LMFDB - Embedded newform 3380.2.f.g.3041.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3380,2,Mod(3041,3380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3380, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3380.3041");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3380 = 2^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3380.f (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:	$26.9894358832$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.153664.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 5x^{4} + 6x^{2} + 1 $ x^6 + 5x^4 + 6x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3041.6
Root			$-1.24698i$ of defining polynomial
Character	$\chi$	$=$	3380.3041
Dual form			3380.2.f.g.3041.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.24698 q^{3} +1.00000i q^{5} +1.24698i q^{7} -1.44504 q^{9} +O(q^{10})$	$q+1.24698 q^{3} +1.00000i q^{5} +1.24698i q^{7} -1.44504 q^{9} +0.198062i q^{11} +1.24698i q^{15} +4.85086 q^{17} -1.35690i q^{19} +1.55496i q^{21} +1.44504 q^{23} -1.00000 q^{25} -5.54288 q^{27} +1.13706 q^{29} +6.85086i q^{31} +0.246980i q^{33} -1.24698 q^{35} +3.00000i q^{37} +3.54288i q^{41} +7.89977 q^{43} -1.44504i q^{45} +8.87263i q^{47} +5.44504 q^{49} +6.04892 q^{51} -1.86831 q^{53} -0.198062 q^{55} -1.69202i q^{57} +0.878002i q^{59} -8.19806 q^{61} -1.80194i q^{63} +0.207751i q^{67} +1.80194 q^{69} +0.664874i q^{71} +1.72587i q^{73} -1.24698 q^{75} -0.246980 q^{77} -12.5700 q^{79} -2.57673 q^{81} +12.3230i q^{83} +4.85086i q^{85} +1.41789 q^{87} +15.8659i q^{89} +8.54288i q^{93} +1.35690 q^{95} -1.31336i q^{97} -0.286208i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} - 8 q^{9}+O(q^{10}) $ 6 * q - 2 * q^3 - 8 * q^9	$ 6 q - 2 q^{3} - 8 q^{9} + 2 q^{17} + 8 q^{23} - 6 q^{25} + 4 q^{27} - 4 q^{29} + 2 q^{35} + 2 q^{43} + 32 q^{49} + 18 q^{51} - 16 q^{53} - 10 q^{55} - 58 q^{61} + 2 q^{69} + 2 q^{75} + 8 q^{77} - 26 q^{79} - 10 q^{81} + 20 q^{87}+O(q^{100}) $ 6 * q - 2 * q^3 - 8 * q^9 + 2 * q^17 + 8 * q^23 - 6 * q^25 + 4 * q^27 - 4 * q^29 + 2 * q^35 + 2 * q^43 + 32 * q^49 + 18 * q^51 - 16 * q^53 - 10 * q^55 - 58 * q^61 + 2 * q^69 + 2 * q^75 + 8 * q^77 - 26 * q^79 - 10 * q^81 + 20 * q^87

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3380\mathbb{Z}\right)^\times$.

$n$	$677$	$1691$	$1861$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.24698	0.719944	0.359972	−	0.932963i	$-0.382786\pi$
$3$	1.24698	0.719944	0.359972	+	0.932963i	$0.382786\pi$
$4$	0	0
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	1.24698i	0.471314i	0.971836	+	0.235657i	$0.0757242\pi$
$7$	1.24698i	0.471314i	−0.971836	+	0.235657i	$0.924276\pi$
$8$	0	0
$9$	−1.44504	−0.481681
$10$	0	0
$11$	0.198062i	0.0597180i	0.999554	+	0.0298590i	$0.00950583\pi$
$11$	0.198062i	0.0597180i	−0.999554	+	0.0298590i	$0.990494\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	1.24698i	0.321969i
$16$	0	0
$17$	4.85086	1.17651	0.588253	−	0.808677i	$-0.299816\pi$
$17$	4.85086	1.17651	0.588253	+	0.808677i	$0.299816\pi$
$18$	0	0
$19$	− 1.35690i	− 0.311293i	−0.987813	−	0.155647i	$-0.950254\pi$
$19$	− 1.35690i	− 0.311293i	0.987813	−	0.155647i	$-0.0497461\pi$
$20$	0	0
$21$	1.55496i	0.339320i
$22$	0	0
$23$	1.44504	0.301312	0.150656	−	0.988586i	$-0.451861\pi$
$23$	1.44504	0.301312	0.150656	+	0.988586i	$0.451861\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−5.54288	−1.06673
$28$	0	0
$29$	1.13706	0.211147	0.105574	−	0.994411i	$-0.466332\pi$
$29$	1.13706	0.211147	0.105574	+	0.994411i	$0.466332\pi$
$30$	0	0
$31$	6.85086i	1.23045i	0.788352	+	0.615225i	$0.210935\pi$
$31$	6.85086i	1.23045i	−0.788352	+	0.615225i	$0.789065\pi$
$32$	0	0
$33$	0.246980i	0.0429936i
$34$	0	0
$35$	−1.24698	−0.210778
$36$	0	0
$37$	3.00000i	0.493197i	0.969118	+	0.246598i	$0.0793129\pi$
$37$	3.00000i	0.493197i	−0.969118	+	0.246598i	$0.920687\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.54288i	0.553304i	0.960970	+	0.276652i	$0.0892250\pi$
$41$	3.54288i	0.553304i	−0.960970	+	0.276652i	$0.910775\pi$
$42$	0	0
$43$	7.89977	1.20470	0.602352	−	0.798231i	$-0.294230\pi$
$43$	7.89977	1.20470	0.602352	+	0.798231i	$0.294230\pi$
$44$	0	0
$45$	− 1.44504i	− 0.215414i
$46$	0	0
$47$	8.87263i	1.29421i	0.762403	+	0.647103i	$0.224020\pi$
$47$	8.87263i	1.29421i	−0.762403	+	0.647103i	$0.775980\pi$
$48$	0	0
$49$	5.44504	0.777863
$50$	0	0
$51$	6.04892	0.847018
$52$	0	0
$53$	−1.86831	−0.256633	−0.128316	−	0.991733i	$-0.540957\pi$
$53$	−1.86831	−0.256633	−0.128316	+	0.991733i	$0.540957\pi$
$54$	0	0
$55$	−0.198062	−0.0267067
$56$	0	0
$57$	− 1.69202i	− 0.224114i
$58$	0	0
$59$	0.878002i	0.114306i	0.998365	+	0.0571531i	$0.0182023\pi$
$59$	0.878002i	0.114306i	−0.998365	+	0.0571531i	$0.981798\pi$
$60$	0	0
$61$	−8.19806	−1.04965	−0.524827	−	0.851209i	$-0.675870\pi$
$61$	−8.19806	−1.04965	−0.524827	+	0.851209i	$0.675870\pi$
$62$	0	0
$63$	− 1.80194i	− 0.227023i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0.207751i	0.0253808i	0.999919	+	0.0126904i	$0.00403959\pi$
$67$	0.207751i	0.0253808i	−0.999919	+	0.0126904i	$0.995960\pi$
$68$	0	0
$69$	1.80194	0.216928
$70$	0	0
$71$	0.664874i	0.0789061i	0.999221	+	0.0394530i	$0.0125616\pi$
$71$	0.664874i	0.0789061i	−0.999221	+	0.0394530i	$0.987438\pi$
$72$	0	0
$73$	1.72587i	0.201998i	0.994887	+	0.100999i	$0.0322039\pi$
$73$	1.72587i	0.201998i	−0.994887	+	0.100999i	$0.967796\pi$
$74$	0	0
$75$	−1.24698	−0.143989
$76$	0	0
$77$	−0.246980	−0.0281459
$78$	0	0
$79$	−12.5700	−1.41424	−0.707119	−	0.707094i	$-0.750006\pi$
$79$	−12.5700	−1.41424	−0.707119	+	0.707094i	$0.750006\pi$
$80$	0	0
$81$	−2.57673	−0.286303
$82$	0	0
$83$	12.3230i	1.35263i	0.736613	+	0.676315i	$0.236424\pi$
$83$	12.3230i	1.35263i	−0.736613	+	0.676315i	$0.763576\pi$
$84$	0	0
$85$	4.85086i	0.526149i
$86$	0	0
$87$	1.41789	0.152014
$88$	0	0
$89$	15.8659i	1.68178i	0.541203	+	0.840892i	$0.317969\pi$
$89$	15.8659i	1.68178i	−0.541203	+	0.840892i	$0.682031\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	8.54288i	0.885855i
$94$	0	0
$95$	1.35690	0.139215
$96$	0	0
$97$	− 1.31336i	− 0.133351i	−0.997775	−	0.0666755i	$-0.978761\pi$
$97$	− 1.31336i	− 0.133351i	0.997775	−	0.0666755i	$-0.0212392\pi$
$98$	0	0
$99$	− 0.286208i	− 0.0287650i
$100$	0	0
$101$	−13.7506	−1.36824	−0.684119	−	0.729370i	$-0.739813\pi$
$101$	−13.7506	−1.36824	−0.684119	+	0.729370i	$0.739813\pi$
$102$	0	0
$103$	11.9608	1.17853	0.589265	−	0.807940i	$-0.299417\pi$
$103$	11.9608	1.17853	0.589265	+	0.807940i	$0.299417\pi$
$104$	0	0
$105$	−1.55496	−0.151748
$106$	0	0
$107$	−5.47219	−0.529016	−0.264508	−	0.964383i	$-0.585210\pi$
$107$	−5.47219	−0.529016	−0.264508	+	0.964383i	$0.585210\pi$
$108$	0	0
$109$	18.8116i	1.80183i	0.433999	+	0.900914i	$0.357102\pi$
$109$	18.8116i	1.80183i	−0.433999	+	0.900914i	$0.642898\pi$
$110$	0	0
$111$	3.74094i	0.355074i
$112$	0	0
$113$	3.72886	0.350781	0.175391	−	0.984499i	$-0.443881\pi$
$113$	3.72886	0.350781	0.175391	+	0.984499i	$0.443881\pi$
$114$	0	0
$115$	1.44504i	0.134751i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	6.04892i	0.554503i
$120$	0	0
$121$	10.9608	0.996434
$122$	0	0
$123$	4.41789i	0.398348i
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	−7.66786	−0.680412	−0.340206	−	0.940351i	$-0.610497\pi$
$127$	−7.66786	−0.680412	−0.340206	+	0.940351i	$0.610497\pi$
$128$	0	0
$129$	9.85086	0.867319
$130$	0	0
$131$	−9.32304	−0.814558	−0.407279	−	0.913304i	$-0.633522\pi$
$131$	−9.32304	−0.814558	−0.407279	+	0.913304i	$0.633522\pi$
$132$	0	0
$133$	1.69202	0.146717
$134$	0	0
$135$	− 5.54288i	− 0.477055i
$136$	0	0
$137$	4.72348i	0.403554i	0.979431	+	0.201777i	$0.0646717\pi$
$137$	4.72348i	0.403554i	−0.979431	+	0.201777i	$0.935328\pi$
$138$	0	0
$139$	0.818331	0.0694099	0.0347050	−	0.999398i	$-0.488951\pi$
$139$	0.818331	0.0694099	0.0347050	+	0.999398i	$0.488951\pi$
$140$	0	0
$141$	11.0640i	0.931755i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	1.13706i	0.0944280i
$146$	0	0
$147$	6.78986	0.560018
$148$	0	0
$149$	− 23.6233i	− 1.93529i	−0.252311	−	0.967646i	$-0.581191\pi$
$149$	− 23.6233i	− 1.93529i	0.252311	−	0.967646i	$-0.418809\pi$
$150$	0	0
$151$	6.78448i	0.552113i	0.961141	+	0.276057i	$0.0890277\pi$
$151$	6.78448i	0.552113i	−0.961141	+	0.276057i	$0.910972\pi$
$152$	0	0
$153$	−7.00969	−0.566700
$154$	0	0
$155$	−6.85086	−0.550274
$156$	0	0
$157$	−16.4276	−1.31106	−0.655532	−	0.755167i	$-0.727556\pi$
$157$	−16.4276	−1.31106	−0.655532	+	0.755167i	$0.727556\pi$
$158$	0	0
$159$	−2.32975	−0.184761
$160$	0	0
$161$	1.80194i	0.142013i
$162$	0	0
$163$	− 12.7463i	− 0.998368i	−0.866496	−	0.499184i	$-0.833633\pi$
$163$	− 12.7463i	− 0.998368i	0.866496	−	0.499184i	$-0.166367\pi$
$164$	0	0
$165$	−0.246980	−0.0192273
$166$	0	0
$167$	− 15.2054i	− 1.17663i	−0.808633	−	0.588313i	$-0.799792\pi$
$167$	− 15.2054i	− 1.17663i	0.808633	−	0.588313i	$-0.200208\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	1.96077i	0.149944i
$172$	0	0
$173$	13.7845	1.04801	0.524007	−	0.851714i	$-0.324436\pi$
$173$	13.7845	1.04801	0.524007	+	0.851714i	$0.324436\pi$
$174$	0	0
$175$	− 1.24698i	− 0.0942628i
$176$	0	0
$177$	1.09485i	0.0822940i
$178$	0	0
$179$	6.51035	0.486607	0.243303	−	0.969950i	$-0.421769\pi$
$179$	6.51035	0.486607	0.243303	+	0.969950i	$0.421769\pi$
$180$	0	0
$181$	4.45712	0.331295	0.165648	−	0.986185i	$-0.447029\pi$
$181$	4.45712	0.331295	0.165648	+	0.986185i	$0.447029\pi$
$182$	0	0
$183$	−10.2228	−0.755692
$184$	0	0
$185$	−3.00000	−0.220564
$186$	0	0
$187$	0.960771i	0.0702586i
$188$	0	0
$189$	− 6.91185i	− 0.502763i
$190$	0	0
$191$	23.1564	1.67554	0.837771	−	0.546022i	$-0.183859\pi$
$191$	23.1564	1.67554	0.837771	+	0.546022i	$0.183859\pi$
$192$	0	0
$193$	0.692021i	0.0498128i	0.999690	+	0.0249064i	$0.00792877\pi$
$193$	0.692021i	0.0498128i	−0.999690	+	0.0249064i	$0.992071\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.98792i	0.497869i	0.968520	+	0.248934i	$0.0800803\pi$
$197$	6.98792i	0.497869i	−0.968520	+	0.248934i	$0.919920\pi$
$198$	0	0
$199$	−18.1400	−1.28591	−0.642957	−	0.765902i	$-0.722293\pi$
$199$	−18.1400	−1.28591	−0.642957	+	0.765902i	$0.722293\pi$
$200$	0	0
$201$	0.259061i	0.0182728i
$202$	0	0
$203$	1.41789i	0.0995167i
$204$	0	0
$205$	−3.54288	−0.247445
$206$	0	0
$207$	−2.08815	−0.145136
$208$	0	0
$209$	0.268750	0.0185898
$210$	0	0
$211$	16.9095	1.16410	0.582048	−	0.813155i	$-0.302252\pi$
$211$	16.9095	1.16410	0.582048	+	0.813155i	$0.302252\pi$
$212$	0	0
$213$	0.829085i	0.0568080i
$214$	0	0
$215$	7.89977i	0.538760i
$216$	0	0
$217$	−8.54288	−0.579928
$218$	0	0
$219$	2.15213i	0.145427i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	12.0532i	0.807144i	0.914948	+	0.403572i	$0.132231\pi$
$223$	12.0532i	0.807144i	−0.914948	+	0.403572i	$0.867769\pi$
$224$	0	0
$225$	1.44504	0.0963361
$226$	0	0
$227$	− 15.1642i	− 1.00648i	−0.864146	−	0.503242i	$-0.832140\pi$
$227$	− 15.1642i	− 1.00648i	0.864146	−	0.503242i	$-0.167860\pi$
$228$	0	0
$229$	− 20.5375i	− 1.35716i	−0.734528	−	0.678578i	$-0.762597\pi$
$229$	− 20.5375i	− 1.35716i	0.734528	−	0.678578i	$-0.237403\pi$
$230$	0	0
$231$	−0.307979	−0.0202635
$232$	0	0
$233$	17.2198	1.12811	0.564054	−	0.825738i	$-0.309241\pi$
$233$	17.2198	1.12811	0.564054	+	0.825738i	$0.309241\pi$
$234$	0	0
$235$	−8.87263	−0.578786
$236$	0	0
$237$	−15.6746	−1.01817
$238$	0	0
$239$	15.4112i	0.996867i	0.866928	+	0.498434i	$0.166091\pi$
$239$	15.4112i	0.996867i	−0.866928	+	0.498434i	$0.833909\pi$
$240$	0	0
$241$	− 10.2892i	− 0.662785i	−0.943493	−	0.331393i	$-0.892482\pi$
$241$	− 10.2892i	− 0.662785i	0.943493	−	0.331393i	$-0.107518\pi$
$242$	0	0
$243$	13.4155	0.860605
$244$	0	0
$245$	5.44504i	0.347871i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	15.3666i	0.973818i
$250$	0	0
$251$	8.09246	0.510791	0.255396	−	0.966837i	$-0.417794\pi$
$251$	8.09246	0.510791	0.255396	+	0.966837i	$0.417794\pi$
$252$	0	0
$253$	0.286208i	0.0179938i
$254$	0	0
$255$	6.04892i	0.378798i
$256$	0	0
$257$	−31.4228	−1.96010	−0.980050	−	0.198750i	$-0.936312\pi$
$257$	−31.4228	−1.96010	−0.980050	+	0.198750i	$0.936312\pi$
$258$	0	0
$259$	−3.74094	−0.232451
$260$	0	0
$261$	−1.64310	−0.101706
$262$	0	0
$263$	6.84117	0.421844	0.210922	−	0.977503i	$-0.432353\pi$
$263$	6.84117	0.421844	0.210922	+	0.977503i	$0.432353\pi$
$264$	0	0
$265$	− 1.86831i	− 0.114770i
$266$	0	0
$267$	19.7845i	1.21079i
$268$	0	0
$269$	0.735562	0.0448480	0.0224240	−	0.999749i	$-0.492862\pi$
$269$	0.735562	0.0448480	0.0224240	+	0.999749i	$0.492862\pi$
$270$	0	0
$271$	0.994623i	0.0604191i	0.999544	+	0.0302095i	$0.00961745\pi$
$271$	0.994623i	0.0604191i	−0.999544	+	0.0302095i	$0.990383\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 0.198062i	− 0.0119436i
$276$	0	0
$277$	27.3110	1.64096	0.820478	−	0.571678i	$-0.193707\pi$
$277$	27.3110	1.64096	0.820478	+	0.571678i	$0.193707\pi$
$278$	0	0
$279$	− 9.89977i	− 0.592684i
$280$	0	0
$281$	− 13.3327i	− 0.795364i	−0.917523	−	0.397682i	$-0.869815\pi$
$281$	− 13.3327i	− 0.795364i	0.917523	−	0.397682i	$-0.130185\pi$
$282$	0	0
$283$	−0.907542	−0.0539478	−0.0269739	−	0.999636i	$-0.508587\pi$
$283$	−0.907542	−0.0539478	−0.0269739	+	0.999636i	$0.508587\pi$
$284$	0	0
$285$	1.69202	0.100227
$286$	0	0
$287$	−4.41789	−0.260780
$288$	0	0
$289$	6.53079	0.384164
$290$	0	0
$291$	− 1.63773i	− 0.0960053i
$292$	0	0
$293$	− 6.68425i	− 0.390498i	−0.980754	−	0.195249i	$-0.937448\pi$
$293$	− 6.68425i	− 0.390498i	0.980754	−	0.195249i	$-0.0625515\pi$
$294$	0	0
$295$	−0.878002	−0.0511193
$296$	0	0
$297$	− 1.09783i	− 0.0637028i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	9.85086i	0.567794i
$302$	0	0
$303$	−17.1468	−0.985055
$304$	0	0
$305$	− 8.19806i	− 0.469420i
$306$	0	0
$307$	30.9269i	1.76509i	0.470226	+	0.882546i	$0.344173\pi$
$307$	30.9269i	1.76509i	−0.470226	+	0.882546i	$0.655827\pi$
$308$	0	0
$309$	14.9148	0.848475
$310$	0	0
$311$	19.9095	1.12896	0.564481	−	0.825446i	$-0.309076\pi$
$311$	19.9095	1.12896	0.564481	+	0.825446i	$0.309076\pi$
$312$	0	0
$313$	26.9681	1.52433	0.762163	−	0.647386i	$-0.224138\pi$
$313$	26.9681	1.52433	0.762163	+	0.647386i	$0.224138\pi$
$314$	0	0
$315$	1.80194	0.101528
$316$	0	0
$317$	− 33.6829i	− 1.89182i	−0.324427	−	0.945911i	$-0.605171\pi$
$317$	− 33.6829i	− 1.89182i	0.324427	−	0.945911i	$-0.394829\pi$
$318$	0	0
$319$	0.225209i	0.0126093i
$320$	0	0
$321$	−6.82371	−0.380862
$322$	0	0
$323$	− 6.58211i	− 0.366238i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	23.4577i	1.29721i
$328$	0	0
$329$	−11.0640	−0.609977
$330$	0	0
$331$	− 0.115293i	− 0.00633708i	−0.999995	−	0.00316854i	$-0.998991\pi$
$331$	− 0.115293i	− 0.00633708i	0.999995	−	0.00316854i	$-0.00100858\pi$
$332$	0	0
$333$	− 4.33513i	− 0.237563i
$334$	0	0
$335$	−0.207751	−0.0113506
$336$	0	0
$337$	−25.0006	−1.36187	−0.680934	−	0.732344i	$-0.738426\pi$
$337$	−25.0006	−1.36187	−0.680934	+	0.732344i	$0.738426\pi$
$338$	0	0
$339$	4.64981	0.252543
$340$	0	0
$341$	−1.35690	−0.0734800
$342$	0	0
$343$	15.5187i	0.837932i
$344$	0	0
$345$	1.80194i	0.0970131i
$346$	0	0
$347$	5.92931	0.318302	0.159151	−	0.987254i	$-0.449124\pi$
$347$	5.92931	0.318302	0.159151	+	0.987254i	$0.449124\pi$
$348$	0	0
$349$	28.0441i	1.50117i	0.660775	+	0.750584i	$0.270228\pi$
$349$	28.0441i	1.50117i	−0.660775	+	0.750584i	$0.729772\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 22.1957i	− 1.18136i	−0.806907	−	0.590678i	$-0.798860\pi$
$353$	− 22.1957i	− 1.18136i	0.806907	−	0.590678i	$-0.201140\pi$
$354$	0	0
$355$	−0.664874	−0.0352879
$356$	0	0
$357$	7.54288i	0.399211i
$358$	0	0
$359$	− 5.31037i	− 0.280271i	−0.990132	−	0.140135i	$-0.955246\pi$
$359$	− 5.31037i	− 0.280271i	0.990132	−	0.140135i	$-0.0447538\pi$
$360$	0	0
$361$	17.1588	0.903097
$362$	0	0
$363$	13.6679	0.717377
$364$	0	0
$365$	−1.72587	−0.0903363
$366$	0	0
$367$	−3.29696	−0.172100	−0.0860500	−	0.996291i	$-0.527424\pi$
$367$	−3.29696	−0.172100	−0.0860500	+	0.996291i	$0.527424\pi$
$368$	0	0
$369$	− 5.11960i	− 0.266516i
$370$	0	0
$371$	− 2.32975i	− 0.120955i
$372$	0	0
$373$	−8.70410	−0.450681	−0.225341	−	0.974280i	$-0.572350\pi$
$373$	−8.70410	−0.450681	−0.225341	+	0.974280i	$0.572350\pi$
$374$	0	0
$375$	− 1.24698i	− 0.0643937i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 15.9269i	− 0.818111i	−0.912510	−	0.409055i	$-0.865858\pi$
$379$	− 15.9269i	− 0.818111i	0.912510	−	0.409055i	$-0.134142\pi$
$380$	0	0
$381$	−9.56166	−0.489859
$382$	0	0
$383$	4.13169i	0.211119i	0.994413	+	0.105560i	$0.0336634\pi$
$383$	4.13169i	0.211119i	−0.994413	+	0.105560i	$0.966337\pi$
$384$	0	0
$385$	− 0.246980i	− 0.0125872i
$386$	0	0
$387$	−11.4155	−0.580283
$388$	0	0
$389$	15.1347	0.767358	0.383679	−	0.923466i	$-0.374657\pi$
$389$	15.1347	0.767358	0.383679	+	0.923466i	$0.374657\pi$
$390$	0	0
$391$	7.00969	0.354495
$392$	0	0
$393$	−11.6256	−0.586436
$394$	0	0
$395$	− 12.5700i	− 0.632467i
$396$	0	0
$397$	19.5080i	0.979076i	0.871982	+	0.489538i	$0.162835\pi$
$397$	19.5080i	0.979076i	−0.871982	+	0.489538i	$0.837165\pi$
$398$	0	0
$399$	2.10992	0.105628
$400$	0	0
$401$	− 27.8974i	− 1.39313i	−0.717494	−	0.696564i	$-0.754711\pi$
$401$	− 27.8974i	− 1.39313i	0.717494	−	0.696564i	$-0.245289\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 2.57673i	− 0.128039i
$406$	0	0
$407$	−0.594187	−0.0294527
$408$	0	0
$409$	6.46788i	0.319816i	0.987132	+	0.159908i	$0.0511197\pi$
$409$	6.46788i	0.319816i	−0.987132	+	0.159908i	$0.948880\pi$
$410$	0	0
$411$	5.89008i	0.290536i
$412$	0	0
$413$	−1.09485	−0.0538741
$414$	0	0
$415$	−12.3230	−0.604914
$416$	0	0
$417$	1.02044	0.0499713
$418$	0	0
$419$	20.9541	1.02367	0.511837	−	0.859083i	$-0.328965\pi$
$419$	20.9541	1.02367	0.511837	+	0.859083i	$0.328965\pi$
$420$	0	0
$421$	11.5211i	0.561504i	0.959780	+	0.280752i	$0.0905839\pi$
$421$	11.5211i	0.561504i	−0.959780	+	0.280752i	$0.909416\pi$
$422$	0	0
$423$	− 12.8213i	− 0.623394i
$424$	0	0
$425$	−4.85086	−0.235301
$426$	0	0
$427$	− 10.2228i	− 0.494717i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 6.33273i	− 0.305037i	−0.988301	−	0.152519i	$-0.951262\pi$
$431$	− 6.33273i	− 0.305037i	0.988301	−	0.152519i	$-0.0487384\pi$
$432$	0	0
$433$	11.5332	0.554250	0.277125	−	0.960834i	$-0.410618\pi$
$433$	11.5332	0.554250	0.277125	+	0.960834i	$0.410618\pi$
$434$	0	0
$435$	1.41789i	0.0679829i
$436$	0	0
$437$	− 1.96077i	− 0.0937964i
$438$	0	0
$439$	27.7036	1.32222	0.661111	−	0.750288i	$-0.270085\pi$
$439$	27.7036	1.32222	0.661111	+	0.750288i	$0.270085\pi$
$440$	0	0
$441$	−7.86831	−0.374682
$442$	0	0
$443$	−12.2814	−0.583508	−0.291754	−	0.956493i	$-0.594239\pi$
$443$	−12.2814	−0.583508	−0.291754	+	0.956493i	$0.594239\pi$
$444$	0	0
$445$	−15.8659	−0.752117
$446$	0	0
$447$	− 29.4577i	− 1.39330i
$448$	0	0
$449$	− 25.2556i	− 1.19189i	−0.803027	−	0.595943i	$-0.796778\pi$
$449$	− 25.2556i	− 1.19189i	0.803027	−	0.595943i	$-0.203222\pi$
$450$	0	0
$451$	−0.701710	−0.0330422
$452$	0	0
$453$	8.46011i	0.397491i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 31.1608i	− 1.45764i	−0.684706	−	0.728819i	$-0.740069\pi$
$457$	− 31.1608i	− 1.45764i	0.684706	−	0.728819i	$-0.259931\pi$
$458$	0	0
$459$	−26.8877	−1.25501
$460$	0	0
$461$	− 0.605203i	− 0.0281871i	−0.999901	−	0.0140936i	$-0.995514\pi$
$461$	− 0.605203i	− 0.0281871i	0.999901	−	0.0140936i	$-0.00448627\pi$
$462$	0	0
$463$	21.3575i	0.992567i	0.868161	+	0.496283i	$0.165302\pi$
$463$	21.3575i	0.992567i	−0.868161	+	0.496283i	$0.834698\pi$
$464$	0	0
$465$	−8.54288	−0.396166
$466$	0	0
$467$	−26.4886	−1.22575	−0.612873	−	0.790182i	$-0.709986\pi$
$467$	−26.4886	−1.22575	−0.612873	+	0.790182i	$0.709986\pi$
$468$	0	0
$469$	−0.259061	−0.0119623
$470$	0	0
$471$	−20.4849	−0.943893
$472$	0	0
$473$	1.56465i	0.0719425i
$474$	0	0
$475$	1.35690i	0.0622587i
$476$	0	0
$477$	2.69979	0.123615
$478$	0	0
$479$	5.08038i	0.232128i	0.993242	+	0.116064i	$0.0370278\pi$
$479$	5.08038i	0.232128i	−0.993242	+	0.116064i	$0.962972\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	2.24698i	0.102241i
$484$	0	0
$485$	1.31336	0.0596364
$486$	0	0
$487$	23.7646i	1.07688i	0.842664	+	0.538439i	$0.180986\pi$
$487$	23.7646i	1.07688i	−0.842664	+	0.538439i	$0.819014\pi$
$488$	0	0
$489$	− 15.8944i	− 0.718769i
$490$	0	0
$491$	12.9071	0.582488	0.291244	−	0.956649i	$-0.405931\pi$
$491$	12.9071	0.582488	0.291244	+	0.956649i	$0.405931\pi$
$492$	0	0
$493$	5.51573	0.248416
$494$	0	0
$495$	0.286208	0.0128641
$496$	0	0
$497$	−0.829085	−0.0371895
$498$	0	0
$499$	40.3749i	1.80743i	0.428134	+	0.903715i	$0.359171\pi$
$499$	40.3749i	1.80743i	−0.428134	+	0.903715i	$0.640829\pi$
$500$	0	0
$501$	− 18.9608i	− 0.847105i
$502$	0	0
$503$	−10.2634	−0.457621	−0.228811	−	0.973471i	$-0.573484\pi$
$503$	−10.2634	−0.457621	−0.228811	+	0.973471i	$0.573484\pi$
$504$	0	0
$505$	− 13.7506i	− 0.611895i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 2.40342i	− 0.106530i	−0.998580	−	0.0532649i	$-0.983037\pi$
$509$	− 2.40342i	− 0.106530i	0.998580	−	0.0532649i	$-0.0169628\pi$
$510$	0	0
$511$	−2.15213	−0.0952046
$512$	0	0
$513$	7.52111i	0.332065i
$514$	0	0
$515$	11.9608i	0.527055i
$516$	0	0
$517$	−1.75733	−0.0772874
$518$	0	0
$519$	17.1890	0.754512
$520$	0	0
$521$	23.1806	1.01556	0.507780	−	0.861487i	$-0.330466\pi$
$521$	23.1806	1.01556	0.507780	+	0.861487i	$0.330466\pi$
$522$	0	0
$523$	40.0073	1.74940	0.874698	−	0.484668i	$-0.161059\pi$
$523$	40.0073	1.74940	0.874698	+	0.484668i	$0.161059\pi$
$524$	0	0
$525$	− 1.55496i	− 0.0678639i
$526$	0	0
$527$	33.2325i	1.44763i
$528$	0	0
$529$	−20.9119	−0.909211
$530$	0	0
$531$	− 1.26875i	− 0.0550591i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 5.47219i	− 0.236583i
$536$	0	0
$537$	8.11828	0.350330
$538$	0	0
$539$	1.07846i	0.0464524i
$540$	0	0
$541$	3.13813i	0.134919i	0.997722	+	0.0674593i	$0.0214893\pi$
$541$	3.13813i	0.134919i	−0.997722	+	0.0674593i	$0.978511\pi$
$542$	0	0
$543$	5.55794	0.238514
$544$	0	0
$545$	−18.8116	−0.805802
$546$	0	0
$547$	10.9280	0.467247	0.233623	−	0.972327i	$-0.424942\pi$
$547$	10.9280	0.467247	0.233623	+	0.972327i	$0.424942\pi$
$548$	0	0
$549$	11.8465	0.505598
$550$	0	0
$551$	− 1.54288i	− 0.0657288i
$552$	0	0
$553$	− 15.6746i	− 0.666550i
$554$	0	0
$555$	−3.74094	−0.158794
$556$	0	0
$557$	6.19508i	0.262494i	0.991350	+	0.131247i	$0.0418981\pi$
$557$	6.19508i	0.262494i	−0.991350	+	0.131247i	$0.958102\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	1.19806i	0.0505822i
$562$	0	0
$563$	−33.0411	−1.39252	−0.696259	−	0.717790i	$-0.745154\pi$
$563$	−33.0411	−1.39252	−0.696259	+	0.717790i	$0.745154\pi$
$564$	0	0
$565$	3.72886i	0.156874i
$566$	0	0
$567$	− 3.21313i	− 0.134939i
$568$	0	0
$569$	14.4577	0.606099	0.303049	−	0.952975i	$-0.401995\pi$
$569$	14.4577	0.606099	0.303049	+	0.952975i	$0.401995\pi$
$570$	0	0
$571$	−34.6765	−1.45117	−0.725583	−	0.688135i	$-0.758430\pi$
$571$	−34.6765	−1.45117	−0.725583	+	0.688135i	$0.758430\pi$
$572$	0	0
$573$	28.8756	1.20630
$574$	0	0
$575$	−1.44504	−0.0602624
$576$	0	0
$577$	− 36.9842i	− 1.53967i	−0.638242	−	0.769836i	$-0.720338\pi$
$577$	− 36.9842i	− 1.53967i	0.638242	−	0.769836i	$-0.279662\pi$
$578$	0	0
$579$	0.862937i	0.0358624i
$580$	0	0
$581$	−15.3666	−0.637513
$582$	0	0
$583$	− 0.370042i	− 0.0153256i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 36.0043i	− 1.48606i	−0.669260	−	0.743028i	$-0.733389\pi$
$587$	− 36.0043i	− 1.48606i	0.669260	−	0.743028i	$-0.266611\pi$
$588$	0	0
$589$	9.29590	0.383031
$590$	0	0
$591$	8.71379i	0.358437i
$592$	0	0
$593$	− 43.4161i	− 1.78289i	−0.453134	−	0.891443i	$-0.649694\pi$
$593$	− 43.4161i	− 1.78289i	0.453134	−	0.891443i	$-0.350306\pi$
$594$	0	0
$595$	−6.04892	−0.247981
$596$	0	0
$597$	−22.6203	−0.925786
$598$	0	0
$599$	38.7549	1.58348	0.791742	−	0.610856i	$-0.209174\pi$
$599$	38.7549	1.58348	0.791742	+	0.610856i	$0.209174\pi$
$600$	0	0
$601$	−18.1293	−0.739509	−0.369755	−	0.929129i	$-0.620558\pi$
$601$	−18.1293	−0.739509	−0.369755	+	0.929129i	$0.620558\pi$
$602$	0	0
$603$	− 0.300209i	− 0.0122254i
$604$	0	0
$605$	10.9608i	0.445619i
$606$	0	0
$607$	−30.1094	−1.22210	−0.611052	−	0.791590i	$-0.709253\pi$
$607$	−30.1094	−1.22210	−0.611052	+	0.791590i	$0.709253\pi$
$608$	0	0
$609$	1.76809i	0.0716465i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 40.9197i	− 1.65273i	−0.563133	−	0.826366i	$-0.690404\pi$
$613$	− 40.9197i	− 1.65273i	0.563133	−	0.826366i	$-0.309596\pi$
$614$	0	0
$615$	−4.41789	−0.178147
$616$	0	0
$617$	28.7308i	1.15666i	0.815804	+	0.578329i	$0.196295\pi$
$617$	28.7308i	1.15666i	−0.815804	+	0.578329i	$0.803705\pi$
$618$	0	0
$619$	− 32.1118i	− 1.29068i	−0.763894	−	0.645342i	$-0.776715\pi$
$619$	− 32.1118i	− 1.29068i	0.763894	−	0.645342i	$-0.223285\pi$
$620$	0	0
$621$	−8.00969	−0.321418
$622$	0	0
$623$	−19.7845	−0.792648
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0.335126	0.0133836
$628$	0	0
$629$	14.5526i	0.580249i
$630$	0	0
$631$	34.2010i	1.36152i	0.732506	+	0.680761i	$0.238351\pi$
$631$	34.2010i	1.36152i	−0.732506	+	0.680761i	$0.761649\pi$
$632$	0	0
$633$	21.0858	0.838083
$634$	0	0
$635$	− 7.66786i	− 0.304290i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 0.960771i	− 0.0380075i
$640$	0	0
$641$	−4.39506	−0.173594	−0.0867972	−	0.996226i	$-0.527663\pi$
$641$	−4.39506	−0.173594	−0.0867972	+	0.996226i	$0.527663\pi$
$642$	0	0
$643$	36.3773i	1.43458i	0.696774	+	0.717291i	$0.254618\pi$
$643$	36.3773i	1.43458i	−0.696774	+	0.717291i	$0.745382\pi$
$644$	0	0
$645$	9.85086i	0.387877i
$646$	0	0
$647$	−18.1129	−0.712092	−0.356046	−	0.934469i	$-0.615875\pi$
$647$	−18.1129	−0.712092	−0.356046	+	0.934469i	$0.615875\pi$
$648$	0	0
$649$	−0.173899	−0.00682614
$650$	0	0
$651$	−10.6528	−0.417516
$652$	0	0
$653$	−16.6601	−0.651960	−0.325980	−	0.945377i	$-0.605694\pi$
$653$	−16.6601	−0.651960	−0.325980	+	0.945377i	$0.605694\pi$
$654$	0	0
$655$	− 9.32304i	− 0.364281i
$656$	0	0
$657$	− 2.49396i	− 0.0972986i
$658$	0	0
$659$	30.8471	1.20163	0.600817	−	0.799387i	$-0.294842\pi$
$659$	30.8471	1.20163	0.600817	+	0.799387i	$0.294842\pi$
$660$	0	0
$661$	− 22.4896i	− 0.874746i	−0.899280	−	0.437373i	$-0.855909\pi$
$661$	− 22.4896i	− 0.874746i	0.899280	−	0.437373i	$-0.144091\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.69202i	0.0656138i
$666$	0	0
$667$	1.64310	0.0636212
$668$	0	0
$669$	15.0301i	0.581098i
$670$	0	0
$671$	− 1.62373i	− 0.0626833i
$672$	0	0
$673$	48.7405	1.87881	0.939403	−	0.342814i	$-0.111380\pi$
$673$	48.7405	1.87881	0.939403	+	0.342814i	$0.111380\pi$
$674$	0	0
$675$	5.54288	0.213345
$676$	0	0
$677$	−2.81295	−0.108111	−0.0540553	−	0.998538i	$-0.517215\pi$
$677$	−2.81295	−0.108111	−0.0540553	+	0.998538i	$0.517215\pi$
$678$	0	0
$679$	1.63773	0.0628502
$680$	0	0
$681$	− 18.9095i	− 0.724612i
$682$	0	0
$683$	− 0.701710i	− 0.0268502i	−0.999910	−	0.0134251i	$-0.995727\pi$
$683$	− 0.701710i	− 0.0268502i	0.999910	−	0.0134251i	$-0.00427347\pi$
$684$	0	0
$685$	−4.72348	−0.180475
$686$	0	0
$687$	− 25.6098i	− 0.977076i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 35.1377i	− 1.33670i	−0.743847	−	0.668350i	$-0.767001\pi$
$691$	− 35.1377i	− 1.33670i	0.743847	−	0.668350i	$-0.232999\pi$
$692$	0	0
$693$	0.356896	0.0135574
$694$	0	0
$695$	0.818331i	0.0310411i
$696$	0	0
$697$	17.1860i	0.650965i
$698$	0	0
$699$	21.4728	0.812175
$700$	0	0
$701$	−8.23968	−0.311209	−0.155604	−	0.987819i	$-0.549732\pi$
$701$	−8.23968	−0.311209	−0.155604	+	0.987819i	$0.549732\pi$
$702$	0	0
$703$	4.07069	0.153529
$704$	0	0
$705$	−11.0640	−0.416694
$706$	0	0
$707$	− 17.1468i	− 0.644870i
$708$	0	0
$709$	− 35.6359i	− 1.33834i	−0.743111	−	0.669168i	$-0.766651\pi$
$709$	− 35.6359i	− 1.33834i	0.743111	−	0.669168i	$-0.233349\pi$
$710$	0	0
$711$	18.1642	0.681211
$712$	0	0
$713$	9.89977i	0.370749i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	19.2174i	0.717688i
$718$	0	0
$719$	−45.3497	−1.69126	−0.845630	−	0.533770i	$-0.820775\pi$
$719$	−45.3497	−1.69126	−0.845630	+	0.533770i	$0.820775\pi$
$720$	0	0
$721$	14.9148i	0.555458i
$722$	0	0
$723$	− 12.8304i	− 0.477168i
$724$	0	0
$725$	−1.13706	−0.0422295
$726$	0	0
$727$	−35.2664	−1.30796	−0.653978	−	0.756513i	$-0.726901\pi$
$727$	−35.2664	−1.30796	−0.653978	+	0.756513i	$0.726901\pi$
$728$	0	0
$729$	24.4590	0.905890
$730$	0	0
$731$	38.3207	1.41734
$732$	0	0
$733$	15.6722i	0.578865i	0.957198	+	0.289432i	$0.0934665\pi$
$733$	15.6722i	0.578865i	−0.957198	+	0.289432i	$0.906533\pi$
$734$	0	0
$735$	6.78986i	0.250448i
$736$	0	0
$737$	−0.0411476	−0.00151569
$738$	0	0
$739$	− 8.80433i	− 0.323873i	−0.986801	−	0.161936i	$-0.948226\pi$
$739$	− 8.80433i	− 0.323873i	0.986801	−	0.161936i	$-0.0517739\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	26.8974i	0.986769i	0.869811	+	0.493385i	$0.164240\pi$
$743$	26.8974i	0.986769i	−0.869811	+	0.493385i	$0.835760\pi$
$744$	0	0
$745$	23.6233	0.865489
$746$	0	0
$747$	− 17.8073i	− 0.651536i
$748$	0	0
$749$	− 6.82371i	− 0.249333i
$750$	0	0
$751$	37.0200	1.35088	0.675439	−	0.737416i	$-0.263954\pi$
$751$	37.0200	1.35088	0.675439	+	0.737416i	$0.263954\pi$
$752$	0	0
$753$	10.0911	0.367741
$754$	0	0
$755$	−6.78448	−0.246912
$756$	0	0
$757$	−20.3730	−0.740470	−0.370235	−	0.928938i	$-0.620723\pi$
$757$	−20.3730	−0.740470	−0.370235	+	0.928938i	$0.620723\pi$
$758$	0	0
$759$	0.356896i	0.0129545i
$760$	0	0
$761$	− 39.7362i	− 1.44043i	−0.693749	−	0.720217i	$-0.744042\pi$
$761$	− 39.7362i	− 1.44043i	0.693749	−	0.720217i	$-0.255958\pi$
$762$	0	0
$763$	−23.4577	−0.849226
$764$	0	0
$765$	− 7.00969i	− 0.253436i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	− 41.6171i	− 1.50075i	−0.661011	−	0.750376i	$-0.729872\pi$
$769$	− 41.6171i	− 1.50075i	0.661011	−	0.750376i	$-0.270128\pi$
$770$	0	0
$771$	−39.1836	−1.41116
$772$	0	0
$773$	− 38.2948i	− 1.37737i	−0.725061	−	0.688685i	$-0.758188\pi$
$773$	− 38.2948i	− 1.37737i	0.725061	−	0.688685i	$-0.241812\pi$
$774$	0	0
$775$	− 6.85086i	− 0.246090i
$776$	0	0
$777$	−4.66487	−0.167351
$778$	0	0
$779$	4.80731	0.172240
$780$	0	0
$781$	−0.131687	−0.00471211
$782$	0	0
$783$	−6.30260	−0.225237
$784$	0	0
$785$	− 16.4276i	− 0.586326i
$786$	0	0
$787$	52.0847i	1.85662i	0.371809	+	0.928309i	$0.378738\pi$
$787$	52.0847i	1.85662i	−0.371809	+	0.928309i	$0.621262\pi$
$788$	0	0
$789$	8.53079	0.303704
$790$	0	0
$791$	4.64981i	0.165328i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 2.32975i	− 0.0826277i
$796$	0	0
$797$	19.8278	0.702335	0.351168	−	0.936313i	$-0.385785\pi$
$797$	19.8278	0.702335	0.351168	+	0.936313i	$0.385785\pi$
$798$	0	0
$799$	43.0398i	1.52264i
$800$	0	0
$801$	− 22.9269i	− 0.810083i
$802$	0	0
$803$	−0.341830	−0.0120629
$804$	0	0
$805$	−1.80194	−0.0635100
$806$	0	0
$807$	0.917231	0.0322881
$808$	0	0
$809$	−2.59073	−0.0910852	−0.0455426	−	0.998962i	$-0.514502\pi$
$809$	−2.59073	−0.0910852	−0.0455426	+	0.998962i	$0.514502\pi$
$810$	0	0
$811$	0.946297i	0.0332290i	0.999862	+	0.0166145i	$0.00528880\pi$
$811$	0.946297i	0.0332290i	−0.999862	+	0.0166145i	$0.994711\pi$
$812$	0	0
$813$	1.24027i	0.0434983i
$814$	0	0
$815$	12.7463	0.446484
$816$	0	0
$817$	− 10.7192i	− 0.375016i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	5.83148i	0.203520i	0.994809	+	0.101760i	$0.0324474\pi$
$821$	5.83148i	0.203520i	−0.994809	+	0.101760i	$0.967553\pi$
$822$	0	0
$823$	−25.2083	−0.878708	−0.439354	−	0.898314i	$-0.644793\pi$
$823$	−25.2083	−0.878708	−0.439354	+	0.898314i	$0.644793\pi$
$824$	0	0
$825$	− 0.246980i	− 0.00859873i
$826$	0	0
$827$	27.0127i	0.939323i	0.882847	+	0.469661i	$0.155624\pi$
$827$	27.0127i	0.939323i	−0.882847	+	0.469661i	$0.844376\pi$
$828$	0	0
$829$	20.4064	0.708744	0.354372	−	0.935105i	$-0.384695\pi$
$829$	20.4064	0.708744	0.354372	+	0.935105i	$0.384695\pi$
$830$	0	0
$831$	34.0562	1.18140
$832$	0	0
$833$	26.4131	0.915160
$834$	0	0
$835$	15.2054	0.526203
$836$	0	0
$837$	− 37.9734i	− 1.31255i
$838$	0	0
$839$	− 39.1463i	− 1.35148i	−0.737140	−	0.675740i	$-0.763824\pi$
$839$	− 39.1463i	− 1.35148i	0.737140	−	0.675740i	$-0.236176\pi$
$840$	0	0
$841$	−27.7071	−0.955417
$842$	0	0
$843$	− 16.6256i	− 0.572618i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	13.6679i	0.469633i
$848$	0	0
$849$	−1.13169	−0.0388394
$850$	0	0
$851$	4.33513i	0.148606i
$852$	0	0
$853$	4.51009i	0.154423i	0.997015	+	0.0772113i	$0.0246016\pi$
$853$	4.51009i	0.154423i	−0.997015	+	0.0772113i	$0.975398\pi$
$854$	0	0
$855$	−1.96077	−0.0670570
$856$	0	0
$857$	−19.3948	−0.662514	−0.331257	−	0.943541i	$-0.607473\pi$
$857$	−19.3948	−0.662514	−0.331257	+	0.943541i	$0.607473\pi$
$858$	0	0
$859$	39.7429	1.35601	0.678004	−	0.735058i	$-0.262845\pi$
$859$	39.7429	1.35601	0.678004	+	0.735058i	$0.262845\pi$
$860$	0	0
$861$	−5.50902	−0.187747
$862$	0	0
$863$	− 26.7827i	− 0.911693i	−0.890058	−	0.455846i	$-0.849337\pi$
$863$	− 26.7827i	− 0.911693i	0.890058	−	0.455846i	$-0.150663\pi$
$864$	0	0
$865$	13.7845i	0.468686i
$866$	0	0
$867$	8.14377	0.276577
$868$	0	0
$869$	− 2.48965i	− 0.0844555i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	1.89785i	0.0642326i
$874$	0	0
$875$	1.24698	0.0421556
$876$	0	0
$877$	− 12.2185i	− 0.412590i	−0.978490	−	0.206295i	$-0.933859\pi$
$877$	− 12.2185i	− 0.412590i	0.978490	−	0.206295i	$-0.0661406\pi$
$878$	0	0
$879$	− 8.33513i	− 0.281137i
$880$	0	0
$881$	1.72827	0.0582268	0.0291134	−	0.999576i	$-0.490732\pi$
$881$	1.72827	0.0582268	0.0291134	+	0.999576i	$0.490732\pi$
$882$	0	0
$883$	−38.6340	−1.30014	−0.650069	−	0.759875i	$-0.725260\pi$
$883$	−38.6340	−1.30014	−0.650069	+	0.759875i	$0.725260\pi$
$884$	0	0
$885$	−1.09485	−0.0368030
$886$	0	0
$887$	−36.6055	−1.22909	−0.614547	−	0.788880i	$-0.710661\pi$
$887$	−36.6055	−1.22909	−0.614547	+	0.788880i	$0.710661\pi$
$888$	0	0
$889$	− 9.56166i	− 0.320688i
$890$	0	0
$891$	− 0.510353i	− 0.0170975i
$892$	0	0
$893$	12.0392	0.402877
$894$	0	0
$895$	6.51035i	0.217617i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	7.78986i	0.259806i
$900$	0	0
$901$	−9.06292	−0.301930
$902$	0	0
$903$	12.2838i	0.408780i
$904$	0	0
$905$	4.45712i	0.148160i
$906$	0	0
$907$	−49.5881	−1.64654	−0.823272	−	0.567646i	$-0.807854\pi$
$907$	−49.5881	−1.64654	−0.823272	+	0.567646i	$0.807854\pi$
$908$	0	0
$909$	19.8702	0.659054
$910$	0	0
$911$	32.7982	1.08665	0.543327	−	0.839521i	$-0.317164\pi$
$911$	32.7982	1.08665	0.543327	+	0.839521i	$0.317164\pi$
$912$	0	0
$913$	−2.44073	−0.0807764
$914$	0	0
$915$	− 10.2228i	− 0.337956i
$916$	0	0
$917$	− 11.6256i	− 0.383913i
$918$	0	0
$919$	−14.6944	−0.484724	−0.242362	−	0.970186i	$-0.577922\pi$
$919$	−14.6944	−0.484724	−0.242362	+	0.970186i	$0.577922\pi$
$920$	0	0
$921$	38.5652i	1.27077i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 3.00000i	− 0.0986394i
$926$	0	0
$927$	−17.2838	−0.567675
$928$	0	0
$929$	2.53127i	0.0830482i	0.999138	+	0.0415241i	$0.0132213\pi$
$929$	2.53127i	0.0830482i	−0.999138	+	0.0415241i	$0.986779\pi$
$930$	0	0
$931$	− 7.38835i	− 0.242144i
$932$	0	0
$933$	24.8267	0.812789
$934$	0	0
$935$	−0.960771	−0.0314206
$936$	0	0
$937$	−14.3951	−0.470266	−0.235133	−	0.971963i	$-0.575553\pi$
$937$	−14.3951	−0.470266	−0.235133	+	0.971963i	$0.575553\pi$
$938$	0	0
$939$	33.6286	1.09743
$940$	0	0
$941$	− 18.7885i	− 0.612489i	−0.951953	−	0.306244i	$-0.900928\pi$
$941$	− 18.7885i	− 0.612489i	0.951953	−	0.306244i	$-0.0990724\pi$
$942$	0	0
$943$	5.11960i	0.166717i
$944$	0	0
$945$	6.91185	0.224843
$946$	0	0
$947$	− 50.4543i	− 1.63954i	−0.572691	−	0.819772i	$-0.694100\pi$
$947$	− 50.4543i	− 1.63954i	0.572691	−	0.819772i	$-0.305900\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	− 42.0019i	− 1.36201i
$952$	0	0
$953$	−56.6708	−1.83575	−0.917874	−	0.396871i	$-0.870096\pi$
$953$	−56.6708	−1.83575	−0.917874	+	0.396871i	$0.870096\pi$
$954$	0	0
$955$	23.1564i	0.749325i
$956$	0	0
$957$	0.280831i	0.00907799i
$958$	0	0
$959$	−5.89008	−0.190201
$960$	0	0
$961$	−15.9342	−0.514007
$962$	0	0
$963$	7.90754	0.254817
$964$	0	0
$965$	−0.692021	−0.0222770
$966$	0	0
$967$	− 13.4421i	− 0.432267i	−0.976364	−	0.216134i	$-0.930655\pi$
$967$	− 13.4421i	− 0.432267i	0.976364	−	0.216134i	$-0.0693447\pi$
$968$	0	0
$969$	− 8.20775i	− 0.263671i
$970$	0	0
$971$	41.6963	1.33810	0.669050	−	0.743218i	$-0.266701\pi$
$971$	41.6963	1.33810	0.669050	+	0.743218i	$0.266701\pi$
$972$	0	0
$973$	1.02044i	0.0327139i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 15.2228i	− 0.487021i	−0.969898	−	0.243511i	$-0.921701\pi$
$977$	− 15.2228i	− 0.487021i	0.969898	−	0.243511i	$-0.0782990\pi$
$978$	0	0
$979$	−3.14244	−0.100433
$980$	0	0
$981$	− 27.1836i	− 0.867905i
$982$	0	0
$983$	− 42.7888i	− 1.36475i	−0.731002	−	0.682375i	$-0.760947\pi$
$983$	− 42.7888i	− 1.36475i	0.731002	−	0.682375i	$-0.239053\pi$
$984$	0	0
$985$	−6.98792	−0.222654
$986$	0	0
$987$	−13.7966	−0.439149
$988$	0	0
$989$	11.4155	0.362992
$990$	0	0
$991$	18.2857	0.580865	0.290433	−	0.956895i	$-0.406201\pi$
$991$	18.2857	0.580865	0.290433	+	0.956895i	$0.406201\pi$
$992$	0	0
$993$	− 0.143768i	− 0.00456234i
$994$	0	0
$995$	− 18.1400i	− 0.575078i
$996$	0	0
$997$	−29.7415	−0.941924	−0.470962	−	0.882154i	$-0.656093\pi$
$997$	−29.7415	−0.941924	−0.470962	+	0.882154i	$0.656093\pi$
$998$	0	0
$999$	− 16.6286i	− 0.526107i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3380.2.f.g.3041.6		6
13.5	odd	4		3380.2.a.l.1.3	yes	3
13.8	odd	4		3380.2.a.k.1.3	✓	3
13.12	even	2	inner	3380.2.f.g.3041.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3380.2.a.k.1.3	✓	3	13.8	odd	4
3380.2.a.l.1.3	yes	3	13.5	odd	4
3380.2.f.g.3041.5		6	13.12	even	2	inner
3380.2.f.g.3041.6		6	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3380.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.24698 q^{3} +1.00000i q^{5} +1.24698i q^{7} -1.44504 q^{9} +O(q^{10})\)	\(q+1.24698 q^{3} +1.00000i q^{5} +1.24698i q^{7} -1.44504 q^{9} +0.198062i q^{11} +1.24698i q^{15} +4.85086 q^{17} -1.35690i q^{19} +1.55496i q^{21} +1.44504 q^{23} -1.00000 q^{25} -5.54288 q^{27} +1.13706 q^{29} +6.85086i q^{31} +0.246980i q^{33} -1.24698 q^{35} +3.00000i q^{37} +3.54288i q^{41} +7.89977 q^{43} -1.44504i q^{45} +8.87263i q^{47} +5.44504 q^{49} +6.04892 q^{51} -1.86831 q^{53} -0.198062 q^{55} -1.69202i q^{57} +0.878002i q^{59} -8.19806 q^{61} -1.80194i q^{63} +0.207751i q^{67} +1.80194 q^{69} +0.664874i q^{71} +1.72587i q^{73} -1.24698 q^{75} -0.246980 q^{77} -12.5700 q^{79} -2.57673 q^{81} +12.3230i q^{83} +4.85086i q^{85} +1.41789 q^{87} +15.8659i q^{89} +8.54288i q^{93} +1.35690 q^{95} -1.31336i q^{97} -0.286208i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} - 8 q^{9}+O(q^{10}) \) 6 * q - 2 * q^3 - 8 * q^9	\( 6 q - 2 q^{3} - 8 q^{9} + 2 q^{17} + 8 q^{23} - 6 q^{25} + 4 q^{27} - 4 q^{29} + 2 q^{35} + 2 q^{43} + 32 q^{49} + 18 q^{51} - 16 q^{53} - 10 q^{55} - 58 q^{61} + 2 q^{69} + 2 q^{75} + 8 q^{77} - 26 q^{79} - 10 q^{81} + 20 q^{87}+O(q^{100}) \) 6 * q - 2 * q^3 - 8 * q^9 + 2 * q^17 + 8 * q^23 - 6 * q^25 + 4 * q^27 - 4 * q^29 + 2 * q^35 + 2 * q^43 + 32 * q^49 + 18 * q^51 - 16 * q^53 - 10 * q^55 - 58 * q^61 + 2 * q^69 + 2 * q^75 + 8 * q^77 - 26 * q^79 - 10 * q^81 + 20 * q^87

Introduction

L-functions

Modular forms

Varieties

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Database

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