LMFDB - Embedded newform 3100.2.c.h.249.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3100,2,Mod(249,3100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3100, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3100.249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3100 = 2^{2} \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3100.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:	$24.7536246266$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 20x^{8} + 136x^{6} + 381x^{4} + 378x^{2} + 9 $ x^10 + 20x^8 + 136x^6 + 381x^4 + 378x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			249.7
Root			$2.18729i$ of defining polynomial
Character	$\chi$	$=$	3100.249
Dual form			3100.2.c.h.249.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.18729i q^{3} +2.76870i q^{7} +1.59033 q^{9} +O(q^{10})$	$q+1.18729i q^{3} +2.76870i q^{7} +1.59033 q^{9} -1.40304 q^{11} +3.57478i q^{13} -1.66582i q^{17} +2.85311 q^{19} -3.28726 q^{21} +3.04041i q^{23} +5.45007i q^{27} -8.24060 q^{29} -1.00000 q^{31} -1.66582i q^{33} +8.45900i q^{37} -4.24431 q^{39} +6.20031 q^{41} -6.55631i q^{43} +4.20285i q^{47} -0.665703 q^{49} +1.97782 q^{51} -9.42789i q^{53} +3.38748i q^{57} +12.9558 q^{59} -13.0338 q^{61} +4.40316i q^{63} +9.44345i q^{67} -3.60986 q^{69} -3.82863 q^{71} +7.81909i q^{73} -3.88460i q^{77} -0.515903 q^{79} -1.69984 q^{81} -5.13133i q^{83} -9.78401i q^{87} -5.03148 q^{89} -9.89749 q^{91} -1.18729i q^{93} +5.28089i q^{97} -2.23130 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 12 q^{9}+O(q^{10}) $ 10 * q - 12 * q^9	$ 10 q - 12 q^{9} - 4 q^{11} - 12 q^{19} - 6 q^{21} - 20 q^{29} - 10 q^{31} - 6 q^{39} - 12 q^{41} - 54 q^{49} - 30 q^{51} - 40 q^{59} - 34 q^{61} - 60 q^{69} - 12 q^{71} - 30 q^{81} - 18 q^{89} - 40 q^{91} - 42 q^{99}+O(q^{100}) $ 10 * q - 12 * q^9 - 4 * q^11 - 12 * q^19 - 6 * q^21 - 20 * q^29 - 10 * q^31 - 6 * q^39 - 12 * q^41 - 54 * q^49 - 30 * q^51 - 40 * q^59 - 34 * q^61 - 60 * q^69 - 12 * q^71 - 30 * q^81 - 18 * q^89 - 40 * q^91 - 42 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3100\mathbb{Z}\right)^\times$.

$n$	$1551$	$1801$	$2977$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	1.18729i	0.685484i	0.939429	+	0.342742i	$0.111356\pi$
$3$	1.18729i	0.685484i	−0.939429	+	0.342742i	$0.888644\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.76870i	1.04647i	0.852188	+	0.523235i	$0.175275\pi$
$7$	2.76870i	1.04647i	−0.852188	+	0.523235i	$0.824725\pi$
$8$	0	0
$9$	1.59033	0.530111
$10$	0	0
$11$	−1.40304	−0.423032	−0.211516	−	0.977375i	$-0.567840\pi$
$11$	−1.40304	−0.423032	−0.211516	+	0.977375i	$0.567840\pi$
$12$	0	0
$13$	3.57478i	0.991465i	0.868475	+	0.495733i	$0.165100\pi$
$13$	3.57478i	0.991465i	−0.868475	+	0.495733i	$0.834900\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 1.66582i	− 0.404021i	−0.979383	−	0.202010i	$-0.935253\pi$
$17$	− 1.66582i	− 0.404021i	0.979383	−	0.202010i	$-0.0647475\pi$
$18$	0	0
$19$	2.85311	0.654549	0.327275	−	0.944929i	$-0.393870\pi$
$19$	2.85311	0.654549	0.327275	+	0.944929i	$0.393870\pi$
$20$	0	0
$21$	−3.28726	−0.717339
$22$	0	0
$23$	3.04041i	0.633969i	0.948431	+	0.316984i	$0.102670\pi$
$23$	3.04041i	0.633969i	−0.948431	+	0.316984i	$0.897330\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.45007i	1.04887i
$28$	0	0
$29$	−8.24060	−1.53024	−0.765120	−	0.643887i	$-0.777321\pi$
$29$	−8.24060	−1.53024	−0.765120	+	0.643887i	$0.777321\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	− 1.66582i	− 0.289982i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.45900i	1.39065i	0.718695	+	0.695326i	$0.244740\pi$
$37$	8.45900i	1.39065i	−0.718695	+	0.695326i	$0.755260\pi$
$38$	0	0
$39$	−4.24431	−0.679634
$40$	0	0
$41$	6.20031	0.968325	0.484163	−	0.874978i	$-0.339124\pi$
$41$	6.20031	0.968325	0.484163	+	0.874978i	$0.339124\pi$
$42$	0	0
$43$	− 6.55631i	− 0.999828i	−0.866075	−	0.499914i	$-0.833365\pi$
$43$	− 6.55631i	− 0.999828i	0.866075	−	0.499914i	$-0.166635\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.20285i	0.613048i	0.951863	+	0.306524i	$0.0991660\pi$
$47$	4.20285i	0.613048i	−0.951863	+	0.306524i	$0.900834\pi$
$48$	0	0
$49$	−0.665703	−0.0951004
$50$	0	0
$51$	1.97782	0.276950
$52$	0	0
$53$	− 9.42789i	− 1.29502i	−0.762057	−	0.647510i	$-0.775810\pi$
$53$	− 9.42789i	− 1.29502i	0.762057	−	0.647510i	$-0.224190\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.38748i	0.448683i
$58$	0	0
$59$	12.9558	1.68670	0.843348	−	0.537368i	$-0.180581\pi$
$59$	12.9558	1.68670	0.843348	+	0.537368i	$0.180581\pi$
$60$	0	0
$61$	−13.0338	−1.66880	−0.834402	−	0.551156i	$-0.814187\pi$
$61$	−13.0338	−1.66880	−0.834402	+	0.551156i	$0.814187\pi$
$62$	0	0
$63$	4.40316i	0.554746i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	9.44345i	1.15370i	0.816850	+	0.576850i	$0.195718\pi$
$67$	9.44345i	1.15370i	−0.816850	+	0.576850i	$0.804282\pi$
$68$	0	0
$69$	−3.60986	−0.434576
$70$	0	0
$71$	−3.82863	−0.454375	−0.227188	−	0.973851i	$-0.572953\pi$
$71$	−3.82863	−0.454375	−0.227188	+	0.973851i	$0.572953\pi$
$72$	0	0
$73$	7.81909i	0.915155i	0.889170	+	0.457578i	$0.151283\pi$
$73$	7.81909i	0.915155i	−0.889170	+	0.457578i	$0.848717\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 3.88460i	− 0.442691i
$78$	0	0
$79$	−0.515903	−0.0580437	−0.0290218	−	0.999579i	$-0.509239\pi$
$79$	−0.515903	−0.0580437	−0.0290218	+	0.999579i	$0.509239\pi$
$80$	0	0
$81$	−1.69984	−0.188871
$82$	0	0
$83$	− 5.13133i	− 0.563237i	−0.959526	−	0.281618i	$-0.909129\pi$
$83$	− 5.13133i	− 0.563237i	0.959526	−	0.281618i	$-0.0908712\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 9.78401i	− 1.04896i
$88$	0	0
$89$	−5.03148	−0.533336	−0.266668	−	0.963788i	$-0.585923\pi$
$89$	−5.03148	−0.533336	−0.266668	+	0.963788i	$0.585923\pi$
$90$	0	0
$91$	−9.89749	−1.03754
$92$	0	0
$93$	− 1.18729i	− 0.123117i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	5.28089i	0.536193i	0.963392	+	0.268097i	$0.0863946\pi$
$97$	5.28089i	0.536193i	−0.963392	+	0.268097i	$0.913605\pi$
$98$	0	0
$99$	−2.23130	−0.224254
$100$	0	0
$101$	−14.5749	−1.45026	−0.725128	−	0.688614i	$-0.758220\pi$
$101$	−14.5749	−1.45026	−0.725128	+	0.688614i	$0.758220\pi$
$102$	0	0
$103$	− 9.86613i	− 0.972138i	−0.873920	−	0.486069i	$-0.838430\pi$
$103$	− 9.86613i	− 0.972138i	0.873920	−	0.486069i	$-0.161570\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.8776i	1.05158i	0.850615	+	0.525788i	$0.176230\pi$
$107$	10.8776i	1.05158i	−0.850615	+	0.525788i	$0.823770\pi$
$108$	0	0
$109$	−7.89315	−0.756027	−0.378013	−	0.925800i	$-0.623393\pi$
$109$	−7.89315	−0.756027	−0.378013	+	0.925800i	$0.623393\pi$
$110$	0	0
$111$	−10.0433	−0.953270
$112$	0	0
$113$	− 16.0586i	− 1.51067i	−0.655339	−	0.755335i	$-0.727474\pi$
$113$	− 16.0586i	− 1.51067i	0.655339	−	0.755335i	$-0.272526\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	5.68509i	0.525587i
$118$	0	0
$119$	4.61216	0.422796
$120$	0	0
$121$	−9.03148	−0.821044
$122$	0	0
$123$	7.36159i	0.663772i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	11.6903i	1.03735i	0.854973	+	0.518673i	$0.173574\pi$
$127$	11.6903i	1.03735i	−0.854973	+	0.518673i	$0.826426\pi$
$128$	0	0
$129$	7.78427	0.685367
$130$	0	0
$131$	18.5779	1.62316	0.811580	−	0.584241i	$-0.198608\pi$
$131$	18.5779	1.62316	0.811580	+	0.584241i	$0.198608\pi$
$132$	0	0
$133$	7.89942i	0.684966i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.70994i	0.402398i	0.979550	+	0.201199i	$0.0644837\pi$
$137$	4.70994i	0.402398i	−0.979550	+	0.201199i	$0.935516\pi$
$138$	0	0
$139$	−10.8114	−0.917012	−0.458506	−	0.888691i	$-0.651615\pi$
$139$	−10.8114	−0.917012	−0.458506	+	0.888691i	$0.651615\pi$
$140$	0	0
$141$	−4.99002	−0.420235
$142$	0	0
$143$	− 5.01555i	− 0.419422i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 0.790385i	− 0.0651899i
$148$	0	0
$149$	−11.8898	−0.974051	−0.487025	−	0.873388i	$-0.661918\pi$
$149$	−11.8898	−0.974051	−0.487025	+	0.873388i	$0.661918\pi$
$150$	0	0
$151$	12.9431	1.05330	0.526648	−	0.850084i	$-0.323449\pi$
$151$	12.9431	1.05330	0.526648	+	0.850084i	$0.323449\pi$
$152$	0	0
$153$	− 2.64921i	− 0.214176i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 10.6615i	− 0.850880i	−0.904987	−	0.425440i	$-0.860119\pi$
$157$	− 10.6615i	− 0.850880i	0.904987	−	0.425440i	$-0.139881\pi$
$158$	0	0
$159$	11.1937	0.887717
$160$	0	0
$161$	−8.41798	−0.663430
$162$	0	0
$163$	2.57887i	0.201992i	0.994887	+	0.100996i	$0.0322030\pi$
$163$	2.57887i	0.201992i	−0.994887	+	0.100996i	$0.967797\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.80874i	0.449493i	0.974417	+	0.224747i	$0.0721555\pi$
$167$	5.80874i	0.449493i	−0.974417	+	0.224747i	$0.927845\pi$
$168$	0	0
$169$	0.220957	0.0169967
$170$	0	0
$171$	4.53740	0.346984
$172$	0	0
$173$	− 3.39411i	− 0.258050i	−0.991641	−	0.129025i	$-0.958815\pi$
$173$	− 3.39411i	− 0.258050i	0.991641	−	0.129025i	$-0.0411847\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	15.3823i	1.15620i
$178$	0	0
$179$	−23.2108	−1.73486	−0.867430	−	0.497559i	$-0.834230\pi$
$179$	−23.2108	−1.73486	−0.867430	+	0.497559i	$0.834230\pi$
$180$	0	0
$181$	0.541116	0.0402208	0.0201104	−	0.999798i	$-0.493598\pi$
$181$	0.541116	0.0402208	0.0201104	+	0.999798i	$0.493598\pi$
$182$	0	0
$183$	− 15.4749i	− 1.14394i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2.33721i	0.170914i
$188$	0	0
$189$	−15.0896	−1.09761
$190$	0	0
$191$	−3.40341	−0.246262	−0.123131	−	0.992390i	$-0.539294\pi$
$191$	−3.40341	−0.246262	−0.123131	+	0.992390i	$0.539294\pi$
$192$	0	0
$193$	26.6223i	1.91632i	0.286241	+	0.958158i	$0.407594\pi$
$193$	26.6223i	1.91632i	−0.286241	+	0.958158i	$0.592406\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 6.26148i	− 0.446112i	−0.974806	−	0.223056i	$-0.928397\pi$
$197$	− 6.26148i	− 0.446112i	0.974806	−	0.223056i	$-0.0716033\pi$
$198$	0	0
$199$	−3.44599	−0.244280	−0.122140	−	0.992513i	$-0.538976\pi$
$199$	−3.44599	−0.244280	−0.122140	+	0.992513i	$0.538976\pi$
$200$	0	0
$201$	−11.2121	−0.790844
$202$	0	0
$203$	− 22.8158i	− 1.60135i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	4.83526i	0.336074i
$208$	0	0
$209$	−4.00303	−0.276895
$210$	0	0
$211$	15.8928	1.09411	0.547054	−	0.837097i	$-0.315749\pi$
$211$	15.8928	1.09411	0.547054	+	0.837097i	$0.315749\pi$
$212$	0	0
$213$	− 4.54571i	− 0.311467i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 2.76870i	− 0.187952i
$218$	0	0
$219$	−9.28356	−0.627325
$220$	0	0
$221$	5.95494	0.400572
$222$	0	0
$223$	9.52718i	0.637987i	0.947757	+	0.318993i	$0.103345\pi$
$223$	9.52718i	0.637987i	−0.947757	+	0.318993i	$0.896655\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	11.1351i	0.739060i	0.929219	+	0.369530i	$0.120481\pi$
$227$	11.1351i	0.739060i	−0.929219	+	0.369530i	$0.879519\pi$
$228$	0	0
$229$	−12.6866	−0.838353	−0.419177	−	0.907905i	$-0.637681\pi$
$229$	−12.6866	−0.838353	−0.419177	+	0.907905i	$0.637681\pi$
$230$	0	0
$231$	4.61216	0.303458
$232$	0	0
$233$	4.10252i	0.268765i	0.990930	+	0.134383i	$0.0429051\pi$
$233$	4.10252i	0.268765i	−0.990930	+	0.134383i	$0.957095\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 0.612529i	− 0.0397880i
$238$	0	0
$239$	−0.261724	−0.0169295	−0.00846474	−	0.999964i	$-0.502694\pi$
$239$	−0.261724	−0.0169295	−0.00846474	+	0.999964i	$0.502694\pi$
$240$	0	0
$241$	2.16220	0.139279	0.0696397	−	0.997572i	$-0.477815\pi$
$241$	2.16220	0.139279	0.0696397	+	0.997572i	$0.477815\pi$
$242$	0	0
$243$	14.3320i	0.919399i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	10.1993i	0.648963i
$248$	0	0
$249$	6.09240	0.386090
$250$	0	0
$251$	−3.96295	−0.250139	−0.125070	−	0.992148i	$-0.539915\pi$
$251$	−3.96295	−0.250139	−0.125070	+	0.992148i	$0.539915\pi$
$252$	0	0
$253$	− 4.26581i	− 0.268189i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 15.4965i	− 0.966645i	−0.875442	−	0.483322i	$-0.839430\pi$
$257$	− 15.4965i	− 0.966645i	0.875442	−	0.483322i	$-0.160570\pi$
$258$	0	0
$259$	−23.4204	−1.45528
$260$	0	0
$261$	−13.1053	−0.811197
$262$	0	0
$263$	6.74590i	0.415970i	0.978132	+	0.207985i	$0.0666906\pi$
$263$	6.74590i	0.415970i	−0.978132	+	0.207985i	$0.933309\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 5.97385i	− 0.365594i
$268$	0	0
$269$	−11.1614	−0.680522	−0.340261	−	0.940331i	$-0.610515\pi$
$269$	−11.1614	−0.680522	−0.340261	+	0.940331i	$0.610515\pi$
$270$	0	0
$271$	−5.01917	−0.304893	−0.152446	−	0.988312i	$-0.548715\pi$
$271$	−5.01917	−0.304893	−0.152446	+	0.988312i	$0.548715\pi$
$272$	0	0
$273$	− 11.7512i	− 0.711217i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 23.6117i	− 1.41869i	−0.704862	−	0.709345i	$-0.748991\pi$
$277$	− 23.6117i	− 1.41869i	0.704862	−	0.709345i	$-0.251009\pi$
$278$	0	0
$279$	−1.59033	−0.0952107
$280$	0	0
$281$	4.49662	0.268246	0.134123	−	0.990965i	$-0.457178\pi$
$281$	4.49662	0.268246	0.134123	+	0.990965i	$0.457178\pi$
$282$	0	0
$283$	− 7.83986i	− 0.466031i	−0.972473	−	0.233016i	$-0.925141\pi$
$283$	− 7.83986i	− 0.466031i	0.972473	−	0.233016i	$-0.0748593\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	17.1668i	1.01332i
$288$	0	0
$289$	14.2250	0.836767
$290$	0	0
$291$	−6.26997	−0.367552
$292$	0	0
$293$	− 12.4989i	− 0.730195i	−0.930969	−	0.365098i	$-0.881036\pi$
$293$	− 12.4989i	− 0.730195i	0.930969	−	0.365098i	$-0.118964\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 7.64667i	− 0.443705i
$298$	0	0
$299$	−10.8688	−0.628558
$300$	0	0
$301$	18.1525	1.04629
$302$	0	0
$303$	− 17.3047i	− 0.994128i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	19.4794i	1.11175i	0.831266	+	0.555874i	$0.187616\pi$
$307$	19.4794i	1.11175i	−0.831266	+	0.555874i	$0.812384\pi$
$308$	0	0
$309$	11.7140	0.666386
$310$	0	0
$311$	14.9501	0.847740	0.423870	−	0.905723i	$-0.360671\pi$
$311$	14.9501	0.847740	0.423870	+	0.905723i	$0.360671\pi$
$312$	0	0
$313$	− 19.5462i	− 1.10482i	−0.833574	−	0.552408i	$-0.813709\pi$
$313$	− 19.5462i	− 1.10482i	0.833574	−	0.552408i	$-0.186291\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	30.7211i	1.72547i	0.505657	+	0.862735i	$0.331250\pi$
$317$	30.7211i	1.72547i	−0.505657	+	0.862735i	$0.668750\pi$
$318$	0	0
$319$	11.5619	0.647341
$320$	0	0
$321$	−12.9149	−0.720840
$322$	0	0
$323$	− 4.75277i	− 0.264451i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 9.37149i	− 0.518244i
$328$	0	0
$329$	−11.6364	−0.641537
$330$	0	0
$331$	12.2843	0.675205	0.337603	−	0.941289i	$-0.390384\pi$
$331$	12.2843	0.675205	0.337603	+	0.941289i	$0.390384\pi$
$332$	0	0
$333$	13.4526i	0.737200i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 13.3971i	− 0.729787i	−0.931049	−	0.364894i	$-0.881105\pi$
$337$	− 13.3971i	− 0.729787i	0.931049	−	0.364894i	$-0.118895\pi$
$338$	0	0
$339$	19.0663	1.03554
$340$	0	0
$341$	1.40304	0.0759788
$342$	0	0
$343$	17.5378i	0.946951i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	22.5649i	1.21135i	0.795713	+	0.605674i	$0.207096\pi$
$347$	22.5649i	1.21135i	−0.795713	+	0.605674i	$0.792904\pi$
$348$	0	0
$349$	24.7586	1.32530	0.662649	−	0.748930i	$-0.269432\pi$
$349$	24.7586	1.32530	0.662649	+	0.748930i	$0.269432\pi$
$350$	0	0
$351$	−19.4828	−1.03992
$352$	0	0
$353$	10.1173i	0.538491i	0.963072	+	0.269245i	$0.0867743\pi$
$353$	10.1173i	0.538491i	−0.963072	+	0.269245i	$0.913226\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	5.47599i	0.289820i
$358$	0	0
$359$	7.78623	0.410941	0.205471	−	0.978663i	$-0.434127\pi$
$359$	7.78623	0.410941	0.205471	+	0.978663i	$0.434127\pi$
$360$	0	0
$361$	−10.8597	−0.571565
$362$	0	0
$363$	− 10.7230i	− 0.562813i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 10.5784i	− 0.552187i	−0.961131	−	0.276093i	$-0.910960\pi$
$367$	− 10.5784i	− 0.552187i	0.961131	−	0.276093i	$-0.0890399\pi$
$368$	0	0
$369$	9.86055	0.513320
$370$	0	0
$371$	26.1030	1.35520
$372$	0	0
$373$	0.602415i	0.0311918i	0.999878	+	0.0155959i	$0.00496454\pi$
$373$	0.602415i	0.0311918i	−0.999878	+	0.0155959i	$0.995035\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 29.4583i	− 1.51718i
$378$	0	0
$379$	−4.30380	−0.221072	−0.110536	−	0.993872i	$-0.535257\pi$
$379$	−4.30380	−0.221072	−0.110536	+	0.993872i	$0.535257\pi$
$380$	0	0
$381$	−13.8798	−0.711085
$382$	0	0
$383$	− 22.8069i	− 1.16538i	−0.812695	−	0.582690i	$-0.802000\pi$
$383$	− 22.8069i	− 1.16538i	0.812695	−	0.582690i	$-0.198000\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 10.4267i	− 0.530020i
$388$	0	0
$389$	22.3464	1.13301	0.566503	−	0.824060i	$-0.308296\pi$
$389$	22.3464	1.13301	0.566503	+	0.824060i	$0.308296\pi$
$390$	0	0
$391$	5.06477	0.256137
$392$	0	0
$393$	22.0575i	1.11265i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	26.8027i	1.34519i	0.740011	+	0.672595i	$0.234820\pi$
$397$	26.8027i	1.34519i	−0.740011	+	0.672595i	$0.765180\pi$
$398$	0	0
$399$	−9.37893	−0.469534
$400$	0	0
$401$	10.5205	0.525369	0.262684	−	0.964882i	$-0.415392\pi$
$401$	10.5205	0.525369	0.262684	+	0.964882i	$0.415392\pi$
$402$	0	0
$403$	− 3.57478i	− 0.178072i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 11.8683i	− 0.588290i
$408$	0	0
$409$	−24.3043	−1.20177	−0.600884	−	0.799336i	$-0.705185\pi$
$409$	−24.3043	−1.20177	−0.600884	+	0.799336i	$0.705185\pi$
$410$	0	0
$411$	−5.59209	−0.275837
$412$	0	0
$413$	35.8706i	1.76508i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 12.8363i	− 0.628597i
$418$	0	0
$419$	21.3479	1.04291	0.521456	−	0.853278i	$-0.325389\pi$
$419$	21.3479	1.04291	0.521456	+	0.853278i	$0.325389\pi$
$420$	0	0
$421$	−26.5186	−1.29244	−0.646219	−	0.763152i	$-0.723651\pi$
$421$	−26.5186	−1.29244	−0.646219	+	0.763152i	$0.723651\pi$
$422$	0	0
$423$	6.68393i	0.324984i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 36.0866i	− 1.74635i
$428$	0	0
$429$	5.95494	0.287507
$430$	0	0
$431$	−13.4498	−0.647852	−0.323926	−	0.946082i	$-0.605003\pi$
$431$	−13.4498	−0.647852	−0.323926	+	0.946082i	$0.605003\pi$
$432$	0	0
$433$	22.2731i	1.07038i	0.844733	+	0.535188i	$0.179759\pi$
$433$	22.2731i	1.07038i	−0.844733	+	0.535188i	$0.820241\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	8.67463i	0.414964i
$438$	0	0
$439$	29.6126	1.41333	0.706667	−	0.707546i	$-0.250198\pi$
$439$	29.6126	1.41333	0.706667	+	0.707546i	$0.250198\pi$
$440$	0	0
$441$	−1.05869	−0.0504138
$442$	0	0
$443$	− 11.0037i	− 0.522802i	−0.965230	−	0.261401i	$-0.915815\pi$
$443$	− 11.0037i	− 0.522802i	0.965230	−	0.261401i	$-0.0841845\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 14.1167i	− 0.667697i
$448$	0	0
$449$	6.68465	0.315468	0.157734	−	0.987482i	$-0.449581\pi$
$449$	6.68465	0.315468	0.157734	+	0.987482i	$0.449581\pi$
$450$	0	0
$451$	−8.69927	−0.409633
$452$	0	0
$453$	15.3673i	0.722018i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	25.2057i	1.17907i	0.807743	+	0.589535i	$0.200689\pi$
$457$	25.2057i	1.17907i	−0.807743	+	0.589535i	$0.799311\pi$
$458$	0	0
$459$	9.07884	0.423764
$460$	0	0
$461$	34.0021	1.58364	0.791818	−	0.610757i	$-0.209135\pi$
$461$	34.0021	1.58364	0.791818	+	0.610757i	$0.209135\pi$
$462$	0	0
$463$	− 28.5486i	− 1.32677i	−0.748279	−	0.663384i	$-0.769120\pi$
$463$	− 28.5486i	− 1.32677i	0.748279	−	0.663384i	$-0.230880\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 28.5511i	− 1.32119i	−0.750743	−	0.660594i	$-0.770305\pi$
$467$	− 28.5511i	− 1.32119i	0.750743	−	0.660594i	$-0.229695\pi$
$468$	0	0
$469$	−26.1461	−1.20731
$470$	0	0
$471$	12.6583	0.583265
$472$	0	0
$473$	9.19876i	0.422959i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 14.9935i	− 0.686505i
$478$	0	0
$479$	35.9971	1.64475	0.822374	−	0.568947i	$-0.192649\pi$
$479$	35.9971	1.64475	0.822374	+	0.568947i	$0.192649\pi$
$480$	0	0
$481$	−30.2391	−1.37878
$482$	0	0
$483$	− 9.99462i	− 0.454771i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 31.5036i	− 1.42757i	−0.700367	−	0.713783i	$-0.746980\pi$
$487$	− 31.5036i	− 1.42757i	0.700367	−	0.713783i	$-0.253020\pi$
$488$	0	0
$489$	−3.06187	−0.138463
$490$	0	0
$491$	29.5694	1.33445	0.667224	−	0.744857i	$-0.267482\pi$
$491$	29.5694	1.33445	0.667224	+	0.744857i	$0.267482\pi$
$492$	0	0
$493$	13.7274i	0.618249i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 10.6003i	− 0.475490i
$498$	0	0
$499$	−12.8839	−0.576761	−0.288380	−	0.957516i	$-0.593117\pi$
$499$	−12.8839	−0.576761	−0.288380	+	0.957516i	$0.593117\pi$
$500$	0	0
$501$	−6.89668	−0.308121
$502$	0	0
$503$	− 25.8536i	− 1.15275i	−0.817184	−	0.576377i	$-0.804466\pi$
$503$	− 25.8536i	− 1.15275i	0.817184	−	0.576377i	$-0.195534\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0.262341i	0.0116510i
$508$	0	0
$509$	2.02461	0.0897391	0.0448695	−	0.998993i	$-0.485713\pi$
$509$	2.02461	0.0897391	0.0448695	+	0.998993i	$0.485713\pi$
$510$	0	0
$511$	−21.6487	−0.957683
$512$	0	0
$513$	15.5497i	0.686535i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 5.89676i	− 0.259339i
$518$	0	0
$519$	4.02981	0.176889
$520$	0	0
$521$	26.4387	1.15830	0.579150	−	0.815221i	$-0.303385\pi$
$521$	26.4387	1.15830	0.579150	+	0.815221i	$0.303385\pi$
$522$	0	0
$523$	12.0293i	0.526005i	0.964795	+	0.263002i	$0.0847127\pi$
$523$	12.0293i	0.526005i	−0.964795	+	0.263002i	$0.915287\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.66582i	0.0725643i
$528$	0	0
$529$	13.7559	0.598084
$530$	0	0
$531$	20.6040	0.894136
$532$	0	0
$533$	22.1647i	0.960061i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 27.5581i	− 1.18922i
$538$	0	0
$539$	0.934007	0.0402305
$540$	0	0
$541$	−29.2871	−1.25915	−0.629575	−	0.776940i	$-0.716771\pi$
$541$	−29.2871	−1.25915	−0.629575	+	0.776940i	$0.716771\pi$
$542$	0	0
$543$	0.642464i	0.0275707i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	19.4869i	0.833199i	0.909090	+	0.416600i	$0.136778\pi$
$547$	19.4869i	0.833199i	−0.909090	+	0.416600i	$0.863222\pi$
$548$	0	0
$549$	−20.7281	−0.884652
$550$	0	0
$551$	−23.5114	−1.00162
$552$	0	0
$553$	− 1.42838i	− 0.0607410i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	12.9493i	0.548679i	0.961633	+	0.274340i	$0.0884593\pi$
$557$	12.9493i	0.548679i	−0.961633	+	0.274340i	$0.911541\pi$
$558$	0	0
$559$	23.4374	0.991295
$560$	0	0
$561$	−2.77496	−0.117159
$562$	0	0
$563$	− 14.4115i	− 0.607372i	−0.952772	−	0.303686i	$-0.901783\pi$
$563$	− 14.4115i	− 0.607372i	0.952772	−	0.303686i	$-0.0982174\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 4.70635i	− 0.197648i
$568$	0	0
$569$	−0.231111	−0.00968867	−0.00484434	−	0.999988i	$-0.501542\pi$
$569$	−0.231111	−0.00968867	−0.00484434	+	0.999988i	$0.501542\pi$
$570$	0	0
$571$	46.0866	1.92867	0.964333	−	0.264693i	$-0.0852705\pi$
$571$	46.0866	1.92867	0.964333	+	0.264693i	$0.0852705\pi$
$572$	0	0
$573$	− 4.04085i	− 0.168809i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 8.36752i	− 0.348344i	−0.984715	−	0.174172i	$-0.944275\pi$
$577$	− 8.36752i	− 0.348344i	0.984715	−	0.174172i	$-0.0557249\pi$
$578$	0	0
$579$	−31.6085	−1.31360
$580$	0	0
$581$	14.2071	0.589411
$582$	0	0
$583$	13.2277i	0.547835i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	17.8902i	0.738410i	0.929348	+	0.369205i	$0.120370\pi$
$587$	17.8902i	0.738410i	−0.929348	+	0.369205i	$0.879630\pi$
$588$	0	0
$589$	−2.85311	−0.117561
$590$	0	0
$591$	7.43422	0.305803
$592$	0	0
$593$	20.0274i	0.822428i	0.911539	+	0.411214i	$0.134895\pi$
$593$	20.0274i	0.822428i	−0.911539	+	0.411214i	$0.865105\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 4.09140i	− 0.167450i
$598$	0	0
$599$	11.2706	0.460503	0.230252	−	0.973131i	$-0.426045\pi$
$599$	11.2706	0.460503	0.230252	+	0.973131i	$0.426045\pi$
$600$	0	0
$601$	41.3857	1.68816	0.844080	−	0.536218i	$-0.180147\pi$
$601$	41.3857	1.68816	0.844080	+	0.536218i	$0.180147\pi$
$602$	0	0
$603$	15.0182i	0.611589i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	26.3612i	1.06997i	0.844862	+	0.534984i	$0.179682\pi$
$607$	26.3612i	1.06997i	−0.844862	+	0.534984i	$0.820318\pi$
$608$	0	0
$609$	27.0890	1.09770
$610$	0	0
$611$	−15.0243	−0.607816
$612$	0	0
$613$	− 15.2023i	− 0.614014i	−0.951707	−	0.307007i	$-0.900672\pi$
$613$	− 15.2023i	− 0.614014i	0.951707	−	0.307007i	$-0.0993276\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 4.33885i	− 0.174675i	−0.996179	−	0.0873377i	$-0.972164\pi$
$617$	− 4.33885i	− 0.174675i	0.996179	−	0.0873377i	$-0.0278359\pi$
$618$	0	0
$619$	26.3637	1.05965	0.529824	−	0.848107i	$-0.322258\pi$
$619$	26.3637	1.05965	0.529824	+	0.848107i	$0.322258\pi$
$620$	0	0
$621$	−16.5704	−0.664949
$622$	0	0
$623$	− 13.9307i	− 0.558120i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 4.75277i	− 0.189807i
$628$	0	0
$629$	14.0912	0.561852
$630$	0	0
$631$	1.08262	0.0430986	0.0215493	−	0.999768i	$-0.493140\pi$
$631$	1.08262	0.0430986	0.0215493	+	0.999768i	$0.493140\pi$
$632$	0	0
$633$	18.8695i	0.749994i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 2.37974i	− 0.0942888i
$638$	0	0
$639$	−6.08880	−0.240869
$640$	0	0
$641$	41.5471	1.64101	0.820505	−	0.571639i	$-0.193692\pi$
$641$	41.5471	1.64101	0.820505	+	0.571639i	$0.193692\pi$
$642$	0	0
$643$	9.63971i	0.380153i	0.981769	+	0.190077i	$0.0608736\pi$
$643$	9.63971i	0.380153i	−0.981769	+	0.190077i	$0.939126\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	33.8932i	1.33248i	0.745737	+	0.666240i	$0.232097\pi$
$647$	33.8932i	1.33248i	−0.745737	+	0.666240i	$0.767903\pi$
$648$	0	0
$649$	−18.1774	−0.713526
$650$	0	0
$651$	3.28726	0.128838
$652$	0	0
$653$	10.6859i	0.418172i	0.977897	+	0.209086i	$0.0670488\pi$
$653$	10.6859i	0.418172i	−0.977897	+	0.209086i	$0.932951\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	12.4350i	0.485134i
$658$	0	0
$659$	34.2886	1.33569	0.667847	−	0.744298i	$-0.267216\pi$
$659$	34.2886	1.33569	0.667847	+	0.744298i	$0.267216\pi$
$660$	0	0
$661$	3.40224	0.132332	0.0661659	−	0.997809i	$-0.478923\pi$
$661$	3.40224	0.132332	0.0661659	+	0.997809i	$0.478923\pi$
$662$	0	0
$663$	7.07026i	0.274586i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 25.0548i	− 0.970125i
$668$	0	0
$669$	−11.3116	−0.437330
$670$	0	0
$671$	18.2869	0.705958
$672$	0	0
$673$	41.3344i	1.59333i	0.604424	+	0.796663i	$0.293403\pi$
$673$	41.3344i	1.59333i	−0.604424	+	0.796663i	$0.706597\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 10.0729i	− 0.387134i	−0.981087	−	0.193567i	$-0.937994\pi$
$677$	− 10.0729i	− 0.387134i	0.981087	−	0.193567i	$-0.0620057\pi$
$678$	0	0
$679$	−14.6212	−0.561110
$680$	0	0
$681$	−13.2206	−0.506614
$682$	0	0
$683$	5.91429i	0.226304i	0.993578	+	0.113152i	$0.0360947\pi$
$683$	5.91429i	0.226304i	−0.993578	+	0.113152i	$0.963905\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 15.0627i	− 0.574678i
$688$	0	0
$689$	33.7026	1.28397
$690$	0	0
$691$	19.6565	0.747770	0.373885	−	0.927475i	$-0.378025\pi$
$691$	19.6565	0.747770	0.373885	+	0.927475i	$0.378025\pi$
$692$	0	0
$693$	− 6.17780i	− 0.234675i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 10.3286i	− 0.391223i
$698$	0	0
$699$	−4.87090	−0.184234
$700$	0	0
$701$	−10.3363	−0.390397	−0.195198	−	0.980764i	$-0.562535\pi$
$701$	−10.3363	−0.390397	−0.195198	+	0.980764i	$0.562535\pi$
$702$	0	0
$703$	24.1345i	0.910250i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 40.3535i	− 1.51765i
$708$	0	0
$709$	45.7547	1.71835	0.859177	−	0.511678i	$-0.170976\pi$
$709$	45.7547	1.71835	0.859177	+	0.511678i	$0.170976\pi$
$710$	0	0
$711$	−0.820458	−0.0307696
$712$	0	0
$713$	− 3.04041i	− 0.113864i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 0.310743i	− 0.0116049i
$718$	0	0
$719$	48.1601	1.79607	0.898035	−	0.439924i	$-0.144995\pi$
$719$	48.1601	1.79607	0.898035	+	0.439924i	$0.144995\pi$
$720$	0	0
$721$	27.3164	1.01731
$722$	0	0
$723$	2.56717i	0.0954739i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 12.8283i	− 0.475774i	−0.971293	−	0.237887i	$-0.923545\pi$
$727$	− 12.8283i	− 0.475774i	0.971293	−	0.237887i	$-0.0764548\pi$
$728$	0	0
$729$	−22.1158	−0.819105
$730$	0	0
$731$	−10.9216	−0.403951
$732$	0	0
$733$	23.1228i	0.854058i	0.904238	+	0.427029i	$0.140440\pi$
$733$	23.1228i	0.854058i	−0.904238	+	0.427029i	$0.859560\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 13.2495i	− 0.488053i
$738$	0	0
$739$	30.9000	1.13668	0.568338	−	0.822795i	$-0.307586\pi$
$739$	30.9000	1.13668	0.568338	+	0.822795i	$0.307586\pi$
$740$	0	0
$741$	−12.1095	−0.444854
$742$	0	0
$743$	19.3986i	0.711667i	0.934549	+	0.355833i	$0.115803\pi$
$743$	19.3986i	0.711667i	−0.934549	+	0.355833i	$0.884197\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 8.16053i	− 0.298578i
$748$	0	0
$749$	−30.1168	−1.10044
$750$	0	0
$751$	19.3054	0.704465	0.352232	−	0.935913i	$-0.385423\pi$
$751$	19.3054	0.704465	0.352232	+	0.935913i	$0.385423\pi$
$752$	0	0
$753$	− 4.70518i	− 0.171466i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 43.3214i	− 1.57454i	−0.616607	−	0.787271i	$-0.711493\pi$
$757$	− 43.3214i	− 1.57454i	0.616607	−	0.787271i	$-0.288507\pi$
$758$	0	0
$759$	5.06477	0.183840
$760$	0	0
$761$	−31.0269	−1.12472	−0.562361	−	0.826892i	$-0.690107\pi$
$761$	−31.0269	−1.12472	−0.562361	+	0.826892i	$0.690107\pi$
$762$	0	0
$763$	− 21.8538i	− 0.791159i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	46.3139i	1.67230i
$768$	0	0
$769$	−34.3285	−1.23792	−0.618958	−	0.785424i	$-0.712445\pi$
$769$	−34.3285	−1.23792	−0.618958	+	0.785424i	$0.712445\pi$
$770$	0	0
$771$	18.3989	0.662620
$772$	0	0
$773$	− 14.8797i	− 0.535185i	−0.963532	−	0.267593i	$-0.913772\pi$
$773$	− 14.8797i	− 0.535185i	0.963532	−	0.267593i	$-0.0862281\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 27.8069i	− 0.997569i
$778$	0	0
$779$	17.6902	0.633817
$780$	0	0
$781$	5.37172	0.192215
$782$	0	0
$783$	− 44.9119i	− 1.60502i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	12.3320i	0.439589i	0.975546	+	0.219795i	$0.0705387\pi$
$787$	12.3320i	0.439589i	−0.975546	+	0.219795i	$0.929461\pi$
$788$	0	0
$789$	−8.00937	−0.285141
$790$	0	0
$791$	44.4615	1.58087
$792$	0	0
$793$	− 46.5929i	− 1.65456i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	32.1782i	1.13981i	0.821710	+	0.569906i	$0.193020\pi$
$797$	32.1782i	1.13981i	−0.821710	+	0.569906i	$0.806980\pi$
$798$	0	0
$799$	7.00119	0.247684
$800$	0	0
$801$	−8.00173	−0.282727
$802$	0	0
$803$	− 10.9705i	− 0.387140i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 13.2519i	− 0.466488i
$808$	0	0
$809$	2.64407	0.0929604	0.0464802	−	0.998919i	$-0.485200\pi$
$809$	2.64407	0.0929604	0.0464802	+	0.998919i	$0.485200\pi$
$810$	0	0
$811$	−30.2848	−1.06344	−0.531721	−	0.846919i	$-0.678455\pi$
$811$	−30.2848	−1.06344	−0.531721	+	0.846919i	$0.678455\pi$
$812$	0	0
$813$	− 5.95923i	− 0.208999i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 18.7059i	− 0.654437i
$818$	0	0
$819$	−15.7403	−0.550011
$820$	0	0
$821$	57.1572	1.99480	0.997401	−	0.0720522i	$-0.0229548\pi$
$821$	57.1572	1.99480	0.997401	+	0.0720522i	$0.0229548\pi$
$822$	0	0
$823$	40.2662i	1.40359i	0.712378	+	0.701796i	$0.247618\pi$
$823$	40.2662i	1.40359i	−0.712378	+	0.701796i	$0.752382\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.09558i	0.0728704i	0.999336	+	0.0364352i	$0.0116003\pi$
$827$	2.09558i	0.0728704i	−0.999336	+	0.0364352i	$0.988400\pi$
$828$	0	0
$829$	−34.3922	−1.19449	−0.597245	−	0.802059i	$-0.703738\pi$
$829$	−34.3922	−1.19449	−0.597245	+	0.802059i	$0.703738\pi$
$830$	0	0
$831$	28.0340	0.972490
$832$	0	0
$833$	1.10894i	0.0384225i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 5.45007i	− 0.188382i
$838$	0	0
$839$	48.2326	1.66517	0.832587	−	0.553894i	$-0.186859\pi$
$839$	48.2326	1.66517	0.832587	+	0.553894i	$0.186859\pi$
$840$	0	0
$841$	38.9075	1.34164
$842$	0	0
$843$	5.33881i	0.183878i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 25.0055i	− 0.859198i
$848$	0	0
$849$	9.30822	0.319457
$850$	0	0
$851$	−25.7188	−0.881630
$852$	0	0
$853$	8.14503i	0.278881i	0.990230	+	0.139440i	$0.0445303\pi$
$853$	8.14503i	0.278881i	−0.990230	+	0.139440i	$0.955470\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 7.32650i	− 0.250268i	−0.992140	−	0.125134i	$-0.960064\pi$
$857$	− 7.32650i	− 0.250268i	0.992140	−	0.125134i	$-0.0399361\pi$
$858$	0	0
$859$	−36.7117	−1.25259	−0.626294	−	0.779587i	$-0.715429\pi$
$859$	−36.7117	−1.25259	−0.626294	+	0.779587i	$0.715429\pi$
$860$	0	0
$861$	−20.3820	−0.694618
$862$	0	0
$863$	− 31.9983i	− 1.08924i	−0.838684	−	0.544618i	$-0.816675\pi$
$863$	− 31.9983i	− 1.08924i	0.838684	−	0.544618i	$-0.183325\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	16.8893i	0.573591i
$868$	0	0
$869$	0.723833	0.0245543
$870$	0	0
$871$	−33.7582	−1.14385
$872$	0	0
$873$	8.39837i	0.284242i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 10.7897i	− 0.364344i	−0.983267	−	0.182172i	$-0.941687\pi$
$877$	− 10.7897i	− 0.364344i	0.983267	−	0.182172i	$-0.0583127\pi$
$878$	0	0
$879$	14.8399	0.500538
$880$	0	0
$881$	32.4394	1.09291	0.546455	−	0.837489i	$-0.315977\pi$
$881$	32.4394	1.09291	0.546455	+	0.837489i	$0.315977\pi$
$882$	0	0
$883$	− 13.5500i	− 0.455993i	−0.973662	−	0.227997i	$-0.926782\pi$
$883$	− 13.5500i	− 0.455993i	0.973662	−	0.227997i	$-0.0732175\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 26.3939i	− 0.886221i	−0.896467	−	0.443111i	$-0.853875\pi$
$887$	− 26.3939i	− 0.886221i	0.896467	−	0.443111i	$-0.146125\pi$
$888$	0	0
$889$	−32.3669	−1.08555
$890$	0	0
$891$	2.38494	0.0798986
$892$	0	0
$893$	11.9912i	0.401270i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 12.9044i	− 0.430867i
$898$	0	0
$899$	8.24060	0.274839
$900$	0	0
$901$	−15.7052	−0.523215
$902$	0	0
$903$	21.5523i	0.717216i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 53.7213i	− 1.78379i	−0.452246	−	0.891893i	$-0.649377\pi$
$907$	− 53.7213i	− 1.78379i	0.452246	−	0.891893i	$-0.350623\pi$
$908$	0	0
$909$	−23.1789	−0.768797
$910$	0	0
$911$	30.8467	1.02200	0.510999	−	0.859581i	$-0.329276\pi$
$911$	30.8467	1.02200	0.510999	+	0.859581i	$0.329276\pi$
$912$	0	0
$913$	7.19946i	0.238267i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	51.4367i	1.69859i
$918$	0	0
$919$	−5.79252	−0.191078	−0.0955388	−	0.995426i	$-0.530457\pi$
$919$	−5.79252	−0.191078	−0.0955388	+	0.995426i	$0.530457\pi$
$920$	0	0
$921$	−23.1278	−0.762086
$922$	0	0
$923$	− 13.6865i	− 0.450497i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 15.6904i	− 0.515341i
$928$	0	0
$929$	5.29659	0.173776	0.0868878	−	0.996218i	$-0.472308\pi$
$929$	5.29659	0.173776	0.0868878	+	0.996218i	$0.472308\pi$
$930$	0	0
$931$	−1.89933	−0.0622479
$932$	0	0
$933$	17.7501i	0.581113i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 19.5112i	− 0.637403i	−0.947855	−	0.318701i	$-0.896753\pi$
$937$	− 19.5112i	− 0.637403i	0.947855	−	0.318701i	$-0.103247\pi$
$938$	0	0
$939$	23.2071	0.757335
$940$	0	0
$941$	−32.9560	−1.07434	−0.537168	−	0.843476i	$-0.680506\pi$
$941$	−32.9560	−1.07434	−0.537168	+	0.843476i	$0.680506\pi$
$942$	0	0
$943$	18.8515i	0.613888i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 47.0379i	− 1.52853i	−0.644905	−	0.764263i	$-0.723103\pi$
$947$	− 47.0379i	− 1.52853i	0.644905	−	0.764263i	$-0.276897\pi$
$948$	0	0
$949$	−27.9515	−0.907345
$950$	0	0
$951$	−36.4750	−1.18278
$952$	0	0
$953$	− 9.00688i	− 0.291762i	−0.989302	−	0.145881i	$-0.953398\pi$
$953$	− 9.00688i	− 0.291762i	0.989302	−	0.145881i	$-0.0466016\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	13.7274i	0.443742i
$958$	0	0
$959$	−13.0404	−0.421097
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	17.2990i	0.557452i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	32.7556i	1.05335i	0.850067	+	0.526675i	$0.176561\pi$
$967$	32.7556i	1.05335i	−0.850067	+	0.526675i	$0.823439\pi$
$968$	0	0
$969$	5.64294	0.181277
$970$	0	0
$971$	−46.9171	−1.50564	−0.752821	−	0.658225i	$-0.771308\pi$
$971$	−46.9171	−1.50564	−0.752821	+	0.658225i	$0.771308\pi$
$972$	0	0
$973$	− 29.9335i	− 0.959626i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 31.9605i	− 1.02251i	−0.859430	−	0.511253i	$-0.829182\pi$
$977$	− 31.9605i	− 1.02251i	0.859430	−	0.511253i	$-0.170818\pi$
$978$	0	0
$979$	7.05936	0.225618
$980$	0	0
$981$	−12.5527	−0.400778
$982$	0	0
$983$	− 36.8703i	− 1.17598i	−0.808868	−	0.587990i	$-0.799919\pi$
$983$	− 36.8703i	− 1.17598i	0.808868	−	0.587990i	$-0.200081\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 13.8159i	− 0.439764i
$988$	0	0
$989$	19.9339	0.633860
$990$	0	0
$991$	26.8105	0.851665	0.425832	−	0.904802i	$-0.359981\pi$
$991$	26.8105	0.851665	0.425832	+	0.904802i	$0.359981\pi$
$992$	0	0
$993$	14.5850i	0.462843i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 32.5656i	− 1.03136i	−0.856781	−	0.515681i	$-0.827539\pi$
$997$	− 32.5656i	− 1.03136i	0.856781	−	0.515681i	$-0.172461\pi$
$998$	0	0
$999$	−46.1022	−1.45861

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3100.2.c.h.249.7		10
5.2	odd	4		3100.2.a.j.1.4	✓	5
5.3	odd	4		3100.2.a.m.1.2	yes	5
5.4	even	2	inner	3100.2.c.h.249.4		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3100.2.a.j.1.4	✓	5	5.2	odd	4
3100.2.a.m.1.2	yes	5	5.3	odd	4
3100.2.c.h.249.4		10	5.4	even	2	inner
3100.2.c.h.249.7		10	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:	\( N \)	\(=\)	\( 3100 = 2^{2} \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3100.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.18729i q^{3} +2.76870i q^{7} +1.59033 q^{9} +O(q^{10})\)	\(q+1.18729i q^{3} +2.76870i q^{7} +1.59033 q^{9} -1.40304 q^{11} +3.57478i q^{13} -1.66582i q^{17} +2.85311 q^{19} -3.28726 q^{21} +3.04041i q^{23} +5.45007i q^{27} -8.24060 q^{29} -1.00000 q^{31} -1.66582i q^{33} +8.45900i q^{37} -4.24431 q^{39} +6.20031 q^{41} -6.55631i q^{43} +4.20285i q^{47} -0.665703 q^{49} +1.97782 q^{51} -9.42789i q^{53} +3.38748i q^{57} +12.9558 q^{59} -13.0338 q^{61} +4.40316i q^{63} +9.44345i q^{67} -3.60986 q^{69} -3.82863 q^{71} +7.81909i q^{73} -3.88460i q^{77} -0.515903 q^{79} -1.69984 q^{81} -5.13133i q^{83} -9.78401i q^{87} -5.03148 q^{89} -9.89749 q^{91} -1.18729i q^{93} +5.28089i q^{97} -2.23130 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 12 q^{9}+O(q^{10}) \) 10 * q - 12 * q^9	\( 10 q - 12 q^{9} - 4 q^{11} - 12 q^{19} - 6 q^{21} - 20 q^{29} - 10 q^{31} - 6 q^{39} - 12 q^{41} - 54 q^{49} - 30 q^{51} - 40 q^{59} - 34 q^{61} - 60 q^{69} - 12 q^{71} - 30 q^{81} - 18 q^{89} - 40 q^{91} - 42 q^{99}+O(q^{100}) \) 10 * q - 12 * q^9 - 4 * q^11 - 12 * q^19 - 6 * q^21 - 20 * q^29 - 10 * q^31 - 6 * q^39 - 12 * q^41 - 54 * q^49 - 30 * q^51 - 40 * q^59 - 34 * q^61 - 60 * q^69 - 12 * q^71 - 30 * q^81 - 18 * q^89 - 40 * q^91 - 42 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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