LMFDB - Embedded newform 1900.3.e.e.1101.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,3,Mod(1101,1900)]

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, 0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1900.1101");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1900 = 2^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1900.e (of order $2$, degree $1$, 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:	$51.7712502285$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 20x^{10} + 44x^{8} + 270x^{6} + 36676x^{4} + 71664x^{2} + 687241 $ x^12 - 20x^10 + 44x^8 + 270x^6 + 36676x^4 + 71664*x^2 + 687241
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{22} $
Twist minimal:	no (minimal twist has level 380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1101.7
Root			$4.19028 - 0.975196i$ of defining polynomial
Character	$\chi$	$=$	1900.1101
Dual form			1900.3.e.e.1101.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.95039i q^{3} -2.18749 q^{7} +5.19597 q^{9} +O(q^{10})$	$q+1.95039i q^{3} -2.18749 q^{7} +5.19597 q^{9} -9.50946 q^{11} -20.4976i q^{13} -6.15792 q^{17} +(-9.68651 - 16.3454i) q^{19} -4.26647i q^{21} +10.9375 q^{23} +27.6877i q^{27} +28.4243i q^{29} +24.8098i q^{31} -18.5472i q^{33} +37.4397i q^{37} +39.9783 q^{39} +81.6583i q^{41} +70.8743 q^{43} +46.1193 q^{47} -44.2149 q^{49} -12.0104i q^{51} +32.1999i q^{53} +(31.8799 - 18.8925i) q^{57} -94.3206i q^{59} -102.566 q^{61} -11.3661 q^{63} +69.0282i q^{67} +21.3323i q^{69} +40.5718i q^{71} -102.209 q^{73} +20.8018 q^{77} +73.9145i q^{79} -7.23816 q^{81} +91.9574 q^{83} -55.4385 q^{87} +57.3634i q^{89} +44.8382i q^{91} -48.3888 q^{93} -84.3229i q^{97} -49.4109 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 76 q^{9}+O(q^{10}) $ 12 * q - 76 * q^9	$ 12 q - 76 q^{9} - 24 q^{11} - 68 q^{19} + 264 q^{39} - 212 q^{49} - 600 q^{61} + 492 q^{81} - 136 q^{99}+O(q^{100}) $ 12 * q - 76 * q^9 - 24 * q^11 - 68 * q^19 + 264 * q^39 - 212 * q^49 - 600 * q^61 + 492 * q^81 - 136 * q^99

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
$3$			1.95039i			0.650131i	0.945692	+	0.325065i	$0.105386\pi$
$3$			1.95039i			0.650131i	−0.945692	+	0.325065i	$0.894614\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−2.18749			−0.312499			−0.156249	−	0.987718i	$-0.549940\pi$
$7$	−2.18749			−0.312499			−0.156249	+	0.987718i	$0.549940\pi$
$8$		0			0
$9$	5.19597			0.577330
$10$		0			0
$11$	−9.50946			−0.864496			−0.432248	−	0.901755i	$-0.642280\pi$
$11$	−9.50946			−0.864496			−0.432248	+	0.901755i	$0.642280\pi$
$12$		0			0
$13$		−	20.4976i		−	1.57674i	−0.615204	−	0.788368i	$-0.710926\pi$
$13$		−	20.4976i		−	1.57674i	0.615204	−	0.788368i	$-0.289074\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−6.15792			−0.362231			−0.181115	−	0.983462i	$-0.557971\pi$
$17$	−6.15792			−0.362231			−0.181115	+	0.983462i	$0.557971\pi$
$18$		0			0
$19$	−9.68651	−	16.3454i	−0.509816	−	0.860283i
$19$	−9.68651	−	16.3454i	−0.509816	−	0.860283i
$20$		0			0
$21$		−	4.26647i		−	0.203165i
$22$		0			0
$23$	10.9375			0.475541			0.237771	−	0.971321i	$-0.423583\pi$
$23$	10.9375			0.475541			0.237771	+	0.971321i	$0.423583\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$			27.6877i			1.02547i
$28$		0			0
$29$			28.4243i			0.980148i	0.871681	+	0.490074i	$0.163030\pi$
$29$			28.4243i			0.980148i	−0.871681	+	0.490074i	$0.836970\pi$
$30$		0			0
$31$			24.8098i			0.800315i	0.916446	+	0.400157i	$0.131045\pi$
$31$			24.8098i			0.800315i	−0.916446	+	0.400157i	$0.868955\pi$
$32$		0			0
$33$		−	18.5472i		−	0.562035i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			37.4397i			1.01188i	0.862568	+	0.505941i	$0.168855\pi$
$37$			37.4397i			1.01188i	−0.862568	+	0.505941i	$0.831145\pi$
$38$		0			0
$39$	39.9783			1.02508
$40$		0			0
$41$			81.6583i			1.99167i	0.0911903	+	0.995833i	$0.470933\pi$
$41$			81.6583i			1.99167i	−0.0911903	+	0.995833i	$0.529067\pi$
$42$		0			0
$43$	70.8743			1.64824			0.824120	−	0.566415i	$-0.191670\pi$
$43$	70.8743			1.64824			0.824120	+	0.566415i	$0.191670\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	46.1193			0.981261			0.490631	−	0.871368i	$-0.336766\pi$
$47$	46.1193			0.981261			0.490631	+	0.871368i	$0.336766\pi$
$48$		0			0
$49$	−44.2149			−0.902345
$50$		0			0
$51$		−	12.0104i		−	0.235497i
$52$		0			0
$53$			32.1999i			0.607546i	0.952745	+	0.303773i	$0.0982464\pi$
$53$			32.1999i			0.607546i	−0.952745	+	0.303773i	$0.901754\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	31.8799	−	18.8925i	0.559297	−	0.331447i
$58$		0			0
$59$		−	94.3206i		−	1.59865i	−0.600896	−	0.799327i	$-0.705189\pi$
$59$		−	94.3206i		−	1.59865i	0.600896	−	0.799327i	$-0.294811\pi$
$60$		0			0
$61$	−102.566			−1.68141			−0.840707	−	0.541491i	$-0.817860\pi$
$61$	−102.566			−1.68141			−0.840707	+	0.541491i	$0.817860\pi$
$62$		0			0
$63$	−11.3661			−0.180415
$64$		0			0
$65$		0			0
$66$		0			0
$67$			69.0282i			1.03027i	0.857109	+	0.515136i	$0.172258\pi$
$67$			69.0282i			1.03027i	−0.857109	+	0.515136i	$0.827742\pi$
$68$		0			0
$69$			21.3323i			0.309164i
$70$		0			0
$71$			40.5718i			0.571433i	0.958314	+	0.285717i	$0.0922316\pi$
$71$			40.5718i			0.571433i	−0.958314	+	0.285717i	$0.907768\pi$
$72$		0			0
$73$	−102.209			−1.40012			−0.700062	−	0.714082i	$-0.746844\pi$
$73$	−102.209			−1.40012			−0.700062	+	0.714082i	$0.746844\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	20.8018			0.270154
$78$		0			0
$79$			73.9145i			0.935626i	0.883827	+	0.467813i	$0.154958\pi$
$79$			73.9145i			0.935626i	−0.883827	+	0.467813i	$0.845042\pi$
$80$		0			0
$81$	−7.23816			−0.0893600
$82$		0			0
$83$	91.9574			1.10792			0.553960	−	0.832543i	$-0.313116\pi$
$83$	91.9574			1.10792			0.553960	+	0.832543i	$0.313116\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−55.4385			−0.637224
$88$		0			0
$89$			57.3634i			0.644533i	0.946649	+	0.322266i	$0.104445\pi$
$89$			57.3634i			0.644533i	−0.946649	+	0.322266i	$0.895555\pi$
$90$		0			0
$91$			44.8382i			0.492728i
$92$		0			0
$93$	−48.3888			−0.520309
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	84.3229i		−	0.869308i	−0.900598	−	0.434654i	$-0.856871\pi$
$97$		−	84.3229i		−	0.869308i	0.900598	−	0.434654i	$-0.143129\pi$
$98$		0			0
$99$	−49.4109			−0.499100
$100$		0			0
$101$	69.2688			0.685830			0.342915	−	0.939366i	$-0.388586\pi$
$101$	69.2688			0.685830			0.342915	+	0.939366i	$0.388586\pi$
$102$		0			0
$103$		−	100.192i		−	0.972739i	−0.873753	−	0.486369i	$-0.838321\pi$
$103$		−	100.192i		−	0.972739i	0.873753	−	0.486369i	$-0.161679\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	52.7344i		−	0.492845i	−0.969163	−	0.246422i	$-0.920745\pi$
$107$		−	52.7344i		−	0.492845i	0.969163	−	0.246422i	$-0.0792550\pi$
$108$		0			0
$109$			76.7400i			0.704036i	0.935993	+	0.352018i	$0.114504\pi$
$109$			76.7400i			0.704036i	−0.935993	+	0.352018i	$0.885496\pi$
$110$		0			0
$111$	−73.0220			−0.657856
$112$		0			0
$113$			210.252i			1.86064i	0.366749	+	0.930320i	$0.380471\pi$
$113$			210.252i			1.86064i	−0.366749	+	0.930320i	$0.619529\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		−	106.505i		−	0.910297i
$118$		0			0
$119$	13.4704			0.113197
$120$		0			0
$121$	−30.5702			−0.252647
$122$		0			0
$123$	−159.266			−1.29484
$124$		0			0
$125$		0			0
$126$		0			0
$127$			19.9230i			0.156874i	0.996919	+	0.0784371i	$0.0249930\pi$
$127$			19.9230i			0.156874i	−0.996919	+	0.0784371i	$0.975007\pi$
$128$		0			0
$129$			138.233i			1.07157i
$130$		0			0
$131$	−3.29737			−0.0251708			−0.0125854	−	0.999921i	$-0.504006\pi$
$131$	−3.29737			−0.0251708			−0.0125854	+	0.999921i	$0.504006\pi$
$132$		0			0
$133$	21.1892	+	35.7554i	0.159317	+	0.268837i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	148.188			1.08166			0.540831	−	0.841131i	$-0.318110\pi$
$137$	148.188			1.08166			0.540831	+	0.841131i	$0.318110\pi$
$138$		0			0
$139$	54.2729			0.390452			0.195226	−	0.980758i	$-0.437456\pi$
$139$	54.2729			0.390452			0.195226	+	0.980758i	$0.437456\pi$
$140$		0			0
$141$			89.9507i			0.637948i
$142$		0			0
$143$			194.921i			1.36308i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		−	86.2364i		−	0.586642i
$148$		0			0
$149$	−160.700			−1.07852			−0.539262	−	0.842138i	$-0.681297\pi$
$149$	−160.700			−1.07852			−0.539262	+	0.842138i	$0.681297\pi$
$150$		0			0
$151$			285.273i			1.88922i	0.328192	+	0.944611i	$0.393561\pi$
$151$			285.273i			1.88922i	−0.328192	+	0.944611i	$0.606439\pi$
$152$		0			0
$153$	−31.9964			−0.209127
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−189.486			−1.20692			−0.603459	−	0.797394i	$-0.706211\pi$
$157$	−189.486			−1.20692			−0.603459	+	0.797394i	$0.706211\pi$
$158$		0			0
$159$	−62.8025			−0.394984
$160$		0			0
$161$	−23.9256			−0.148606
$162$		0			0
$163$	152.943			0.938302			0.469151	−	0.883118i	$-0.344560\pi$
$163$	152.943			0.938302			0.469151	+	0.883118i	$0.344560\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			89.7126i			0.537201i	0.963252	+	0.268601i	$0.0865612\pi$
$167$			89.7126i			0.537201i	−0.963252	+	0.268601i	$0.913439\pi$
$168$		0			0
$169$	−251.150			−1.48610
$170$		0			0
$171$	−50.3308	−	84.9301i	−0.294332	−	0.496667i
$172$		0			0
$173$		−	154.463i		−	0.892851i	−0.894821	−	0.446425i	$-0.852697\pi$
$173$		−	154.463i		−	0.892851i	0.894821	−	0.446425i	$-0.147303\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	183.962			1.03933
$178$		0			0
$179$			138.885i			0.775892i	0.921682	+	0.387946i	$0.126815\pi$
$179$			138.885i			0.775892i	−0.921682	+	0.387946i	$0.873185\pi$
$180$		0			0
$181$			106.605i			0.588979i	0.955655	+	0.294490i	$0.0951496\pi$
$181$			106.605i			0.588979i	−0.955655	+	0.294490i	$0.904850\pi$
$182$		0			0
$183$		−	200.044i		−	1.09314i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	58.5585			0.313147
$188$		0			0
$189$		−	60.5666i		−	0.320458i
$190$		0			0
$191$	−262.954			−1.37672			−0.688362	−	0.725367i	$-0.741670\pi$
$191$	−262.954			−1.37672			−0.688362	+	0.725367i	$0.741670\pi$
$192$		0			0
$193$			34.0711i			0.176534i	0.996097	+	0.0882670i	$0.0281329\pi$
$193$			34.0711i			0.176534i	−0.996097	+	0.0882670i	$0.971867\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−293.544			−1.49007			−0.745035	−	0.667026i	$-0.767567\pi$
$197$	−293.544			−1.49007			−0.745035	+	0.667026i	$0.767567\pi$
$198$		0			0
$199$	144.876			0.728018			0.364009	−	0.931395i	$-0.381408\pi$
$199$	144.876			0.728018			0.364009	+	0.931395i	$0.381408\pi$
$200$		0			0
$201$	−134.632			−0.669811
$202$		0			0
$203$		−	62.1779i		−	0.306295i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	56.8307			0.274544
$208$		0			0
$209$	92.1135	+	155.436i	0.440734	+	0.743711i
$210$		0			0
$211$			355.298i			1.68388i	0.539573	+	0.841939i	$0.318586\pi$
$211$			355.298i			1.68388i	−0.539573	+	0.841939i	$0.681414\pi$
$212$		0			0
$213$	−79.1309			−0.371506
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	54.2711i		−	0.250097i
$218$		0			0
$219$		−	199.348i		−	0.910264i
$220$		0			0
$221$			126.222i			0.571142i
$222$		0			0
$223$		−	330.367i		−	1.48147i	−0.671798	−	0.740734i	$-0.734478\pi$
$223$		−	330.367i		−	1.48147i	0.671798	−	0.740734i	$-0.265522\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	23.4416i		−	0.103267i	−0.998666	−	0.0516335i	$-0.983557\pi$
$227$		−	23.4416i		−	0.103267i	0.998666	−	0.0516335i	$-0.0164428\pi$
$228$		0			0
$229$	183.710			0.802229			0.401114	−	0.916028i	$-0.368623\pi$
$229$	183.710			0.802229			0.401114	+	0.916028i	$0.368623\pi$
$230$		0			0
$231$			40.5718i			0.175635i
$232$		0			0
$233$	143.648			0.616515			0.308258	−	0.951303i	$-0.400254\pi$
$233$	143.648			0.616515			0.308258	+	0.951303i	$0.400254\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−144.162			−0.608279
$238$		0			0
$239$	79.5051			0.332657			0.166329	−	0.986070i	$-0.446809\pi$
$239$	79.5051			0.332657			0.166329	+	0.986070i	$0.446809\pi$
$240$		0			0
$241$			122.504i			0.508317i	0.967163	+	0.254158i	$0.0817984\pi$
$241$			122.504i			0.508317i	−0.967163	+	0.254158i	$0.918202\pi$
$242$		0			0
$243$			235.072i			0.967375i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−335.040	+	198.550i	−1.35644	+	0.803846i
$248$		0			0
$249$			179.353i			0.720293i
$250$		0			0
$251$	19.2335			0.0766274			0.0383137	−	0.999266i	$-0.487801\pi$
$251$	19.2335			0.0766274			0.0383137	+	0.999266i	$0.487801\pi$
$252$		0			0
$253$	−104.009			−0.411104
$254$		0			0
$255$		0			0
$256$		0			0
$257$			60.4198i			0.235097i	0.993067	+	0.117548i	$0.0375035\pi$
$257$			60.4198i			0.235097i	−0.993067	+	0.117548i	$0.962497\pi$
$258$		0			0
$259$		−	81.8989i		−	0.316212i
$260$		0			0
$261$			147.692i			0.565869i
$262$		0			0
$263$	23.1886			0.0881694			0.0440847	−	0.999028i	$-0.485963\pi$
$263$	23.1886			0.0881694			0.0440847	+	0.999028i	$0.485963\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−111.881			−0.419030
$268$		0			0
$269$		−	211.221i		−	0.785208i	−0.919708	−	0.392604i	$-0.871574\pi$
$269$		−	211.221i		−	0.785208i	0.919708	−	0.392604i	$-0.128426\pi$
$270$		0			0
$271$	−371.216			−1.36980			−0.684901	−	0.728637i	$-0.740154\pi$
$271$	−371.216			−1.36980			−0.684901	+	0.728637i	$0.740154\pi$
$272$		0			0
$273$	−87.4521			−0.320337
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−6.90841			−0.0249401			−0.0124701	−	0.999922i	$-0.503969\pi$
$277$	−6.90841			−0.0249401			−0.0124701	+	0.999922i	$0.503969\pi$
$278$		0			0
$279$			128.911i			0.462046i
$280$		0			0
$281$		−	7.46967i		−	0.0265825i	−0.999912	−	0.0132912i	$-0.995769\pi$
$281$		−	7.46967i		−	0.0265825i	0.999912	−	0.0132912i	$-0.00423086\pi$
$282$		0			0
$283$	−64.0548			−0.226342			−0.113171	−	0.993576i	$-0.536101\pi$
$283$	−64.0548			−0.226342			−0.113171	+	0.993576i	$0.536101\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	178.627i		−	0.622393i
$288$		0			0
$289$	−251.080			−0.868789
$290$		0			0
$291$	164.463			0.565164
$292$		0			0
$293$		−	527.510i		−	1.80038i	−0.435502	−	0.900188i	$-0.643429\pi$
$293$		−	527.510i		−	1.80038i	0.435502	−	0.900188i	$-0.356571\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		−	263.295i		−	0.886515i
$298$		0			0
$299$		−	224.191i		−	0.749803i
$300$		0			0
$301$	−155.037			−0.515073
$302$		0			0
$303$			135.101i			0.445879i
$304$		0			0
$305$		0			0
$306$		0			0
$307$			385.063i			1.25428i	0.778908	+	0.627139i	$0.215774\pi$
$307$			385.063i			1.25428i	−0.778908	+	0.627139i	$0.784226\pi$
$308$		0			0
$309$	195.414			0.632407
$310$		0			0
$311$	107.903			0.346956			0.173478	−	0.984838i	$-0.444499\pi$
$311$	107.903			0.346956			0.173478	+	0.984838i	$0.444499\pi$
$312$		0			0
$313$	311.071			0.993838			0.496919	−	0.867797i	$-0.334465\pi$
$313$	311.071			0.993838			0.496919	+	0.867797i	$0.334465\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			213.771i			0.674356i	0.941441	+	0.337178i	$0.109472\pi$
$317$			213.771i			0.674356i	−0.941441	+	0.337178i	$0.890528\pi$
$318$		0			0
$319$		−	270.300i		−	0.847334i
$320$		0			0
$321$	102.853			0.320413
$322$		0			0
$323$	59.6488	+	100.654i	0.184671	+	0.311621i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−149.673			−0.457716
$328$		0			0
$329$	−100.886			−0.306643
$330$		0			0
$331$		−	248.075i		−	0.749471i	−0.927132	−	0.374735i	$-0.877734\pi$
$331$		−	248.075i		−	0.749471i	0.927132	−	0.374735i	$-0.122266\pi$
$332$		0			0
$333$			194.535i			0.584190i
$334$		0			0
$335$		0			0
$336$		0			0
$337$			160.776i			0.477080i	0.971133	+	0.238540i	$0.0766688\pi$
$337$			160.776i			0.477080i	−0.971133	+	0.238540i	$0.923331\pi$
$338$		0			0
$339$	−410.074			−1.20966
$340$		0			0
$341$		−	235.927i		−	0.691869i
$342$		0			0
$343$	203.907			0.594480
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−245.342			−0.707038			−0.353519	−	0.935427i	$-0.615015\pi$
$347$	−245.342			−0.707038			−0.353519	+	0.935427i	$0.615015\pi$
$348$		0			0
$349$	220.492			0.631782			0.315891	−	0.948796i	$-0.397697\pi$
$349$	220.492			0.631782			0.315891	+	0.948796i	$0.397697\pi$
$350$		0			0
$351$	567.531			1.61690
$352$		0			0
$353$	−414.946			−1.17548			−0.587742	−	0.809048i	$-0.699983\pi$
$353$	−414.946			−1.17548			−0.587742	+	0.809048i	$0.699983\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$			26.2726i			0.0735926i
$358$		0			0
$359$	130.150			0.362533			0.181267	−	0.983434i	$-0.441980\pi$
$359$	130.150			0.362533			0.181267	+	0.983434i	$0.441980\pi$
$360$		0			0
$361$	−173.343	+	316.659i	−0.480174	+	0.877173i
$362$		0			0
$363$		−	59.6239i		−	0.164253i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	625.628			1.70471			0.852354	−	0.522965i	$-0.175174\pi$
$367$	625.628			1.70471			0.852354	+	0.522965i	$0.175174\pi$
$368$		0			0
$369$			424.294i			1.14985i
$370$		0			0
$371$		−	70.4370i		−	0.189857i
$372$		0			0
$373$		−	282.836i		−	0.758273i	−0.925341	−	0.379137i	$-0.876221\pi$
$373$		−	282.836i		−	0.758273i	0.925341	−	0.379137i	$-0.123779\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	582.629			1.54543
$378$		0			0
$379$			60.1889i			0.158810i	0.996842	+	0.0794049i	$0.0253020\pi$
$379$			60.1889i			0.158810i	−0.996842	+	0.0794049i	$0.974698\pi$
$380$		0			0
$381$	−38.8577			−0.101989
$382$		0			0
$383$			747.259i			1.95107i	0.219848	+	0.975534i	$0.429444\pi$
$383$			747.259i			1.95107i	−0.219848	+	0.975534i	$0.570556\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	368.261			0.951578
$388$		0			0
$389$	−160.723			−0.413170			−0.206585	−	0.978429i	$-0.566235\pi$
$389$	−160.723			−0.413170			−0.206585	+	0.978429i	$0.566235\pi$
$390$		0			0
$391$	−67.3520			−0.172256
$392$		0			0
$393$		−	6.43117i		−	0.0163643i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	394.796			0.994449			0.497224	−	0.867622i	$-0.334353\pi$
$397$	394.796			0.994449			0.497224	+	0.867622i	$0.334353\pi$
$398$		0			0
$399$	−69.7370	+	41.3272i	−0.174779	+	0.103577i
$400$		0			0
$401$		−	206.496i		−	0.514953i	−0.966285	−	0.257477i	$-0.917109\pi$
$401$		−	206.496i		−	0.514953i	0.966285	−	0.257477i	$-0.0828910\pi$
$402$		0			0
$403$	508.540			1.26188
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	356.031i		−	0.874769i
$408$		0			0
$409$			588.126i			1.43796i	0.695031	+	0.718980i	$0.255391\pi$
$409$			588.126i			1.43796i	−0.695031	+	0.718980i	$0.744609\pi$
$410$		0			0
$411$			289.024i			0.703222i
$412$		0			0
$413$			206.326i			0.499578i
$414$		0			0
$415$		0			0
$416$		0			0
$417$			105.853i			0.253845i
$418$		0			0
$419$	573.699			1.36921			0.684605	−	0.728914i	$-0.259975\pi$
$419$	573.699			1.36921			0.684605	+	0.728914i	$0.259975\pi$
$420$		0			0
$421$		−	17.8321i		−	0.0423566i	−0.999776	−	0.0211783i	$-0.993258\pi$
$421$		−	17.8321i		−	0.0423566i	0.999776	−	0.0211783i	$-0.00674176\pi$
$422$		0			0
$423$	239.634			0.566512
$424$		0			0
$425$		0			0
$426$		0			0
$427$	224.363			0.525439
$428$		0			0
$429$	−380.172			−0.886181
$430$		0			0
$431$		−	424.809i		−	0.985636i	−0.870132	−	0.492818i	$-0.835967\pi$
$431$		−	424.809i		−	0.985636i	0.870132	−	0.492818i	$-0.164033\pi$
$432$		0			0
$433$			309.312i			0.714346i	0.934038	+	0.357173i	$0.116259\pi$
$433$			309.312i			0.714346i	−0.934038	+	0.357173i	$0.883741\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−105.946	−	178.777i	−0.242439	−	0.409100i
$438$		0			0
$439$		−	126.062i		−	0.287158i	−0.989639	−	0.143579i	$-0.954139\pi$
$439$		−	126.062i		−	0.287158i	0.989639	−	0.143579i	$-0.0458612\pi$
$440$		0			0
$441$	−229.739			−0.520951
$442$		0			0
$443$	−602.717			−1.36054			−0.680268	−	0.732964i	$-0.738137\pi$
$443$	−602.717			−1.36054			−0.680268	+	0.732964i	$0.738137\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		−	313.428i		−	0.701182i
$448$		0			0
$449$		−	174.732i		−	0.389157i	−0.980887	−	0.194579i	$-0.937666\pi$
$449$		−	174.732i		−	0.389157i	0.980887	−	0.194579i	$-0.0623339\pi$
$450$		0			0
$451$		−	776.527i		−	1.72179i
$452$		0			0
$453$	−556.393			−1.22824
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−766.792			−1.67788			−0.838941	−	0.544223i	$-0.816825\pi$
$457$	−766.792			−1.67788			−0.838941	+	0.544223i	$0.816825\pi$
$458$		0			0
$459$		−	170.499i		−	0.371457i
$460$		0			0
$461$	−192.820			−0.418264			−0.209132	−	0.977887i	$-0.567064\pi$
$461$	−192.820			−0.418264			−0.209132	+	0.977887i	$0.567064\pi$
$462$		0			0
$463$	440.773			0.951994			0.475997	−	0.879447i	$-0.342087\pi$
$463$	440.773			0.951994			0.475997	+	0.879447i	$0.342087\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	450.985			0.965707			0.482853	−	0.875701i	$-0.339600\pi$
$467$	450.985			0.965707			0.482853	+	0.875701i	$0.339600\pi$
$468$		0			0
$469$		−	150.998i		−	0.321958i
$470$		0			0
$471$		−	369.572i		−	0.784654i
$472$		0			0
$473$	−673.976			−1.42490
$474$		0			0
$475$		0			0
$476$		0			0
$477$			167.310i			0.350754i
$478$		0			0
$479$	−493.345			−1.02995			−0.514974	−	0.857206i	$-0.672198\pi$
$479$	−493.345			−1.02995			−0.514974	+	0.857206i	$0.672198\pi$
$480$		0			0
$481$	767.422			1.59547
$482$		0			0
$483$		−	46.6643i		−	0.0966134i
$484$		0			0
$485$		0			0
$486$		0			0
$487$			111.096i			0.228124i	0.993474	+	0.114062i	$0.0363862\pi$
$487$			111.096i			0.228124i	−0.993474	+	0.114062i	$0.963614\pi$
$488$		0			0
$489$			298.299i			0.610019i
$490$		0			0
$491$	671.679			1.36798			0.683990	−	0.729491i	$-0.260243\pi$
$491$	671.679			1.36798			0.683990	+	0.729491i	$0.260243\pi$
$492$		0			0
$493$		−	175.035i		−	0.355040i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	88.7504i		−	0.178572i
$498$		0			0
$499$	813.880			1.63102			0.815511	−	0.578741i	$-0.196456\pi$
$499$	813.880			1.63102			0.815511	+	0.578741i	$0.196456\pi$
$500$		0			0
$501$	−174.975			−0.349251
$502$		0			0
$503$	−901.318			−1.79189			−0.895943	−	0.444170i	$-0.853499\pi$
$503$	−901.318			−1.79189			−0.895943	+	0.444170i	$0.853499\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		−	489.841i		−	0.966156i
$508$		0			0
$509$			92.1471i			0.181035i	0.995895	+	0.0905177i	$0.0288522\pi$
$509$			92.1471i			0.181035i	−0.995895	+	0.0905177i	$0.971148\pi$
$510$		0			0
$511$	223.581			0.437537
$512$		0			0
$513$	452.566	−	268.197i	0.882195	−	0.522802i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−438.569			−0.848297
$518$		0			0
$519$	301.264			0.580470
$520$		0			0
$521$		−	235.653i		−	0.452309i	−0.974091	−	0.226155i	$-0.927385\pi$
$521$		−	235.653i		−	0.452309i	0.974091	−	0.226155i	$-0.0726154\pi$
$522$		0			0
$523$			290.076i			0.554639i	0.960778	+	0.277320i	$0.0894461\pi$
$523$			290.076i			0.554639i	−0.960778	+	0.277320i	$0.910554\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	152.776i		−	0.289898i
$528$		0			0
$529$	−409.372			−0.773860
$530$		0			0
$531$		−	490.087i		−	0.922951i
$532$		0			0
$533$	1673.80			3.14033
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−270.880			−0.504431
$538$		0			0
$539$	420.460			0.780073
$540$		0			0
$541$	352.993			0.652483			0.326242	−	0.945286i	$-0.394218\pi$
$541$	352.993			0.652483			0.326242	+	0.945286i	$0.394218\pi$
$542$		0			0
$543$	−207.922			−0.382913
$544$		0			0
$545$		0			0
$546$		0			0
$547$			571.004i			1.04388i	0.852981	+	0.521942i	$0.174792\pi$
$547$			571.004i			1.04388i	−0.852981	+	0.521942i	$0.825208\pi$
$548$		0			0
$549$	−532.931			−0.970730
$550$		0			0
$551$	464.606	−	275.332i	0.843205	−	0.499696i
$552$		0			0
$553$		−	161.687i		−	0.292382i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	245.730			0.441166			0.220583	−	0.975368i	$-0.429204\pi$
$557$	245.730			0.441166			0.220583	+	0.975368i	$0.429204\pi$
$558$		0			0
$559$		−	1452.75i		−	2.59884i
$560$		0			0
$561$			114.212i			0.203586i
$562$		0			0
$563$			623.923i			1.10821i	0.832446	+	0.554106i	$0.186940\pi$
$563$			623.923i			1.10821i	−0.832446	+	0.554106i	$0.813060\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	15.8334			0.0279249
$568$		0			0
$569$		−	286.095i		−	0.502804i	−0.967883	−	0.251402i	$-0.919108\pi$
$569$		−	286.095i		−	0.502804i	0.967883	−	0.251402i	$-0.0808915\pi$
$570$		0			0
$571$	213.683			0.374226			0.187113	−	0.982338i	$-0.440087\pi$
$571$	213.683			0.374226			0.187113	+	0.982338i	$0.440087\pi$
$572$		0			0
$573$		−	512.864i		−	0.895051i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	521.531			0.903866			0.451933	−	0.892052i	$-0.350735\pi$
$577$	521.531			0.903866			0.451933	+	0.892052i	$0.350735\pi$
$578$		0			0
$579$	−66.4519			−0.114770
$580$		0			0
$581$	−201.156			−0.346224
$582$		0			0
$583$		−	306.204i		−	0.525221i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	195.026			0.332243			0.166121	−	0.986105i	$-0.446876\pi$
$587$	195.026			0.332243			0.166121	+	0.986105i	$0.446876\pi$
$588$		0			0
$589$	405.525	−	240.320i	0.688497	−	0.408014i
$590$		0			0
$591$		−	572.525i		−	0.968740i
$592$		0			0
$593$	333.768			0.562847			0.281423	−	0.959584i	$-0.409193\pi$
$593$	333.768			0.562847			0.281423	+	0.959584i	$0.409193\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$			282.564i			0.473307i
$598$		0			0
$599$		−	556.121i		−	0.928415i	−0.885726	−	0.464208i	$-0.846339\pi$
$599$		−	556.121i		−	0.928415i	0.885726	−	0.464208i	$-0.153661\pi$
$600$		0			0
$601$		−	1015.58i		−	1.68981i	−0.534916	−	0.844905i	$-0.679657\pi$
$601$		−	1015.58i		−	1.68981i	0.534916	−	0.844905i	$-0.320343\pi$
$602$		0			0
$603$			358.668i			0.594806i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		−	916.408i		−	1.50973i	−0.655878	−	0.754867i	$-0.727702\pi$
$607$		−	916.408i		−	1.50973i	0.655878	−	0.754867i	$-0.272298\pi$
$608$		0			0
$609$	121.271			0.199132
$610$		0			0
$611$		−	945.333i		−	1.54719i
$612$		0			0
$613$	265.921			0.433802			0.216901	−	0.976194i	$-0.430405\pi$
$613$	265.921			0.433802			0.216901	+	0.976194i	$0.430405\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−24.9941			−0.0405091			−0.0202545	−	0.999795i	$-0.506448\pi$
$617$	−24.9941			−0.0405091			−0.0202545	+	0.999795i	$0.506448\pi$
$618$		0			0
$619$	−52.7690			−0.0852488			−0.0426244	−	0.999091i	$-0.513572\pi$
$619$	−52.7690			−0.0852488			−0.0426244	+	0.999091i	$0.513572\pi$
$620$		0			0
$621$			302.833i			0.487654i
$622$		0			0
$623$		−	125.482i		−	0.201416i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−303.161	+	179.657i	−0.483510	+	0.286535i
$628$		0			0
$629$		−	230.551i		−	0.366535i
$630$		0			0
$631$	−439.188			−0.696019			−0.348010	−	0.937491i	$-0.613142\pi$
$631$	−439.188			−0.696019			−0.348010	+	0.937491i	$0.613142\pi$
$632$		0			0
$633$	−692.971			−1.09474
$634$		0			0
$635$		0			0
$636$		0			0
$637$			906.297i			1.42276i
$638$		0			0
$639$			210.810i			0.329906i
$640$		0			0
$641$			795.892i			1.24164i	0.783953	+	0.620821i	$0.213201\pi$
$641$			795.892i			1.24164i	−0.783953	+	0.620821i	$0.786799\pi$
$642$		0			0
$643$	−927.013			−1.44170			−0.720850	−	0.693092i	$-0.756248\pi$
$643$	−927.013			−1.44170			−0.720850	+	0.693092i	$0.756248\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−128.287			−0.198280			−0.0991401	−	0.995073i	$-0.531609\pi$
$647$	−128.287			−0.198280			−0.0991401	+	0.995073i	$0.531609\pi$
$648$		0			0
$649$			896.938i			1.38203i
$650$		0			0
$651$	105.850			0.162596
$652$		0			0
$653$	745.488			1.14164			0.570818	−	0.821077i	$-0.306626\pi$
$653$	745.488			1.14164			0.570818	+	0.821077i	$0.306626\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−531.075			−0.808334
$658$		0			0
$659$		−	563.842i		−	0.855602i	−0.903873	−	0.427801i	$-0.859288\pi$
$659$		−	563.842i		−	0.855602i	0.903873	−	0.427801i	$-0.140712\pi$
$660$		0			0
$661$			219.673i			0.332335i	0.986098	+	0.166167i	$0.0531392\pi$
$661$			219.673i			0.332335i	−0.986098	+	0.166167i	$0.946861\pi$
$662$		0			0
$663$	−246.183			−0.371317
$664$		0			0
$665$		0			0
$666$		0			0
$667$			310.889i			0.466101i
$668$		0			0
$669$	644.346			0.963148
$670$		0			0
$671$	975.349			1.45357
$672$		0			0
$673$			145.649i			0.216418i	0.994128	+	0.108209i	$0.0345115\pi$
$673$			145.649i			0.216418i	−0.994128	+	0.108209i	$0.965488\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			1031.23i			1.52323i	0.648029	+	0.761616i	$0.275594\pi$
$677$			1031.23i			1.52323i	−0.648029	+	0.761616i	$0.724406\pi$
$678$		0			0
$679$			184.455i			0.271658i
$680$		0			0
$681$	45.7203			0.0671370
$682$		0			0
$683$		−	1266.00i		−	1.85358i	−0.375577	−	0.926791i	$-0.622555\pi$
$683$		−	1266.00i		−	1.85358i	0.375577	−	0.926791i	$-0.377445\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$			358.307i			0.521553i
$688$		0			0
$689$	660.020			0.957939
$690$		0			0
$691$	−458.754			−0.663899			−0.331949	−	0.943297i	$-0.607706\pi$
$691$	−458.754			−0.663899			−0.331949	+	0.943297i	$0.607706\pi$
$692$		0			0
$693$	108.086			0.155968
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	502.846i		−	0.721443i
$698$		0			0
$699$			280.170i			0.400815i
$700$		0			0
$701$	790.423			1.12756			0.563782	−	0.825923i	$-0.309346\pi$
$701$	790.423			1.12756			0.563782	+	0.825923i	$0.309346\pi$
$702$		0			0
$703$	611.966	−	362.660i	0.870506	−	0.515875i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−151.525			−0.214321
$708$		0			0
$709$	131.077			0.184875			0.0924377	−	0.995718i	$-0.470534\pi$
$709$	131.077			0.184875			0.0924377	+	0.995718i	$0.470534\pi$
$710$		0			0
$711$			384.057i			0.540165i
$712$		0			0
$713$			271.356i			0.380583i
$714$		0			0
$715$		0			0
$716$		0			0
$717$			155.066i			0.216271i
$718$		0			0
$719$	820.977			1.14183			0.570916	−	0.821008i	$-0.306588\pi$
$719$	820.977			1.14183			0.570916	+	0.821008i	$0.306588\pi$
$720$		0			0
$721$			219.169i			0.303980i
$722$		0			0
$723$	−238.931			−0.330472
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−1121.14			−1.54214			−0.771071	−	0.636749i	$-0.780279\pi$
$727$	−1121.14			−1.54214			−0.771071	+	0.636749i	$0.780279\pi$
$728$		0			0
$729$	−523.626			−0.718280
$730$		0			0
$731$	−436.438			−0.597043
$732$		0			0
$733$	97.7494			0.133355			0.0666777	−	0.997775i	$-0.478760\pi$
$733$	97.7494			0.133355			0.0666777	+	0.997775i	$0.478760\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	656.420i		−	0.890665i
$738$		0			0
$739$	−520.032			−0.703697			−0.351849	−	0.936057i	$-0.614447\pi$
$739$	−520.032			−0.703697			−0.351849	+	0.936057i	$0.614447\pi$
$740$		0			0
$741$	−387.250	−	653.460i	−0.522605	−	0.881863i
$742$		0			0
$743$		−	1357.00i		−	1.82638i	−0.407535	−	0.913189i	$-0.633612\pi$
$743$		−	1357.00i		−	1.82638i	0.407535	−	0.913189i	$-0.366388\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	477.808			0.639635
$748$		0			0
$749$			115.356i			0.154013i
$750$		0			0
$751$		−	431.283i		−	0.574278i	−0.957889	−	0.287139i	$-0.907296\pi$
$751$		−	431.283i		−	0.574278i	0.957889	−	0.287139i	$-0.0927042\pi$
$752$		0			0
$753$			37.5128i			0.0498178i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	665.943			0.879713			0.439857	−	0.898068i	$-0.355029\pi$
$757$	665.943			0.879713			0.439857	+	0.898068i	$0.355029\pi$
$758$		0			0
$759$		−	202.859i		−	0.267271i
$760$		0			0
$761$	−792.938			−1.04197			−0.520984	−	0.853566i	$-0.674435\pi$
$761$	−792.938			−1.04197			−0.520984	+	0.853566i	$0.674435\pi$
$762$		0			0
$763$		−	167.868i		−	0.220010i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1933.34			−2.52066
$768$		0			0
$769$	−1263.74			−1.64336			−0.821678	−	0.569953i	$-0.806962\pi$
$769$	−1263.74			−1.64336			−0.821678	+	0.569953i	$0.806962\pi$
$770$		0			0
$771$	−117.842			−0.152843
$772$		0			0
$773$			441.297i			0.570889i	0.958395	+	0.285445i	$0.0921412\pi$
$773$			441.297i			0.570889i	−0.958395	+	0.285445i	$0.907859\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	159.735			0.205579
$778$		0			0
$779$	1334.74	−	790.985i	1.71340	−	1.01538i
$780$		0			0
$781$		−	385.815i		−	0.494002i
$782$		0			0
$783$	−787.004			−1.00511
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	367.069i		−	0.466415i	−0.972427	−	0.233207i	$-0.925078\pi$
$787$		−	367.069i		−	0.466415i	0.972427	−	0.233207i	$-0.0749221\pi$
$788$		0			0
$789$			45.2268i			0.0573216i
$790$		0			0
$791$		−	459.925i		−	0.581447i
$792$		0			0
$793$			2102.36i			2.65114i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			923.535i			1.15876i	0.815056	+	0.579382i	$0.196706\pi$
$797$			923.535i			1.15876i	−0.815056	+	0.579382i	$0.803294\pi$
$798$		0			0
$799$	−283.999			−0.355443
$800$		0			0
$801$			298.059i			0.372108i
$802$		0			0
$803$	971.953			1.21040
$804$		0			0
$805$		0			0
$806$		0			0
$807$	411.964			0.510488
$808$		0			0
$809$	−823.083			−1.01741			−0.508704	−	0.860942i	$-0.669875\pi$
$809$	−823.083			−1.01741			−0.508704	+	0.860942i	$0.669875\pi$
$810$		0			0
$811$			125.683i			0.154973i	0.996993	+	0.0774866i	$0.0246895\pi$
$811$			125.683i			0.154973i	−0.996993	+	0.0774866i	$0.975311\pi$
$812$		0			0
$813$		−	724.017i		−	0.890550i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−686.525	−	1158.47i	−0.840300	−	1.41795i
$818$		0			0
$819$			232.978i			0.284467i
$820$		0			0
$821$	713.601			0.869185			0.434592	−	0.900627i	$-0.356892\pi$
$821$	713.601			0.869185			0.434592	+	0.900627i	$0.356892\pi$
$822$		0			0
$823$	−823.797			−1.00097			−0.500484	−	0.865746i	$-0.666845\pi$
$823$	−823.797			−1.00097			−0.500484	+	0.865746i	$0.666845\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	654.261i		−	0.791125i	−0.918439	−	0.395563i	$-0.870550\pi$
$827$		−	654.261i		−	0.791125i	0.918439	−	0.395563i	$-0.129450\pi$
$828$		0			0
$829$			530.752i			0.640231i	0.947379	+	0.320116i	$0.103722\pi$
$829$			530.752i			0.640231i	−0.947379	+	0.320116i	$0.896278\pi$
$830$		0			0
$831$		−	13.4741i		−	0.0162143i
$832$		0			0
$833$	272.272			0.326857
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−686.925			−0.820699
$838$		0			0
$839$		−	917.949i		−	1.09410i	−0.837100	−	0.547049i	$-0.815751\pi$
$839$		−	917.949i		−	1.09410i	0.837100	−	0.547049i	$-0.184249\pi$
$840$		0			0
$841$	33.0594			0.0393097
$842$		0			0
$843$	14.5688			0.0172821
$844$		0			0
$845$		0			0
$846$		0			0
$847$	66.8721			0.0789517
$848$		0			0
$849$		−	124.932i		−	0.147152i
$850$		0			0
$851$			409.495i			0.481192i
$852$		0			0
$853$	−600.529			−0.704020			−0.352010	−	0.935996i	$-0.614502\pi$
$853$	−600.529			−0.704020			−0.352010	+	0.935996i	$0.614502\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	1059.28i		−	1.23603i	−0.786167	−	0.618015i	$-0.787937\pi$
$857$		−	1059.28i		−	1.23603i	0.786167	−	0.618015i	$-0.212063\pi$
$858$		0			0
$859$	−734.054			−0.854545			−0.427272	−	0.904123i	$-0.640525\pi$
$859$	−734.054			−0.854545			−0.427272	+	0.904123i	$0.640525\pi$
$860$		0			0
$861$	348.392			0.404637
$862$		0			0
$863$			714.413i			0.827825i	0.910317	+	0.413912i	$0.135838\pi$
$863$			714.413i			0.827825i	−0.910317	+	0.413912i	$0.864162\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		−	489.705i		−	0.564826i
$868$		0			0
$869$		−	702.886i		−	0.808845i
$870$		0			0
$871$	1414.91			1.62447
$872$		0			0
$873$		−	438.139i		−	0.501878i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	1129.51i		−	1.28792i	−0.765059	−	0.643961i	$-0.777290\pi$
$877$		−	1129.51i		−	1.28792i	0.765059	−	0.643961i	$-0.222710\pi$
$878$		0			0
$879$	1028.85			1.17048
$880$		0			0
$881$	161.933			0.183806			0.0919029	−	0.995768i	$-0.470705\pi$
$881$	161.933			0.183806			0.0919029	+	0.995768i	$0.470705\pi$
$882$		0			0
$883$	199.206			0.225602			0.112801	−	0.993618i	$-0.464018\pi$
$883$	199.206			0.225602			0.112801	+	0.993618i	$0.464018\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			727.670i			0.820373i	0.912002	+	0.410186i	$0.134536\pi$
$887$			727.670i			0.820373i	−0.912002	+	0.410186i	$0.865464\pi$
$888$		0			0
$889$		−	43.5814i		−	0.0490230i
$890$		0			0
$891$	68.8310			0.0772514
$892$		0			0
$893$	−446.735	−	753.837i	−0.500263	−	0.844163i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	437.261			0.487470
$898$		0			0
$899$	−705.200			−0.784427
$900$		0			0
$901$		−	198.285i		−	0.220072i
$902$		0			0
$903$		−	302.383i		−	0.334865i
$904$		0			0
$905$		0			0
$906$		0			0
$907$			1249.33i			1.37743i	0.725031	+	0.688716i	$0.241826\pi$
$907$			1249.33i			1.37743i	−0.725031	+	0.688716i	$0.758174\pi$
$908$		0			0
$909$	359.919			0.395950
$910$		0			0
$911$		−	813.909i		−	0.893423i	−0.894678	−	0.446712i	$-0.852595\pi$
$911$		−	813.909i		−	0.893423i	0.894678	−	0.446712i	$-0.147405\pi$
$912$		0			0
$913$	−874.465			−0.957793
$914$		0			0
$915$		0			0
$916$		0			0
$917$	7.21297			0.00786584
$918$		0			0
$919$	442.339			0.481327			0.240663	−	0.970609i	$-0.422635\pi$
$919$	442.339			0.481327			0.240663	+	0.970609i	$0.422635\pi$
$920$		0			0
$921$	−751.024			−0.815444
$922$		0			0
$923$	831.622			0.900999
$924$		0			0
$925$		0			0
$926$		0			0
$927$		−	520.595i		−	0.561591i
$928$		0			0
$929$	514.468			0.553786			0.276893	−	0.960901i	$-0.410695\pi$
$929$	514.468			0.553786			0.276893	+	0.960901i	$0.410695\pi$
$930$		0			0
$931$	428.288	+	722.709i	0.460030	+	0.776272i
$932$		0			0
$933$			210.454i			0.225567i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1147.17			1.22430			0.612149	−	0.790743i	$-0.290305\pi$
$937$	1147.17			1.22430			0.612149	+	0.790743i	$0.290305\pi$
$938$		0			0
$939$			606.711i			0.646124i
$940$		0			0
$941$		−	1355.11i		−	1.44008i	−0.693934	−	0.720039i	$-0.744124\pi$
$941$		−	1355.11i		−	1.44008i	0.693934	−	0.720039i	$-0.255876\pi$
$942$		0			0
$943$			893.134i			0.947120i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1560.15			1.64746			0.823732	−	0.566980i	$-0.191888\pi$
$947$	1560.15			1.64746			0.823732	+	0.566980i	$0.191888\pi$
$948$		0			0
$949$			2095.04i			2.20763i
$950$		0			0
$951$	−416.937			−0.438420
$952$		0			0
$953$			643.715i			0.675461i	0.941243	+	0.337731i	$0.109659\pi$
$953$			643.715i			0.675461i	−0.941243	+	0.337731i	$0.890341\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	527.190			0.550878
$958$		0			0
$959$	−324.159			−0.338018
$960$		0			0
$961$	345.476			0.359496
$962$		0			0
$963$		−	274.006i		−	0.284534i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−570.639			−0.590113			−0.295056	−	0.955480i	$-0.595338\pi$
$967$	−570.639			−0.590113			−0.295056	+	0.955480i	$0.595338\pi$
$968$		0			0
$969$	−196.314	+	116.339i	−0.202594	+	0.120060i
$970$		0			0
$971$		−	516.280i		−	0.531699i	−0.964015	−	0.265850i	$-0.914348\pi$
$971$		−	516.280i		−	0.531699i	0.964015	−	0.265850i	$-0.0856525\pi$
$972$		0			0
$973$	−118.721			−0.122016
$974$		0			0
$975$		0			0
$976$		0			0
$977$			432.740i			0.442927i	0.975169	+	0.221464i	$0.0710834\pi$
$977$			432.740i			0.442927i	−0.975169	+	0.221464i	$0.928917\pi$
$978$		0			0
$979$		−	545.495i		−	0.557196i
$980$		0			0
$981$			398.739i			0.406461i
$982$		0			0
$983$			1116.57i			1.13588i	0.823070	+	0.567940i	$0.192260\pi$
$983$			1116.57i			1.13588i	−0.823070	+	0.567940i	$0.807740\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		−	196.766i		−	0.199358i
$988$		0			0
$989$	775.185			0.783806
$990$		0			0
$991$			1158.19i			1.16871i	0.811499	+	0.584354i	$0.198652\pi$
$991$			1158.19i			1.16871i	−0.811499	+	0.584354i	$0.801348\pi$
$992$		0			0
$993$	483.843			0.487254
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−76.0503			−0.0762792			−0.0381396	−	0.999272i	$-0.512143\pi$
$997$	−76.0503			−0.0762792			−0.0381396	+	0.999272i	$0.512143\pi$
$998$		0			0
$999$	−1036.62			−1.03766

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1900.3.e.e.1101.7		12
5.2	odd	4		380.3.g.c.189.8	yes	12
5.3	odd	4		380.3.g.c.189.5	✓	12
5.4	even	2	inner	1900.3.e.e.1101.6		12
15.2	even	4		3420.3.h.e.2089.9		12
15.8	even	4		3420.3.h.e.2089.12		12
19.18	odd	2	inner	1900.3.e.e.1101.5		12
95.18	even	4		380.3.g.c.189.7	yes	12
95.37	even	4		380.3.g.c.189.6	yes	12
95.94	odd	2	inner	1900.3.e.e.1101.8		12
285.113	odd	4		3420.3.h.e.2089.11		12
285.227	odd	4		3420.3.h.e.2089.10		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.3.g.c.189.5	✓	12	5.3	odd	4
380.3.g.c.189.6	yes	12	95.37	even	4
380.3.g.c.189.7	yes	12	95.18	even	4
380.3.g.c.189.8	yes	12	5.2	odd	4
1900.3.e.e.1101.5		12	19.18	odd	2	inner
1900.3.e.e.1101.6		12	5.4	even	2	inner
1900.3.e.e.1101.7		12	1.1	even	1	trivial
1900.3.e.e.1101.8		12	95.94	odd	2	inner
3420.3.h.e.2089.9		12	15.2	even	4
3420.3.h.e.2089.10		12	285.227	odd	4
3420.3.h.e.2089.11		12	285.113	odd	4
3420.3.h.e.2089.12		12	15.8	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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1900.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.95039i q^{3} -2.18749 q^{7} +5.19597 q^{9} +O(q^{10})\)	\(q+1.95039i q^{3} -2.18749 q^{7} +5.19597 q^{9} -9.50946 q^{11} -20.4976i q^{13} -6.15792 q^{17} +(-9.68651 - 16.3454i) q^{19} -4.26647i q^{21} +10.9375 q^{23} +27.6877i q^{27} +28.4243i q^{29} +24.8098i q^{31} -18.5472i q^{33} +37.4397i q^{37} +39.9783 q^{39} +81.6583i q^{41} +70.8743 q^{43} +46.1193 q^{47} -44.2149 q^{49} -12.0104i q^{51} +32.1999i q^{53} +(31.8799 - 18.8925i) q^{57} -94.3206i q^{59} -102.566 q^{61} -11.3661 q^{63} +69.0282i q^{67} +21.3323i q^{69} +40.5718i q^{71} -102.209 q^{73} +20.8018 q^{77} +73.9145i q^{79} -7.23816 q^{81} +91.9574 q^{83} -55.4385 q^{87} +57.3634i q^{89} +44.8382i q^{91} -48.3888 q^{93} -84.3229i q^{97} -49.4109 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 76 q^{9}+O(q^{10}) \) 12 * q - 76 * q^9	\( 12 q - 76 q^{9} - 24 q^{11} - 68 q^{19} + 264 q^{39} - 212 q^{49} - 600 q^{61} + 492 q^{81} - 136 q^{99}+O(q^{100}) \) 12 * q - 76 * q^9 - 24 * q^11 - 68 * q^19 + 264 * q^39 - 212 * q^49 - 600 * q^61 + 492 * q^81 - 136 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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