
$41$	−140.775	−0.536229	−0.268114	−	0.963387i	$-0.586401\pi$
$41$	−140.775	−0.536229	−0.268114	+	0.963387i	$0.586401\pi$
$42$	0	0
$43$	− 5.68738i	− 0.0201702i	−0.999949	−	0.0100851i	$-0.996790\pi$
$43$	− 5.68738i	− 0.0201702i	0.999949	−	0.0100851i	$-0.00321024\pi$
$44$	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	$-0.0973688\pi$
$59$	864.300	1.90716	0.953578	+	0.301145i	$0.0973688\pi$
$60$	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	$-0.880432\pi$
$67$	− 402.388i	− 0.733723i	0.930275	−	0.366862i	$-0.119568\pi$
$68$	222.150i	0.396171i
$69$	0	0
$70$	0	0
$71$	−927.225	−1.54988	−0.774939	−	0.632036i	$-0.782219\pi$
$71$	−927.225	−1.54988	−0.774939	+	0.632036i	$0.782219\pi$
$72$	0	0
$73$	− 1019.61i	− 1.63475i	−0.576107	−	0.817374i	$-0.695429\pi$
$73$	− 1019.61i	− 1.63475i	0.576107	−	0.817374i	$-0.304571\pi$
$74$	−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	$-0.178040\pi$
$89$	1423.35	1.69522	0.847611	+	0.530619i	$0.178040\pi$
$90$	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	$0.0208164\pi$
$97$	124.863i	0.130700i	−0.997862	+	0.0653500i	$0.979184\pi$
$98$	1179.07i	1.21535i
$99$	0	0
$100$	0	0
$101$	−571.537	−0.563070	−0.281535	−	0.959551i	$-0.590844\pi$
$101$	−571.537	−0.563070	−0.281535	+	0.959551i	$0.590844\pi$
$102$	0	0
$103$	− 1200.51i	− 1.14845i	−0.818699	−	0.574223i	$-0.805304\pi$
$103$	− 1200.51i	− 1.14845i	0.818699	−	0.574223i	$-0.194696\pi$
$104$	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	$0.324659\pi$
$127$	2439.02i	1.70416i	−0.523411	+	0.852080i	$0.675341\pi$
$128$	128.000i	0.0883883i
$129$	0	0
$130$	0	0
$131$	−85.6874	−0.0571492	−0.0285746	−	0.999592i	$-0.509097\pi$
$131$	−85.6874	−0.0571492	−0.0285746	+	0.999592i	$0.509097\pi$
$132$	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	$-0.519678\pi$
$149$	−224.737	−0.123565	−0.0617824	+	0.998090i	$0.519678\pi$
$150$	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	$-0.847541\pi$
$157$	− 1813.21i	− 0.921720i	0.887473	−	0.460860i	$-0.152459\pi$
$158$	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	$-0.983276\pi$
$163$	− 218.576i	− 0.105032i	0.998620	−	0.0525159i	$-0.0167240\pi$
$164$	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	$-0.734909\pi$
$179$	−3222.52	−1.34560	−0.672801	+	0.739824i	$0.734909\pi$
$180$	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	$-0.572094\pi$
$191$	−1185.52	−0.449118	−0.224559	+	0.974460i	$0.572094\pi$
$192$	0	0
$193$	1401.16i	0.522580i	0.965260	+	0.261290i	$0.0841479\pi$
$193$	1401.16i	0.522580i	−0.965260	+	0.261290i	$0.915852\pi$
$194$	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	$0.343065\pi$
$223$	5867.00i	1.76181i	−0.473295	+	0.880904i	$0.656935\pi$
$224$	−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	$-0.336431\pi$
$239$	3632.40	0.983098	0.491549	+	0.870850i	$0.336431\pi$
$240$	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	$-0.489938\pi$
$251$	251.362	0.0632105	0.0316052	+	0.999500i	$0.489938\pi$
$252$	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	$-0.342921\pi$
$269$	4179.79	0.947383	0.473692	+	0.880691i	$0.342921\pi$
$270$	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	$0.0564582\pi$
$277$	1626.85i	0.352880i	−0.984311	+	0.176440i	$0.943542\pi$
$278$	6156.62i	1.32824i
$279$	0	0
$280$	0	0
$281$	−906.227	−0.192388	−0.0961939	−	0.995363i	$-0.530667\pi$
$281$	−906.227	−0.192388	−0.0961939	+	0.995363i	$0.530667\pi$
$282$	0	0
$283$	− 1643.95i	− 0.345310i	−0.984982	−	0.172655i	$-0.944765\pi$
$283$	− 1643.95i	− 0.345310i	0.984982	−	0.172655i	$-0.0552345\pi$
$284$	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	$-0.855458\pi$
$307$	− 4719.01i	− 0.877291i	0.898660	−	0.438646i	$-0.144542\pi$
$308$	− 1653.60i	− 0.305918i
$309$	0	0
$310$	0	0
$311$	5386.65	0.982150	0.491075	−	0.871117i	$-0.336604\pi$
$311$	5386.65	0.982150	0.491075	+	0.871117i	$0.336604\pi$
$312$	0	0
$313$	− 2706.96i	− 0.488838i	−0.969670	−	0.244419i	$-0.921403\pi$
$313$	− 2706.96i	− 0.488838i	0.969670	−	0.244419i	$-0.0785972\pi$
$314$	−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	$-0.827754\pi$
$337$	− 6373.36i	− 1.03020i	0.857129	−	0.515102i	$-0.172246\pi$
$338$	− 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	$-0.663528\pi$
$359$	−6685.57	−0.982872	−0.491436	+	0.870914i	$0.663528\pi$
$360$	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	$0.0906898\pi$
$367$	3952.26i	0.562143i	−0.959687	+	0.281071i	$0.909310\pi$
$368$	− 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	$-0.765727\pi$
$373$	− 9672.17i	− 1.34264i	0.741166	−	0.671322i	$-0.234273\pi$
$374$	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	$-0.860748\pi$
$389$	−13899.5	−1.81165	−0.905825	+	0.423653i	$0.860748\pi$
$390$	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	$-0.870964\pi$
$397$	− 6239.04i	− 0.788736i	0.918953	−	0.394368i	$-0.129036\pi$
$398$	− 415.250i	− 0.0522981i
$399$	0	0
$400$	0	0
$401$	−10106.0	−1.25853	−0.629265	−	0.777191i	$-0.716644\pi$
$401$	−10106.0	−1.25853	−0.629265	+	0.777191i	$0.716644\pi$
$402$	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	$-0.361196\pi$
$419$	7245.19	0.844751	0.422376	+	0.906421i	$0.361196\pi$
$420$	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	$-0.385620\pi$
$431$	6293.02	0.703305	0.351652	+	0.936131i	$0.385620\pi$
$432$	0	0
$433$	3761.61i	0.417485i	0.977971	+	0.208743i	$0.0669371\pi$
$433$	3761.61i	0.417485i	−0.977971	+	0.208743i	$0.933063\pi$
$434$	−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	$-0.433306\pi$
$449$	3957.82	0.415994	0.207997	+	0.978129i	$0.433306\pi$
$450$	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	$0.0454154\pi$
$457$	2778.33i	0.284386i	−0.989839	+	0.142193i	$0.954585\pi$
$458$	− 639.649i	− 0.0652595i
$459$	0	0
$460$	0	0
$461$	3117.07	0.314916	0.157458	−	0.987526i	$-0.449670\pi$
$461$	3117.07	0.314916	0.157458	+	0.987526i	$0.449670\pi$
$462$	0	0
$463$	− 1171.27i	− 0.117567i	−0.998271	−	0.0587835i	$-0.981278\pi$
$463$	− 1171.27i	− 0.117567i	0.998271	−	0.0587835i	$-0.0187221\pi$
$464$	−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	$-0.722678\pi$
$479$	−13500.2	−1.28777	−0.643884	+	0.765123i	$0.722678\pi$
$480$	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	$-0.917295\pi$
$487$	− 5522.12i	− 0.513821i	0.966435	−	0.256911i	$-0.0827046\pi$
$488$	− 380.899i	− 0.0353330i
$489$	0	0
$490$	0	0
$491$	4908.83	0.451186	0.225593	−	0.974222i	$-0.427568\pi$
$491$	4908.83	0.451186	0.225593	+	0.974222i	$0.427568\pi$
$492$	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	$-0.287145\pi$
$509$	14238.9	1.23994	0.619970	+	0.784626i	$0.287145\pi$
$510$	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	$-0.356941\pi$
$521$	10333.1	0.868910	0.434455	+	0.900694i	$0.356941\pi$
$522$	0	0
$523$	− 8396.57i	− 0.702020i	−0.936372	−	0.351010i	$-0.885838\pi$
$523$	− 8396.57i	− 0.702020i	0.936372	−	0.351010i	$-0.114162\pi$
$524$	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	$-0.645228\pi$
$547$	− 22969.3i	− 1.79542i	0.440583	−	0.897712i	$-0.354772\pi$
$548$	− 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	$-0.602059\pi$
$569$	−8555.25	−0.630324	−0.315162	+	0.949038i	$0.602059\pi$
$570$	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	$0.0867811\pi$
$577$	7464.06i	0.538532i	−0.963066	+	0.269266i	$0.913219\pi$
$578$	− 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	$-0.256980\pi$
$599$	20273.1	1.38287	0.691433	+	0.722441i	$0.256980\pi$
$600$	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	$-0.821377\pi$
$607$	− 15917.0i	− 1.06434i	0.846638	−	0.532169i	$-0.178623\pi$
$608$	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	$0.144946\pi$
$613$	13349.4i	0.879574i	−0.898102	+	0.439787i	$0.855054\pi$
$614$	−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	$-0.740297\pi$
$641$	−22240.9	−1.37045	−0.685227	+	0.728329i	$0.740297\pi$
$642$	0	0
$643$	− 25250.1i	− 1.54862i	−0.632804	−	0.774312i	$-0.718096\pi$
$643$	− 25250.1i	− 1.54862i	0.632804	−	0.774312i	$-0.281904\pi$
$644$	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	$-0.506639\pi$
$659$	−705.594	−0.0417087	−0.0208544	+	0.999783i	$0.506639\pi$
$660$	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	$-0.837043\pi$
$673$	− 17105.5i	− 0.979747i	0.871793	−	0.489874i	$-0.162957\pi$
$674$	−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	$-0.581281\pi$
$701$	−9375.93	−0.505170	−0.252585	+	0.967575i	$0.581281\pi$
$702$	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	$-0.684850\pi$
$719$	−21154.5	−1.09726	−0.548630	+	0.836065i	$0.684850\pi$
$720$	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	$-0.778485\pi$
$727$	− 25133.1i	− 1.28217i	0.767471	−	0.641084i	$-0.221515\pi$
$728$	− 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	$-0.988154\pi$
$733$	− 1476.70i	− 0.0744108i	0.999308	−	0.0372054i	$-0.0118456\pi$
$734$	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	$0.112853\pi$
$757$	14461.0i	0.694312i	−0.937807	+	0.347156i	$0.887147\pi$
$758$	− 8099.06i	− 0.388089i
$759$	0	0
$760$	0	0
$761$	24463.5	1.16531	0.582655	−	0.812719i	$-0.302014\pi$
$761$	24463.5	1.16531	0.582655	+	0.812719i	$0.302014\pi$
$762$	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	$-0.786044\pi$
$787$	− 27495.2i	− 1.24536i	0.782477	−	0.622679i	$-0.213956\pi$
$788$	− 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	$-0.551903\pi$
$809$	−7470.89	−0.324675	−0.162338	+	0.986735i	$0.551903\pi$
$810$	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	$-0.423926\pi$
$821$	11137.6	0.473452	0.236726	+	0.971576i	$0.423926\pi$
$822$	0	0
$823$	2369.19i	0.100346i	0.998741	+	0.0501730i	$0.0159773\pi$
$823$	2369.19i	0.100346i	−0.998741	+	0.0501730i	$0.984023\pi$
$824$	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	$-0.481470\pi$
$839$	2827.86	0.116363	0.0581816	+	0.998306i	$0.481470\pi$
$840$	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	$0.165296\pi$
$853$	24726.8i	0.992532i	−0.868171	+	0.496266i	$0.834704\pi$
$854$	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	$-0.986228\pi$
$877$	− 2246.67i	− 0.0865048i	0.999064	−	0.0432524i	$-0.0137720\pi$
$878$	29288.2i	1.12577i
$879$	0	0
$880$	0	0
$881$	−20270.2	−0.775167	−0.387583	−	0.921835i	$-0.626690\pi$
$881$	−20270.2	−0.775167	−0.387583	+	0.921835i	$0.626690\pi$
$882$	0	0
$883$	− 18547.6i	− 0.706882i	−0.935457	−	0.353441i	$-0.885011\pi$
$883$	− 18547.6i	− 0.706882i	0.935457	−	0.353441i	$-0.114989\pi$
$884$	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	$-0.825953\pi$
$907$	− 28405.1i	− 1.03989i	0.854201	−	0.519943i	$-0.174047\pi$
$908$	24636.3i	0.900423i
$909$	0	0
$910$	0	0
$911$	−25264.2	−0.918814	−0.459407	−	0.888226i	$-0.651938\pi$
$911$	−25264.2	−0.918814	−0.459407	+	0.888226i	$0.651938\pi$
$912$	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	$-0.216830\pi$
$929$	43992.2	1.55365	0.776823	+	0.629719i	$0.216830\pi$
$930$	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	$-0.974963\pi$
$937$	− 4507.35i	− 0.157149i	0.996908	−	0.0785745i	$-0.0250369\pi$
$938$	24575.8i	0.855467i
$939$	0	0
$940$	0	0
$941$	40420.2	1.40028	0.700138	−	0.714008i	$-0.253122\pi$
$941$	40420.2	1.40028	0.700138	+	0.714008i	$0.253122\pi$
$942$	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	$0.314068\pi$
$967$	50169.3i	1.66839i	−0.551468	+	0.834196i	$0.685932\pi$
$968$	− 9181.89i	− 0.304873i
$969$	0	0
$970$	0	0
$971$	−24543.7	−0.811169	−0.405585	−	0.914057i	$-0.632932\pi$
$971$	−24543.7	−0.811169	−0.405585	+	0.914057i	$0.632932\pi$
$972$	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	$-0.989351\pi$
$997$	− 2105.88i	− 0.0668945i	0.999440	−	0.0334473i	$-0.0106486\pi$
$998$	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.bb.649.1		4
3.2	odd	2		1350.4.c.u.649.3		4
5.2	odd	4		270.4.a.n.1.2	yes	2
5.3	odd	4		1350.4.a.bf.1.1		2
5.4	even	2	inner	1350.4.c.bb.649.4		4
15.2	even	4		270.4.a.m.1.2	✓	2
15.8	even	4		1350.4.a.bm.1.1		2
15.14	odd	2		1350.4.c.u.649.2		4
20.7	even	4		2160.4.a.bb.1.1		2
45.2	even	12		810.4.e.bd.271.1		4
45.7	odd	12		810.4.e.z.271.1		4
45.22	odd	12		810.4.e.z.541.1		4
45.32	even	12		810.4.e.bd.541.1		4
60.47	odd	4		2160.4.a.w.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.4.a.m.1.2	✓	2	15.2	even	4
270.4.a.n.1.2	yes	2	5.2	odd	4
810.4.e.z.271.1		4	45.7	odd	12
810.4.e.z.541.1		4	45.22	odd	12
810.4.e.bd.271.1		4	45.2	even	12
810.4.e.bd.541.1		4	45.32	even	12
1350.4.a.bf.1.1		2	5.3	odd	4
1350.4.a.bm.1.1		2	15.8	even	4
1350.4.c.u.649.2		4	15.14	odd	2
1350.4.c.u.649.3		4	3.2	odd	2
1350.4.c.bb.649.1		4	1.1	even	1	trivial
1350.4.c.bb.649.4		4	5.4	even	2	inner
2160.4.a.w.1.1		2	60.47	odd	4
2160.4.a.bb.1.1		2	20.7	even	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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