LMFDB - Embedded newform 1900.2.c.g.1749.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,2,Mod(1749,1900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1900, 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("1900.1749");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1900 = 2^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1900.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:	$15.1715763840$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.4227136.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 9x^{4} + 22x^{2} + 9 $ x^6 + 9x^4 + 22x^2 + 9
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1749.6
Root			$1.91223i$ of defining polynomial
Character	$\chi$	$=$	1900.1749
Dual form			1900.2.c.g.1749.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.91223i q^{3} -2.91223i q^{7} -5.48108 q^{9} +O(q^{10})$	$q+2.91223i q^{3} -2.91223i q^{7} -5.48108 q^{9} +0.598988 q^{11} -2.56885i q^{13} -2.91223i q^{17} +1.00000 q^{19} +8.48108 q^{21} -1.16784i q^{23} -7.22547i q^{27} +1.40101 q^{29} +7.88209 q^{31} +1.74439i q^{33} +6.53871i q^{37} +7.48108 q^{39} +9.48108 q^{41} -0.824458i q^{43} -6.05763i q^{47} -1.48108 q^{49} +8.48108 q^{51} -0.824458i q^{53} +2.91223i q^{57} +14.1601 q^{59} -1.88209 q^{61} +15.9622i q^{63} -5.11021i q^{67} +3.40101 q^{69} -11.9622 q^{71} +1.03014i q^{73} -1.74439i q^{77} +15.3632 q^{79} +4.59899 q^{81} -15.7565i q^{83} +4.08007i q^{87} -15.4432 q^{89} -7.48108 q^{91} +22.9545i q^{93} -7.70655i q^{97} -3.28310 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{9}+O(q^{10}) $ 6 * q - 2 * q^9	$ 6 q - 2 q^{9} - 2 q^{11} + 6 q^{19} + 20 q^{21} + 14 q^{29} + 22 q^{31} + 14 q^{39} + 26 q^{41} + 22 q^{49} + 20 q^{51} + 12 q^{59} + 14 q^{61} + 26 q^{69} - 10 q^{71} + 36 q^{79} + 22 q^{81} - 14 q^{91}+O(q^{100}) $ 6 * q - 2 * q^9 - 2 * q^11 + 6 * q^19 + 20 * q^21 + 14 * q^29 + 22 * q^31 + 14 * q^39 + 26 * q^41 + 22 * q^49 + 20 * q^51 + 12 * q^59 + 14 * q^61 + 26 * q^69 - 10 * q^71 + 36 * q^79 + 22 * q^81 - 14 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times$.

$n$	$77$	$401$	$951$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	2.91223i	1.68138i	0.541520	+	0.840688i	$0.317849\pi$
$3$	2.91223i	1.68138i	−0.541520	+	0.840688i	$0.682151\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.91223i	− 1.10072i	−0.834928	−	0.550360i	$-0.814491\pi$
$7$	− 2.91223i	− 1.10072i	0.834928	−	0.550360i	$-0.185509\pi$
$8$	0	0
$9$	−5.48108	−1.82703
$10$	0	0
$11$	0.598988	0.180602	0.0903009	−	0.995915i	$-0.471217\pi$
$11$	0.598988	0.180602	0.0903009	+	0.995915i	$0.471217\pi$
$12$	0	0
$13$	− 2.56885i	− 0.712471i	−0.934396	−	0.356235i	$-0.884060\pi$
$13$	− 2.56885i	− 0.712471i	0.934396	−	0.356235i	$-0.115940\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 2.91223i	− 0.706319i	−0.935563	−	0.353160i	$-0.885107\pi$
$17$	− 2.91223i	− 0.706319i	0.935563	−	0.353160i	$-0.114893\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	8.48108	1.85072
$22$	0	0
$23$	− 1.16784i	− 0.243511i	−0.992560	−	0.121756i	$-0.961148\pi$
$23$	− 1.16784i	− 0.243511i	0.992560	−	0.121756i	$-0.0388524\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 7.22547i	− 1.39054i
$28$	0	0
$29$	1.40101	0.260161	0.130081	−	0.991503i	$-0.458476\pi$
$29$	1.40101	0.260161	0.130081	+	0.991503i	$0.458476\pi$
$30$	0	0
$31$	7.88209	1.41567	0.707833	−	0.706380i	$-0.249673\pi$
$31$	7.88209	1.41567	0.707833	+	0.706380i	$0.249673\pi$
$32$	0	0
$33$	1.74439i	0.303660i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.53871i	1.07496i	0.843277	+	0.537479i	$0.180623\pi$
$37$	6.53871i	1.07496i	−0.843277	+	0.537479i	$0.819377\pi$
$38$	0	0
$39$	7.48108	1.19793
$40$	0	0
$41$	9.48108	1.48070	0.740348	−	0.672224i	$-0.234661\pi$
$41$	9.48108	1.48070	0.740348	+	0.672224i	$0.234661\pi$
$42$	0	0
$43$	− 0.824458i	− 0.125729i	−0.998022	−	0.0628644i	$-0.979976\pi$
$43$	− 0.824458i	− 0.125729i	0.998022	−	0.0628644i	$-0.0200236\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 6.05763i	− 0.883596i	−0.897114	−	0.441798i	$-0.854341\pi$
$47$	− 6.05763i	− 0.883596i	0.897114	−	0.441798i	$-0.145659\pi$
$48$	0	0
$49$	−1.48108	−0.211583
$50$	0	0
$51$	8.48108	1.18759
$52$	0	0
$53$	− 0.824458i	− 0.113248i	−0.998396	−	0.0566240i	$-0.981966\pi$
$53$	− 0.824458i	− 0.113248i	0.998396	−	0.0566240i	$-0.0180336\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.91223i	0.385734i
$58$	0	0
$59$	14.1601	1.84349	0.921746	−	0.387794i	$-0.126763\pi$
$59$	14.1601	1.84349	0.921746	+	0.387794i	$0.126763\pi$
$60$	0	0
$61$	−1.88209	−0.240977	−0.120488	−	0.992715i	$-0.538446\pi$
$61$	−1.88209	−0.240977	−0.120488	+	0.992715i	$0.538446\pi$
$62$	0	0
$63$	15.9622i	2.01104i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 5.11021i	− 0.624311i	−0.950031	−	0.312156i	$-0.898949\pi$
$67$	− 5.11021i	− 0.624311i	0.950031	−	0.312156i	$-0.101051\pi$
$68$	0	0
$69$	3.40101	0.409434
$70$	0	0
$71$	−11.9622	−1.41965	−0.709823	−	0.704380i	$-0.751225\pi$
$71$	−11.9622	−1.41965	−0.709823	+	0.704380i	$0.751225\pi$
$72$	0	0
$73$	1.03014i	0.120569i	0.998181	+	0.0602843i	$0.0192007\pi$
$73$	1.03014i	0.120569i	−0.998181	+	0.0602843i	$0.980799\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 1.74439i	− 0.198792i
$78$	0	0
$79$	15.3632	1.72849	0.864246	−	0.503070i	$-0.167796\pi$
$79$	15.3632	1.72849	0.864246	+	0.503070i	$0.167796\pi$
$80$	0	0
$81$	4.59899	0.510999
$82$	0	0
$83$	− 15.7565i	− 1.72950i	−0.502204	−	0.864749i	$-0.667477\pi$
$83$	− 15.7565i	− 1.72950i	0.502204	−	0.864749i	$-0.332523\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	4.08007i	0.437429i
$88$	0	0
$89$	−15.4432	−1.63698	−0.818490	−	0.574521i	$-0.805188\pi$
$89$	−15.4432	−1.63698	−0.818490	+	0.574521i	$0.805188\pi$
$90$	0	0
$91$	−7.48108	−0.784230
$92$	0	0
$93$	22.9545i	2.38027i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 7.70655i	− 0.782481i	−0.920288	−	0.391241i	$-0.872046\pi$
$97$	− 7.70655i	− 0.782481i	0.920288	−	0.391241i	$-0.127954\pi$
$98$	0	0
$99$	−3.28310	−0.329964
$100$	0	0
$101$	4.40101	0.437917	0.218959	−	0.975734i	$-0.429734\pi$
$101$	4.40101	0.437917	0.218959	+	0.975734i	$0.429734\pi$
$102$	0	0
$103$	17.3632i	1.71084i	0.517932	+	0.855422i	$0.326702\pi$
$103$	17.3632i	1.71084i	−0.517932	+	0.855422i	$0.673298\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 14.1903i	− 1.37183i	−0.727684	−	0.685913i	$-0.759403\pi$
$107$	− 14.1903i	− 1.37183i	0.727684	−	0.685913i	$-0.240597\pi$
$108$	0	0
$109$	10.6791	1.02287	0.511434	−	0.859323i	$-0.329114\pi$
$109$	10.6791	1.02287	0.511434	+	0.859323i	$0.329114\pi$
$110$	0	0
$111$	−19.0422	−1.80741
$112$	0	0
$113$	8.25561i	0.776622i	0.921528	+	0.388311i	$0.126941\pi$
$113$	8.25561i	0.776622i	−0.921528	+	0.388311i	$0.873059\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	14.0801i	1.30170i
$118$	0	0
$119$	−8.48108	−0.777459
$120$	0	0
$121$	−10.6412	−0.967383
$122$	0	0
$123$	27.6111i	2.48961i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	12.1300i	1.07636i	0.842829	+	0.538182i	$0.180889\pi$
$127$	12.1300i	1.07636i	−0.842829	+	0.538182i	$0.819111\pi$
$128$	0	0
$129$	2.40101	0.211397
$130$	0	0
$131$	5.48108	0.478884	0.239442	−	0.970911i	$-0.423035\pi$
$131$	5.48108	0.478884	0.239442	+	0.970911i	$0.423035\pi$
$132$	0	0
$133$	− 2.91223i	− 0.252522i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 7.59634i	− 0.648999i	−0.945886	−	0.324500i	$-0.894804\pi$
$137$	− 7.59634i	− 0.648999i	0.945886	−	0.324500i	$-0.105196\pi$
$138$	0	0
$139$	−7.16013	−0.607315	−0.303657	−	0.952781i	$-0.598208\pi$
$139$	−7.16013	−0.607315	−0.303657	+	0.952781i	$0.598208\pi$
$140$	0	0
$141$	17.6412	1.48566
$142$	0	0
$143$	− 1.53871i	− 0.128673i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 4.31324i	− 0.355750i
$148$	0	0
$149$	18.4432	1.51093	0.755464	−	0.655190i	$-0.227411\pi$
$149$	18.4432	1.51093	0.755464	+	0.655190i	$0.227411\pi$
$150$	0	0
$151$	−9.27804	−0.755036	−0.377518	−	0.926002i	$-0.623222\pi$
$151$	−9.27804	−0.755036	−0.377518	+	0.926002i	$0.623222\pi$
$152$	0	0
$153$	15.9622i	1.29046i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	14.1903i	1.13251i	0.824231	+	0.566254i	$0.191608\pi$
$157$	14.1903i	1.13251i	−0.824231	+	0.566254i	$0.808392\pi$
$158$	0	0
$159$	2.40101	0.190413
$160$	0	0
$161$	−3.40101	−0.268037
$162$	0	0
$163$	− 25.1799i	− 1.97224i	−0.166023	−	0.986122i	$-0.553093\pi$
$163$	− 25.1799i	− 1.97224i	0.166023	−	0.986122i	$-0.446907\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 11.0301i	− 0.853538i	−0.904361	−	0.426769i	$-0.859652\pi$
$167$	− 11.0301i	− 0.853538i	0.904361	−	0.426769i	$-0.140348\pi$
$168$	0	0
$169$	6.40101	0.492386
$170$	0	0
$171$	−5.48108	−0.419149
$172$	0	0
$173$	15.3029i	1.16346i	0.813383	+	0.581729i	$0.197623\pi$
$173$	15.3029i	1.16346i	−0.813383	+	0.581729i	$0.802377\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	41.2376i	3.09960i
$178$	0	0
$179$	14.9243	1.11550	0.557748	−	0.830010i	$-0.311666\pi$
$179$	14.9243	1.11550	0.557748	+	0.830010i	$0.311666\pi$
$180$	0	0
$181$	3.80202	0.282602	0.141301	−	0.989967i	$-0.454871\pi$
$181$	3.80202	0.282602	0.141301	+	0.989967i	$0.454871\pi$
$182$	0	0
$183$	− 5.48108i	− 0.405173i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 1.74439i	− 0.127563i
$188$	0	0
$189$	−21.0422	−1.53060
$190$	0	0
$191$	−10.5990	−0.766916	−0.383458	−	0.923558i	$-0.625267\pi$
$191$	−10.5990	−0.766916	−0.383458	+	0.923558i	$0.625267\pi$
$192$	0	0
$193$	22.2677i	1.60286i	0.598086	+	0.801432i	$0.295928\pi$
$193$	22.2677i	1.60286i	−0.598086	+	0.801432i	$0.704072\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	15.1652i	1.08048i	0.841513	+	0.540238i	$0.181666\pi$
$197$	15.1652i	1.08048i	−0.841513	+	0.540238i	$0.818334\pi$
$198$	0	0
$199$	−16.3632	−1.15995	−0.579977	−	0.814633i	$-0.696939\pi$
$199$	−16.3632	−1.15995	−0.579977	+	0.814633i	$0.696939\pi$
$200$	0	0
$201$	14.8821	1.04970
$202$	0	0
$203$	− 4.08007i	− 0.286365i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	6.40101i	0.444901i
$208$	0	0
$209$	0.598988	0.0414329
$210$	0	0
$211$	−5.80202	−0.399428	−0.199714	−	0.979854i	$-0.564001\pi$
$211$	−5.80202	−0.399428	−0.199714	+	0.979854i	$0.564001\pi$
$212$	0	0
$213$	− 34.8365i	− 2.38696i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 22.9545i	− 1.55825i
$218$	0	0
$219$	−3.00000	−0.202721
$220$	0	0
$221$	−7.48108	−0.503232
$222$	0	0
$223$	− 10.1377i	− 0.678871i	−0.940629	−	0.339435i	$-0.889764\pi$
$223$	− 10.1377i	− 0.678871i	0.940629	−	0.339435i	$-0.110236\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 14.5189i	− 0.963655i	−0.876266	−	0.481827i	$-0.839973\pi$
$227$	− 14.5189i	− 0.963655i	0.876266	−	0.481827i	$-0.160027\pi$
$228$	0	0
$229$	27.2024	1.79758	0.898791	−	0.438377i	$-0.144446\pi$
$229$	27.2024	1.79758	0.898791	+	0.438377i	$0.144446\pi$
$230$	0	0
$231$	5.08007	0.334244
$232$	0	0
$233$	− 1.40101i	− 0.0917833i	−0.998946	−	0.0458917i	$-0.985387\pi$
$233$	− 1.40101i	− 0.0917833i	0.998946	−	0.0458917i	$-0.0146129\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	44.7411i	2.90624i
$238$	0	0
$239$	−9.88209	−0.639219	−0.319610	−	0.947549i	$-0.603552\pi$
$239$	−9.88209	−0.639219	−0.319610	+	0.947549i	$0.603552\pi$
$240$	0	0
$241$	−3.08513	−0.198730	−0.0993652	−	0.995051i	$-0.531681\pi$
$241$	−3.08513	−0.198730	−0.0993652	+	0.995051i	$0.531681\pi$
$242$	0	0
$243$	− 8.28310i	− 0.531361i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 2.56885i	− 0.163452i
$248$	0	0
$249$	45.8865	2.90794
$250$	0	0
$251$	6.08007	0.383770	0.191885	−	0.981417i	$-0.438540\pi$
$251$	6.08007	0.383770	0.191885	+	0.981417i	$0.438540\pi$
$252$	0	0
$253$	− 0.699521i	− 0.0439785i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 19.0121i	− 1.18594i	−0.805224	−	0.592971i	$-0.797955\pi$
$257$	− 19.0121i	− 1.18594i	0.805224	−	0.592971i	$-0.202045\pi$
$258$	0	0
$259$	19.0422	1.18323
$260$	0	0
$261$	−7.67906	−0.475322
$262$	0	0
$263$	− 0.453585i	− 0.0279693i	−0.999902	−	0.0139846i	$-0.995548\pi$
$263$	− 0.453585i	− 0.0279693i	0.999902	−	0.0139846i	$-0.00445159\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 44.9742i	− 2.75238i
$268$	0	0
$269$	−23.8442	−1.45381	−0.726905	−	0.686738i	$-0.759042\pi$
$269$	−23.8442	−1.45381	−0.726905	+	0.686738i	$0.759042\pi$
$270$	0	0
$271$	12.7591	0.775061	0.387531	−	0.921857i	$-0.373328\pi$
$271$	12.7591	0.775061	0.387531	+	0.921857i	$0.373328\pi$
$272$	0	0
$273$	− 21.7866i	− 1.31859i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 8.29783i	− 0.498568i	−0.968430	−	0.249284i	$-0.919805\pi$
$277$	− 8.29783i	− 0.498568i	0.968430	−	0.249284i	$-0.0801953\pi$
$278$	0	0
$279$	−43.2024	−2.58646
$280$	0	0
$281$	−9.19798	−0.548705	−0.274353	−	0.961629i	$-0.588464\pi$
$281$	−9.19798	−0.548705	−0.274353	+	0.961629i	$0.588464\pi$
$282$	0	0
$283$	− 19.8770i	− 1.18157i	−0.806830	−	0.590783i	$-0.798819\pi$
$283$	− 19.8770i	− 1.18157i	0.806830	−	0.590783i	$-0.201181\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 27.6111i	− 1.62983i
$288$	0	0
$289$	8.51892	0.501113
$290$	0	0
$291$	22.4432	1.31565
$292$	0	0
$293$	− 8.14540i	− 0.475860i	−0.971282	−	0.237930i	$-0.923531\pi$
$293$	− 8.14540i	− 0.475860i	0.971282	−	0.237930i	$-0.0764688\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 4.32797i	− 0.251134i
$298$	0	0
$299$	−3.00000	−0.173494
$300$	0	0
$301$	−2.40101	−0.138392
$302$	0	0
$303$	12.8168i	0.736303i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 10.7565i	− 0.613905i	−0.951725	−	0.306952i	$-0.900691\pi$
$307$	− 10.7565i	− 0.613905i	0.951725	−	0.306952i	$-0.0993092\pi$
$308$	0	0
$309$	−50.5655	−2.87657
$310$	0	0
$311$	−30.8442	−1.74902	−0.874508	−	0.485010i	$-0.838816\pi$
$311$	−30.8442	−1.74902	−0.874508	+	0.485010i	$0.838816\pi$
$312$	0	0
$313$	6.53871i	0.369590i	0.982777	+	0.184795i	$0.0591621\pi$
$313$	6.53871i	0.369590i	−0.982777	+	0.184795i	$0.940838\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 11.3055i	− 0.634982i	−0.948261	−	0.317491i	$-0.897160\pi$
$317$	− 11.3055i	− 0.634982i	0.948261	−	0.317491i	$-0.102840\pi$
$318$	0	0
$319$	0.839190	0.0469856
$320$	0	0
$321$	41.3253	2.30655
$322$	0	0
$323$	− 2.91223i	− 0.162041i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	31.0999i	1.71983i
$328$	0	0
$329$	−17.6412	−0.972592
$330$	0	0
$331$	22.8064	1.25355	0.626777	−	0.779199i	$-0.284374\pi$
$331$	22.8064	1.25355	0.626777	+	0.779199i	$0.284374\pi$
$332$	0	0
$333$	− 35.8392i	− 1.96398i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	17.7065i	0.964537i	0.876023	+	0.482269i	$0.160187\pi$
$337$	17.7065i	0.964537i	−0.876023	+	0.482269i	$0.839813\pi$
$338$	0	0
$339$	−24.0422	−1.30579
$340$	0	0
$341$	4.72128	0.255672
$342$	0	0
$343$	− 16.0724i	− 0.867826i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 6.26331i	− 0.336232i	−0.985767	−	0.168116i	$-0.946232\pi$
$347$	− 6.26331i	− 0.336232i	0.985767	−	0.168116i	$-0.0537684\pi$
$348$	0	0
$349$	14.8821	0.796620	0.398310	−	0.917251i	$-0.369597\pi$
$349$	14.8821	0.796620	0.398310	+	0.917251i	$0.369597\pi$
$350$	0	0
$351$	−18.5611	−0.990721
$352$	0	0
$353$	− 5.45359i	− 0.290265i	−0.989412	−	0.145133i	$-0.953639\pi$
$353$	− 5.45359i	− 0.290265i	0.989412	−	0.145133i	$-0.0463609\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 24.6988i	− 1.30720i
$358$	0	0
$359$	−2.11791	−0.111779	−0.0558895	−	0.998437i	$-0.517799\pi$
$359$	−2.11791	−0.111779	−0.0558895	+	0.998437i	$0.517799\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	− 30.9897i	− 1.62653i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	17.5414i	0.915651i	0.889042	+	0.457826i	$0.151372\pi$
$367$	17.5414i	0.915651i	−0.889042	+	0.457826i	$0.848628\pi$
$368$	0	0
$369$	−51.9665	−2.70527
$370$	0	0
$371$	−2.40101	−0.124654
$372$	0	0
$373$	− 29.0801i	− 1.50571i	−0.658187	−	0.752854i	$-0.728676\pi$
$373$	− 29.0801i	− 1.50571i	0.658187	−	0.752854i	$-0.271324\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 3.59899i	− 0.185357i
$378$	0	0
$379$	−26.2780	−1.34981	−0.674906	−	0.737904i	$-0.735816\pi$
$379$	−26.2780	−1.34981	−0.674906	+	0.737904i	$0.735816\pi$
$380$	0	0
$381$	−35.3253	−1.80977
$382$	0	0
$383$	21.6911i	1.10837i	0.832395	+	0.554183i	$0.186969\pi$
$383$	21.6911i	1.10837i	−0.832395	+	0.554183i	$0.813031\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.51892i	0.229710i
$388$	0	0
$389$	−2.43380	−0.123398	−0.0616992	−	0.998095i	$-0.519652\pi$
$389$	−2.43380	−0.123398	−0.0616992	+	0.998095i	$0.519652\pi$
$390$	0	0
$391$	−3.40101	−0.171997
$392$	0	0
$393$	15.9622i	0.805184i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	31.2780i	1.56980i	0.619622	+	0.784900i	$0.287286\pi$
$397$	31.2780i	1.56980i	−0.619622	+	0.784900i	$0.712714\pi$
$398$	0	0
$399$	8.48108	0.424585
$400$	0	0
$401$	−8.95710	−0.447296	−0.223648	−	0.974670i	$-0.571797\pi$
$401$	−8.95710	−0.447296	−0.223648	+	0.974670i	$0.571797\pi$
$402$	0	0
$403$	− 20.2479i	− 1.00862i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.91661i	0.194139i
$408$	0	0
$409$	24.6412	1.21843	0.609215	−	0.793005i	$-0.291485\pi$
$409$	24.6412	1.21843	0.609215	+	0.793005i	$0.291485\pi$
$410$	0	0
$411$	22.1223	1.09121
$412$	0	0
$413$	− 41.2376i	− 2.02917i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 20.8520i	− 1.02112i
$418$	0	0
$419$	−13.9199	−0.680033	−0.340017	−	0.940419i	$-0.610433\pi$
$419$	−13.9199	−0.680033	−0.340017	+	0.940419i	$0.610433\pi$
$420$	0	0
$421$	4.91993	0.239783	0.119891	−	0.992787i	$-0.461745\pi$
$421$	4.91993	0.239783	0.119891	+	0.992787i	$0.461745\pi$
$422$	0	0
$423$	33.2024i	1.61435i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	5.48108i	0.265248i
$428$	0	0
$429$	4.48108	0.216349
$430$	0	0
$431$	20.2875	0.977214	0.488607	−	0.872504i	$-0.337505\pi$
$431$	20.2875	0.977214	0.488607	+	0.872504i	$0.337505\pi$
$432$	0	0
$433$	− 20.9769i	− 1.00808i	−0.863679	−	0.504042i	$-0.831845\pi$
$433$	− 20.9769i	− 1.00808i	0.863679	−	0.504042i	$-0.168155\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 1.16784i	− 0.0558653i
$438$	0	0
$439$	18.7591	0.895324	0.447662	−	0.894203i	$-0.352257\pi$
$439$	18.7591	0.895324	0.447662	+	0.894203i	$0.352257\pi$
$440$	0	0
$441$	8.11791	0.386567
$442$	0	0
$443$	40.4828i	1.92340i	0.274110	+	0.961698i	$0.411617\pi$
$443$	40.4828i	1.92340i	−0.274110	+	0.961698i	$0.588383\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	53.7109i	2.54044i
$448$	0	0
$449$	19.2402	0.908001	0.454001	−	0.891001i	$-0.349996\pi$
$449$	19.2402	0.908001	0.454001	+	0.891001i	$0.349996\pi$
$450$	0	0
$451$	5.67906	0.267416
$452$	0	0
$453$	− 27.0198i	− 1.26950i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	8.44829i	0.395195i	0.980283	+	0.197597i	$0.0633138\pi$
$457$	8.44829i	0.395195i	−0.980283	+	0.197597i	$0.936686\pi$
$458$	0	0
$459$	−21.0422	−0.982167
$460$	0	0
$461$	−25.2780	−1.17732	−0.588658	−	0.808382i	$-0.700343\pi$
$461$	−25.2780	−1.17732	−0.588658	+	0.808382i	$0.700343\pi$
$462$	0	0
$463$	7.15749i	0.332637i	0.986072	+	0.166318i	$0.0531879\pi$
$463$	7.15749i	0.332637i	−0.986072	+	0.166318i	$0.946812\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 1.71690i	− 0.0794486i	−0.999211	−	0.0397243i	$-0.987352\pi$
$467$	− 1.71690i	− 0.0794486i	0.999211	−	0.0397243i	$-0.0126480\pi$
$468$	0	0
$469$	−14.8821	−0.687191
$470$	0	0
$471$	−41.3253	−1.90417
$472$	0	0
$473$	− 0.493841i	− 0.0227068i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	4.51892i	0.206907i
$478$	0	0
$479$	31.6885	1.44788	0.723942	−	0.689861i	$-0.242328\pi$
$479$	31.6885	1.44788	0.723942	+	0.689861i	$0.242328\pi$
$480$	0	0
$481$	16.7970	0.765876
$482$	0	0
$483$	− 9.90453i	− 0.450672i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 1.08513i	− 0.0491717i	−0.999698	−	0.0245859i	$-0.992173\pi$
$487$	− 1.08513i	− 0.0491717i	0.999698	−	0.0245859i	$-0.00782671\pi$
$488$	0	0
$489$	73.3297	3.31608
$490$	0	0
$491$	−10.9149	−0.492581	−0.246291	−	0.969196i	$-0.579212\pi$
$491$	−10.9149	−0.492581	−0.246291	+	0.969196i	$0.579212\pi$
$492$	0	0
$493$	− 4.08007i	− 0.183757i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	34.8365i	1.56263i
$498$	0	0
$499$	−38.8814	−1.74057	−0.870286	−	0.492547i	$-0.836066\pi$
$499$	−38.8814	−1.74057	−0.870286	+	0.492547i	$0.836066\pi$
$500$	0	0
$501$	32.1223	1.43512
$502$	0	0
$503$	20.3306i	0.906497i	0.891384	+	0.453249i	$0.149735\pi$
$503$	20.3306i	0.906497i	−0.891384	+	0.453249i	$0.850265\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	18.6412i	0.827885i
$508$	0	0
$509$	−10.5189	−0.466243	−0.233121	−	0.972448i	$-0.574894\pi$
$509$	−10.5189	−0.466243	−0.233121	+	0.972448i	$0.574894\pi$
$510$	0	0
$511$	3.00000	0.132712
$512$	0	0
$513$	− 7.22547i	− 0.319012i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 3.62845i	− 0.159579i
$518$	0	0
$519$	−44.5655	−1.95621
$520$	0	0
$521$	−25.7642	−1.12875	−0.564375	−	0.825519i	$-0.690883\pi$
$521$	−25.7642	−1.12875	−0.564375	+	0.825519i	$0.690883\pi$
$522$	0	0
$523$	30.8814i	1.35035i	0.737658	+	0.675175i	$0.235932\pi$
$523$	30.8814i	1.35035i	−0.737658	+	0.675175i	$0.764068\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 22.9545i	− 0.999912i
$528$	0	0
$529$	21.6362	0.940702
$530$	0	0
$531$	−77.6128	−3.36811
$532$	0	0
$533$	− 24.3555i	− 1.05495i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	43.4630i	1.87557i
$538$	0	0
$539$	−0.887149	−0.0382122
$540$	0	0
$541$	13.3253	0.572901	0.286450	−	0.958095i	$-0.407525\pi$
$541$	13.3253	0.572901	0.286450	+	0.958095i	$0.407525\pi$
$542$	0	0
$543$	11.0724i	0.475161i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 26.8821i	− 1.14940i	−0.818366	−	0.574698i	$-0.805120\pi$
$547$	− 26.8821i	− 1.14940i	0.818366	−	0.574698i	$-0.194880\pi$
$548$	0	0
$549$	10.3159	0.440271
$550$	0	0
$551$	1.40101	0.0596851
$552$	0	0
$553$	− 44.7411i	− 1.90258i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	27.3082i	1.15708i	0.815652	+	0.578542i	$0.196378\pi$
$557$	27.3082i	1.15708i	−0.815652	+	0.578542i	$0.803622\pi$
$558$	0	0
$559$	−2.11791	−0.0895780
$560$	0	0
$561$	5.08007	0.214481
$562$	0	0
$563$	− 40.3324i	− 1.69981i	−0.526939	−	0.849903i	$-0.676660\pi$
$563$	− 40.3324i	− 1.69981i	0.526939	−	0.849903i	$-0.323340\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 13.3933i	− 0.562466i
$568$	0	0
$569$	−11.8399	−0.496353	−0.248176	−	0.968715i	$-0.579831\pi$
$569$	−11.8399	−0.496353	−0.248176	+	0.968715i	$0.579831\pi$
$570$	0	0
$571$	−1.71690	−0.0718499	−0.0359250	−	0.999354i	$-0.511438\pi$
$571$	−1.71690	−0.0718499	−0.0359250	+	0.999354i	$0.511438\pi$
$572$	0	0
$573$	− 30.8667i	− 1.28947i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	0.0552190i	0.00229880i	0.999999	+	0.00114940i	$0.000365865\pi$
$577$	0.0552190i	0.00229880i	−0.999999	+	0.00114940i	$0.999634\pi$
$578$	0	0
$579$	−64.8486	−2.69502
$580$	0	0
$581$	−45.8865	−1.90369
$582$	0	0
$583$	− 0.493841i	− 0.0204528i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	11.4010i	0.470570i	0.971926	+	0.235285i	$0.0756024\pi$
$587$	11.4010i	0.470570i	−0.971926	+	0.235285i	$0.924398\pi$
$588$	0	0
$589$	7.88209	0.324776
$590$	0	0
$591$	−44.1645	−1.81669
$592$	0	0
$593$	36.8288i	1.51238i	0.654353	+	0.756190i	$0.272941\pi$
$593$	36.8288i	1.51238i	−0.654353	+	0.756190i	$0.727059\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 47.6533i	− 1.95032i
$598$	0	0
$599$	22.7591	0.929913	0.464956	−	0.885334i	$-0.346070\pi$
$599$	22.7591	0.929913	0.464956	+	0.885334i	$0.346070\pi$
$600$	0	0
$601$	−36.5277	−1.49000	−0.744998	−	0.667067i	$-0.767549\pi$
$601$	−36.5277	−1.49000	−0.744998	+	0.667067i	$0.767549\pi$
$602$	0	0
$603$	28.0094i	1.14063i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 10.1230i	− 0.410879i	−0.978670	−	0.205439i	$-0.934138\pi$
$607$	− 10.1230i	− 0.410879i	0.978670	−	0.205439i	$-0.0658623\pi$
$608$	0	0
$609$	11.8821	0.481487
$610$	0	0
$611$	−15.5611	−0.629537
$612$	0	0
$613$	35.5930i	1.43759i	0.695223	+	0.718794i	$0.255306\pi$
$613$	35.5930i	1.43759i	−0.695223	+	0.718794i	$0.744694\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 13.2134i	− 0.531951i	−0.963980	−	0.265975i	$-0.914306\pi$
$617$	− 13.2134i	− 0.531951i	0.963980	−	0.265975i	$-0.0856940\pi$
$618$	0	0
$619$	23.2453	0.934306	0.467153	−	0.884177i	$-0.345280\pi$
$619$	23.2453	0.934306	0.467153	+	0.884177i	$0.345280\pi$
$620$	0	0
$621$	−8.43818	−0.338612
$622$	0	0
$623$	44.9742i	1.80186i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1.74439i	0.0696643i
$628$	0	0
$629$	19.0422	0.759263
$630$	0	0
$631$	3.88715	0.154745	0.0773725	−	0.997002i	$-0.475347\pi$
$631$	3.88715	0.154745	0.0773725	+	0.997002i	$0.475347\pi$
$632$	0	0
$633$	− 16.8968i	− 0.671588i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	3.80467i	0.150746i
$638$	0	0
$639$	65.5655	2.59373
$640$	0	0
$641$	−34.1223	−1.34775	−0.673875	−	0.738846i	$-0.735371\pi$
$641$	−34.1223	−1.34775	−0.673875	+	0.738846i	$0.735371\pi$
$642$	0	0
$643$	31.9243i	1.25897i	0.777012	+	0.629486i	$0.216735\pi$
$643$	31.9243i	1.25897i	−0.777012	+	0.629486i	$0.783265\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 35.3434i	− 1.38949i	−0.719255	−	0.694746i	$-0.755517\pi$
$647$	− 35.3434i	− 1.38949i	0.719255	−	0.694746i	$-0.244483\pi$
$648$	0	0
$649$	8.48175	0.332938
$650$	0	0
$651$	66.8486	2.62000
$652$	0	0
$653$	27.5431i	1.07784i	0.842356	+	0.538922i	$0.181168\pi$
$653$	27.5431i	1.07784i	−0.842356	+	0.538922i	$0.818832\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 5.64627i	− 0.220282i
$658$	0	0
$659$	−29.6740	−1.15593	−0.577967	−	0.816060i	$-0.696154\pi$
$659$	−29.6740	−1.15593	−0.577967	+	0.816060i	$0.696154\pi$
$660$	0	0
$661$	−16.3159	−0.634614	−0.317307	−	0.948323i	$-0.602779\pi$
$661$	−16.3159	−0.634614	−0.317307	+	0.948323i	$0.602779\pi$
$662$	0	0
$663$	− 21.7866i	− 0.846122i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 1.63615i	− 0.0633522i
$668$	0	0
$669$	29.5233	1.14144
$670$	0	0
$671$	−1.12735	−0.0435209
$672$	0	0
$673$	13.6945i	0.527883i	0.964539	+	0.263941i	$0.0850225\pi$
$673$	13.6945i	0.527883i	−0.964539	+	0.263941i	$0.914977\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 18.2426i	− 0.701121i	−0.936540	−	0.350560i	$-0.885991\pi$
$677$	− 18.2426i	− 0.701121i	0.936540	−	0.350560i	$-0.114009\pi$
$678$	0	0
$679$	−22.4432	−0.861292
$680$	0	0
$681$	42.2824	1.62027
$682$	0	0
$683$	8.92764i	0.341607i	0.985305	+	0.170803i	$0.0546363\pi$
$683$	8.92764i	0.341607i	−0.985305	+	0.170803i	$0.945364\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	79.2195i	3.02241i
$688$	0	0
$689$	−2.11791	−0.0806859
$690$	0	0
$691$	37.5284	1.42765	0.713823	−	0.700326i	$-0.246962\pi$
$691$	37.5284	1.42765	0.713823	+	0.700326i	$0.246962\pi$
$692$	0	0
$693$	9.56115i	0.363198i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 27.6111i	− 1.04584i
$698$	0	0
$699$	4.08007	0.154322
$700$	0	0
$701$	−42.1746	−1.59291	−0.796457	−	0.604695i	$-0.793295\pi$
$701$	−42.1746	−1.59291	−0.796457	+	0.604695i	$0.793295\pi$
$702$	0	0
$703$	6.53871i	0.246612i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 12.8168i	− 0.482024i
$708$	0	0
$709$	4.35373	0.163508	0.0817539	−	0.996653i	$-0.473948\pi$
$709$	4.35373	0.163508	0.0817539	+	0.996653i	$0.473948\pi$
$710$	0	0
$711$	−84.2067	−3.15800
$712$	0	0
$713$	− 9.20500i	− 0.344730i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 28.7789i	− 1.07477i
$718$	0	0
$719$	−14.2780	−0.532481	−0.266241	−	0.963907i	$-0.585782\pi$
$719$	−14.2780	−0.532481	−0.266241	+	0.963907i	$0.585782\pi$
$720$	0	0
$721$	50.5655	1.88316
$722$	0	0
$723$	− 8.98459i	− 0.334141i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	29.7538i	1.10351i	0.834007	+	0.551754i	$0.186041\pi$
$727$	29.7538i	1.10351i	−0.834007	+	0.551754i	$0.813959\pi$
$728$	0	0
$729$	37.9193	1.40442
$730$	0	0
$731$	−2.40101	−0.0888046
$732$	0	0
$733$	16.4580i	0.607889i	0.952690	+	0.303944i	$0.0983037\pi$
$733$	16.4580i	0.607889i	−0.952690	+	0.303944i	$0.901696\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 3.06095i	− 0.112752i
$738$	0	0
$739$	−11.2024	−0.412085	−0.206043	−	0.978543i	$-0.566059\pi$
$739$	−11.2024	−0.412085	−0.206043	+	0.978543i	$0.566059\pi$
$740$	0	0
$741$	7.48108	0.274824
$742$	0	0
$743$	− 28.5053i	− 1.04576i	−0.852407	−	0.522878i	$-0.824858\pi$
$743$	− 28.5053i	− 1.04576i	0.852407	−	0.522878i	$-0.175142\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	86.3625i	3.15984i
$748$	0	0
$749$	−41.3253	−1.50999
$750$	0	0
$751$	−24.9716	−0.911227	−0.455613	−	0.890178i	$-0.650580\pi$
$751$	−24.9716	−0.911227	−0.455613	+	0.890178i	$0.650580\pi$
$752$	0	0
$753$	17.7065i	0.645263i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	12.8717i	0.467831i	0.972257	+	0.233916i	$0.0751540\pi$
$757$	12.8717i	0.467831i	−0.972257	+	0.233916i	$0.924846\pi$
$758$	0	0
$759$	2.03717	0.0739445
$760$	0	0
$761$	−26.6791	−0.967115	−0.483558	−	0.875313i	$-0.660656\pi$
$761$	−26.6791	−0.967115	−0.483558	+	0.875313i	$0.660656\pi$
$762$	0	0
$763$	− 31.0999i	− 1.12589i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 36.3753i	− 1.31343i
$768$	0	0
$769$	−29.7541	−1.07296	−0.536479	−	0.843913i	$-0.680246\pi$
$769$	−29.7541	−1.07296	−0.536479	+	0.843913i	$0.680246\pi$
$770$	0	0
$771$	55.3676	1.99401
$772$	0	0
$773$	50.5655i	1.81872i	0.416015	+	0.909358i	$0.363426\pi$
$773$	50.5655i	1.81872i	−0.416015	+	0.909358i	$0.636574\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	55.4553i	1.98945i
$778$	0	0
$779$	9.48108	0.339695
$780$	0	0
$781$	−7.16519	−0.256391
$782$	0	0
$783$	− 10.1230i	− 0.361765i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	54.8530i	1.95530i	0.210242	+	0.977649i	$0.432575\pi$
$787$	54.8530i	1.95530i	−0.210242	+	0.977649i	$0.567425\pi$
$788$	0	0
$789$	1.32094	0.0470269
$790$	0	0
$791$	24.0422	0.854843
$792$	0	0
$793$	4.83481i	0.171689i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 30.3434i	− 1.07482i	−0.843322	−	0.537409i	$-0.819403\pi$
$797$	− 30.3434i	− 1.07482i	0.843322	−	0.537409i	$-0.180597\pi$
$798$	0	0
$799$	−17.6412	−0.624101
$800$	0	0
$801$	84.6456	2.99081
$802$	0	0
$803$	0.617041i	0.0217749i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 69.4399i	− 2.44440i
$808$	0	0
$809$	11.0801	0.389554	0.194777	−	0.980848i	$-0.437602\pi$
$809$	11.0801	0.389554	0.194777	+	0.980848i	$0.437602\pi$
$810$	0	0
$811$	39.3625	1.38220	0.691102	−	0.722757i	$-0.257126\pi$
$811$	39.3625	1.38220	0.691102	+	0.722757i	$0.257126\pi$
$812$	0	0
$813$	37.1575i	1.30317i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 0.824458i	− 0.0288441i
$818$	0	0
$819$	41.0044	1.43281
$820$	0	0
$821$	34.8493	1.21625	0.608125	−	0.793842i	$-0.291922\pi$
$821$	34.8493	1.21625	0.608125	+	0.793842i	$0.291922\pi$
$822$	0	0
$823$	− 22.7615i	− 0.793417i	−0.917945	−	0.396709i	$-0.870152\pi$
$823$	− 22.7615i	− 0.793417i	0.917945	−	0.396709i	$-0.129848\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 32.7615i	− 1.13923i	−0.821912	−	0.569615i	$-0.807092\pi$
$827$	− 32.7615i	− 1.13923i	0.821912	−	0.569615i	$-0.192908\pi$
$828$	0	0
$829$	−36.4905	−1.26737	−0.633684	−	0.773592i	$-0.718458\pi$
$829$	−36.4905	−1.26737	−0.633684	+	0.773592i	$0.718458\pi$
$830$	0	0
$831$	24.1652	0.838281
$832$	0	0
$833$	4.31324i	0.149445i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 56.9518i	− 1.96854i
$838$	0	0
$839$	−47.4949	−1.63971	−0.819853	−	0.572574i	$-0.805945\pi$
$839$	−47.4949	−1.63971	−0.819853	+	0.572574i	$0.805945\pi$
$840$	0	0
$841$	−27.0372	−0.932316
$842$	0	0
$843$	− 26.7866i	− 0.922580i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	30.9897i	1.06482i
$848$	0	0
$849$	57.8865	1.98666
$850$	0	0
$851$	7.63615	0.261764
$852$	0	0
$853$	1.86736i	0.0639372i	0.999489	+	0.0319686i	$0.0101776\pi$
$853$	1.86736i	0.0639372i	−0.999489	+	0.0319686i	$0.989822\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 16.3608i	− 0.558873i	−0.960164	−	0.279436i	$-0.909852\pi$
$857$	− 16.3608i	− 0.558873i	0.960164	−	0.279436i	$-0.0901476\pi$
$858$	0	0
$859$	21.0851	0.719415	0.359708	−	0.933065i	$-0.382876\pi$
$859$	21.0851	0.719415	0.359708	+	0.933065i	$0.382876\pi$
$860$	0	0
$861$	80.4098	2.74036
$862$	0	0
$863$	25.6868i	0.874387i	0.899367	+	0.437194i	$0.144028\pi$
$863$	25.6868i	0.874387i	−0.899367	+	0.437194i	$0.855972\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	24.8091i	0.842560i
$868$	0	0
$869$	9.20236	0.312169
$870$	0	0
$871$	−13.1274	−0.444803
$872$	0	0
$873$	42.2402i	1.42961i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	43.2978i	1.46206i	0.682343	+	0.731032i	$0.260961\pi$
$877$	43.2978i	1.46206i	−0.682343	+	0.731032i	$0.739039\pi$
$878$	0	0
$879$	23.7213	0.800099
$880$	0	0
$881$	22.0750	0.743726	0.371863	−	0.928288i	$-0.378719\pi$
$881$	22.0750	0.743726	0.371863	+	0.928288i	$0.378719\pi$
$882$	0	0
$883$	− 0.385604i	− 0.0129766i	−0.999979	−	0.00648831i	$-0.997935\pi$
$883$	− 0.385604i	− 0.0129766i	0.999979	−	0.00648831i	$-0.00206531\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	54.9210i	1.84407i	0.387111	+	0.922033i	$0.373473\pi$
$887$	54.9210i	1.84407i	−0.387111	+	0.922033i	$0.626527\pi$
$888$	0	0
$889$	35.3253	1.18477
$890$	0	0
$891$	2.75474	0.0922873
$892$	0	0
$893$	− 6.05763i	− 0.202711i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 8.73669i	− 0.291710i
$898$	0	0
$899$	11.0429	0.368301
$900$	0	0
$901$	−2.40101	−0.0799893
$902$	0	0
$903$	− 6.99230i	− 0.232689i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 41.4727i	− 1.37708i	−0.725199	−	0.688539i	$-0.758252\pi$
$907$	− 41.4727i	− 1.37708i	0.725199	−	0.688539i	$-0.241748\pi$
$908$	0	0
$909$	−24.1223	−0.800086
$910$	0	0
$911$	41.8909	1.38791	0.693953	−	0.720020i	$-0.255868\pi$
$911$	41.8909	1.38791	0.693953	+	0.720020i	$0.255868\pi$
$912$	0	0
$913$	− 9.43795i	− 0.312350i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 15.9622i	− 0.527117i
$918$	0	0
$919$	36.0044	1.18767	0.593837	−	0.804585i	$-0.297612\pi$
$919$	36.0044	1.18767	0.593837	+	0.804585i	$0.297612\pi$
$920$	0	0
$921$	31.3253	1.03220
$922$	0	0
$923$	30.7290i	1.01146i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 95.1689i	− 3.12576i
$928$	0	0
$929$	−59.4898	−1.95180	−0.975899	−	0.218222i	$-0.929974\pi$
$929$	−59.4898	−1.95180	−0.975899	+	0.218222i	$0.929974\pi$
$930$	0	0
$931$	−1.48108	−0.0485404
$932$	0	0
$933$	− 89.8255i	− 2.94076i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	35.7143i	1.16673i	0.812209	+	0.583367i	$0.198265\pi$
$937$	35.7143i	1.16673i	−0.812209	+	0.583367i	$0.801735\pi$
$938$	0	0
$939$	−19.0422	−0.621420
$940$	0	0
$941$	20.2074	0.658743	0.329371	−	0.944200i	$-0.393163\pi$
$941$	20.2074	0.658743	0.329371	+	0.944200i	$0.393163\pi$
$942$	0	0
$943$	− 11.0724i	− 0.360566i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 51.4982i	− 1.67347i	−0.547610	−	0.836734i	$-0.684462\pi$
$947$	− 51.4982i	− 1.67347i	0.547610	−	0.836734i	$-0.315538\pi$
$948$	0	0
$949$	2.64627	0.0859016
$950$	0	0
$951$	32.9243	1.06764
$952$	0	0
$953$	7.95272i	0.257614i	0.991670	+	0.128807i	$0.0411147\pi$
$953$	7.95272i	0.257614i	−0.991670	+	0.128807i	$0.958885\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	2.44391i	0.0790005i
$958$	0	0
$959$	−22.1223	−0.714366
$960$	0	0
$961$	31.1274	1.00411
$962$	0	0
$963$	77.7780i	2.50636i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 32.5706i	− 1.04740i	−0.851903	−	0.523700i	$-0.824551\pi$
$967$	− 32.5706i	− 1.04740i	0.851903	−	0.523700i	$-0.175449\pi$
$968$	0	0
$969$	8.48108	0.272452
$970$	0	0
$971$	21.0378	0.675136	0.337568	−	0.941301i	$-0.390396\pi$
$971$	21.0378	0.675136	0.337568	+	0.941301i	$0.390396\pi$
$972$	0	0
$973$	20.8520i	0.668483i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 46.9648i	− 1.50254i	−0.659997	−	0.751269i	$-0.729442\pi$
$977$	− 46.9648i	− 1.50254i	0.659997	−	0.751269i	$-0.270558\pi$
$978$	0	0
$979$	−9.25032	−0.295641
$980$	0	0
$981$	−58.5327	−1.86881
$982$	0	0
$983$	− 18.5182i	− 0.590640i	−0.955398	−	0.295320i	$-0.904574\pi$
$983$	− 18.5182i	− 0.590640i	0.955398	−	0.295320i	$-0.0954263\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 51.3753i	− 1.63529i
$988$	0	0
$989$	−0.962834	−0.0306163
$990$	0	0
$991$	−5.40539	−0.171708	−0.0858540	−	0.996308i	$-0.527362\pi$
$991$	−5.40539	−0.171708	−0.0858540	+	0.996308i	$0.527362\pi$
$992$	0	0
$993$	66.4175i	2.10770i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 8.55676i	− 0.270995i	−0.990778	−	0.135498i	$-0.956737\pi$
$997$	− 8.55676i	− 0.270995i	0.990778	−	0.135498i	$-0.0432633\pi$
$998$	0	0
$999$	47.2453	1.49477

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1900.2.c.g.1749.6		6
5.2	odd	4		1900.2.a.h.1.3	yes	3
5.3	odd	4		1900.2.a.f.1.1	✓	3
5.4	even	2	inner	1900.2.c.g.1749.1		6
20.3	even	4		7600.2.a.by.1.3		3
20.7	even	4		7600.2.a.bj.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1900.2.a.f.1.1	✓	3	5.3	odd	4
1900.2.a.h.1.3	yes	3	5.2	odd	4
1900.2.c.g.1749.1		6	5.4	even	2	inner
1900.2.c.g.1749.6		6	1.1	even	1	trivial
7600.2.a.bj.1.1		3	20.7	even	4
7600.2.a.by.1.3		3	20.3	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:	\( N \)	\(=\)	\( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1900.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.91223i q^{3} -2.91223i q^{7} -5.48108 q^{9} +O(q^{10})\)	\(q+2.91223i q^{3} -2.91223i q^{7} -5.48108 q^{9} +0.598988 q^{11} -2.56885i q^{13} -2.91223i q^{17} +1.00000 q^{19} +8.48108 q^{21} -1.16784i q^{23} -7.22547i q^{27} +1.40101 q^{29} +7.88209 q^{31} +1.74439i q^{33} +6.53871i q^{37} +7.48108 q^{39} +9.48108 q^{41} -0.824458i q^{43} -6.05763i q^{47} -1.48108 q^{49} +8.48108 q^{51} -0.824458i q^{53} +2.91223i q^{57} +14.1601 q^{59} -1.88209 q^{61} +15.9622i q^{63} -5.11021i q^{67} +3.40101 q^{69} -11.9622 q^{71} +1.03014i q^{73} -1.74439i q^{77} +15.3632 q^{79} +4.59899 q^{81} -15.7565i q^{83} +4.08007i q^{87} -15.4432 q^{89} -7.48108 q^{91} +22.9545i q^{93} -7.70655i q^{97} -3.28310 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{9}+O(q^{10}) \) 6 * q - 2 * q^9	\( 6 q - 2 q^{9} - 2 q^{11} + 6 q^{19} + 20 q^{21} + 14 q^{29} + 22 q^{31} + 14 q^{39} + 26 q^{41} + 22 q^{49} + 20 q^{51} + 12 q^{59} + 14 q^{61} + 26 q^{69} - 10 q^{71} + 36 q^{79} + 22 q^{81} - 14 q^{91}+O(q^{100}) \) 6 * q - 2 * q^9 - 2 * q^11 + 6 * q^19 + 20 * q^21 + 14 * q^29 + 22 * q^31 + 14 * q^39 + 26 * q^41 + 22 * q^49 + 20 * q^51 + 12 * q^59 + 14 * q^61 + 26 * q^69 - 10 * q^71 + 36 * q^79 + 22 * q^81 - 14 * q^91

Introduction

L-functions

Modular forms

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Database

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