LMFDB - Embedded newform 2300.3.d.c.1149.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2300,3,Mod(1149,2300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2300, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2300.1149");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2300 = 2^{2} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2300.d (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:	$62.6704608029$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1149.10
Character	$\chi$	$=$	2300.1149
Dual form			2300.3.d.c.1149.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.89154i q^{3} -6.27459 q^{7} -6.14407 q^{9} +O(q^{10})$	$q+3.89154i q^{3} -6.27459 q^{7} -6.14407 q^{9} +11.1672i q^{11} -15.9681i q^{13} +23.9086 q^{17} -7.55287i q^{19} -24.4178i q^{21} +(18.3975 + 13.8033i) q^{23} +11.1140i q^{27} -10.3458 q^{29} +53.0811 q^{31} -43.4578 q^{33} +26.7604 q^{37} +62.1406 q^{39} +39.5201 q^{41} -11.0919 q^{43} -56.1707i q^{47} -9.62955 q^{49} +93.0412i q^{51} -92.4926 q^{53} +29.3923 q^{57} +78.5148 q^{59} +113.907i q^{61} +38.5515 q^{63} +55.8509 q^{67} +(-53.7159 + 71.5948i) q^{69} +59.4804 q^{71} +9.67213i q^{73} -70.0698i q^{77} +75.4257i q^{79} -98.5470 q^{81} -141.543 q^{83} -40.2612i q^{87} +21.7363i q^{89} +100.193i q^{91} +206.567i q^{93} -19.6534 q^{97} -68.6123i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 128 q^{9}+O(q^{10}) $ 32 * q - 128 * q^9	$ 32 q - 128 q^{9} + 12 q^{29} + 56 q^{31} - 148 q^{39} - 180 q^{41} + 380 q^{49} + 348 q^{59} - 100 q^{69} + 232 q^{71} + 112 q^{81}+O(q^{100}) $ 32 * q - 128 * q^9 + 12 * q^29 + 56 * q^31 - 148 * q^39 - 180 * q^41 + 380 * q^49 + 348 * q^59 - 100 * q^69 + 232 * q^71 + 112 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2300\mathbb{Z}\right)^\times$.

$n$	$277$	$1151$	$1201$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$			3.89154i			1.29718i	0.761138	+	0.648590i	$0.224641\pi$
$3$			3.89154i			1.29718i	−0.761138	+	0.648590i	$0.775359\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−6.27459			−0.896370			−0.448185	−	0.893941i	$-0.647929\pi$
$7$	−6.27459			−0.896370			−0.448185	+	0.893941i	$0.647929\pi$
$8$		0			0
$9$	−6.14407			−0.682675
$10$		0			0
$11$			11.1672i			1.01520i	0.861592	+	0.507602i	$0.169468\pi$
$11$			11.1672i			1.01520i	−0.861592	+	0.507602i	$0.830532\pi$
$12$		0			0
$13$		−	15.9681i		−	1.22832i	−0.789182	−	0.614159i	$-0.789495\pi$
$13$		−	15.9681i		−	1.22832i	0.789182	−	0.614159i	$-0.210505\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	23.9086			1.40639			0.703194	−	0.710998i	$-0.251757\pi$
$17$	23.9086			1.40639			0.703194	+	0.710998i	$0.251757\pi$
$18$		0			0
$19$		−	7.55287i		−	0.397520i	−0.980048	−	0.198760i	$-0.936309\pi$
$19$		−	7.55287i		−	0.397520i	0.980048	−	0.198760i	$-0.0636914\pi$
$20$		0			0
$21$		−	24.4178i		−	1.16275i
$22$		0			0
$23$	18.3975	+	13.8033i	0.799893	+	0.600142i
$23$	18.3975	+	13.8033i	0.799893	+	0.600142i
$24$		0			0
$25$		0			0
$26$		0			0
$27$			11.1140i			0.411628i
$28$		0			0
$29$	−10.3458			−0.356753			−0.178376	−	0.983962i	$-0.557084\pi$
$29$	−10.3458			−0.356753			−0.178376	+	0.983962i	$0.557084\pi$
$30$		0			0
$31$	53.0811			1.71230			0.856148	−	0.516731i	$-0.172851\pi$
$31$	53.0811			1.71230			0.856148	+	0.516731i	$0.172851\pi$
$32$		0			0
$33$	−43.4578			−1.31690
$34$		0			0
$35$		0			0
$36$		0			0
$37$	26.7604			0.723254			0.361627	−	0.932323i	$-0.382221\pi$
$37$	26.7604			0.723254			0.361627	+	0.932323i	$0.382221\pi$
$38$		0			0
$39$	62.1406			1.59335
$40$		0			0
$41$	39.5201			0.963906			0.481953	−	0.876197i	$-0.339928\pi$
$41$	39.5201			0.963906			0.481953	+	0.876197i	$0.339928\pi$
$42$		0			0
$43$	−11.0919			−0.257952			−0.128976	−	0.991648i	$-0.541169\pi$
$43$	−11.0919			−0.257952			−0.128976	+	0.991648i	$0.541169\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	56.1707i		−	1.19512i	−0.801824	−	0.597561i	$-0.796137\pi$
$47$		−	56.1707i		−	1.19512i	0.801824	−	0.597561i	$-0.203863\pi$
$48$		0			0
$49$	−9.62955			−0.196521
$50$		0			0
$51$			93.0412i			1.82434i
$52$		0			0
$53$	−92.4926			−1.74514			−0.872572	−	0.488486i	$-0.837549\pi$
$53$	−92.4926			−1.74514			−0.872572	+	0.488486i	$0.837549\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	29.3923			0.515654
$58$		0			0
$59$	78.5148			1.33076			0.665379	−	0.746505i	$-0.268270\pi$
$59$	78.5148			1.33076			0.665379	+	0.746505i	$0.268270\pi$
$60$		0			0
$61$			113.907i			1.86733i	0.358142	+	0.933667i	$0.383410\pi$
$61$			113.907i			1.86733i	−0.358142	+	0.933667i	$0.616590\pi$
$62$		0			0
$63$	38.5515			0.611929
$64$		0			0
$65$		0			0
$66$		0			0
$67$	55.8509			0.833596			0.416798	−	0.908999i	$-0.363152\pi$
$67$	55.8509			0.833596			0.416798	+	0.908999i	$0.363152\pi$
$68$		0			0
$69$	−53.7159	+	71.5948i	−0.778492	+	1.03761i
$70$		0			0
$71$	59.4804			0.837752			0.418876	−	0.908043i	$-0.362424\pi$
$71$	59.4804			0.837752			0.418876	+	0.908043i	$0.362424\pi$
$72$		0			0
$73$			9.67213i			0.132495i	0.997803	+	0.0662475i	$0.0211027\pi$
$73$			9.67213i			0.132495i	−0.997803	+	0.0662475i	$0.978897\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	70.0698i		−	0.909998i
$78$		0			0
$79$			75.4257i			0.954755i	0.878698	+	0.477378i	$0.158413\pi$
$79$			75.4257i			0.954755i	−0.878698	+	0.477378i	$0.841587\pi$
$80$		0			0
$81$	−98.5470			−1.21663
$82$		0			0
$83$	−141.543			−1.70533			−0.852667	−	0.522454i	$-0.825017\pi$
$83$	−141.543			−1.70533			−0.852667	+	0.522454i	$0.825017\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		−	40.2612i		−	0.462772i
$88$		0			0
$89$			21.7363i			0.244228i	0.992516	+	0.122114i	$0.0389674\pi$
$89$			21.7363i			0.244228i	−0.992516	+	0.122114i	$0.961033\pi$
$90$		0			0
$91$			100.193i			1.10103i
$92$		0			0
$93$			206.567i			2.22115i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−19.6534			−0.202612			−0.101306	−	0.994855i	$-0.532302\pi$
$97$	−19.6534			−0.202612			−0.101306	+	0.994855i	$0.532302\pi$
$98$		0			0
$99$		−	68.6123i		−	0.693054i
$100$		0			0
$101$	9.17328			0.0908245			0.0454123	−	0.998968i	$-0.485540\pi$
$101$	9.17328			0.0908245			0.0454123	+	0.998968i	$0.485540\pi$
$102$		0			0
$103$	146.503			1.42236			0.711179	−	0.703011i	$-0.248162\pi$
$103$	146.503			1.42236			0.711179	+	0.703011i	$0.248162\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−203.207			−1.89913			−0.949565	−	0.313570i	$-0.898475\pi$
$107$	−203.207			−1.89913			−0.949565	+	0.313570i	$0.898475\pi$
$108$		0			0
$109$			30.4560i			0.279413i	0.990193	+	0.139706i	$0.0446159\pi$
$109$			30.4560i			0.279413i	−0.990193	+	0.139706i	$0.955384\pi$
$110$		0			0
$111$			104.139i			0.938190i
$112$		0			0
$113$	−95.2436			−0.842863			−0.421432	−	0.906860i	$-0.638472\pi$
$113$	−95.2436			−0.842863			−0.421432	+	0.906860i	$0.638472\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$			98.1093i			0.838541i
$118$		0			0
$119$	−150.016			−1.26064
$120$		0			0
$121$	−3.70731			−0.0306389
$122$		0			0
$123$			153.794i			1.25036i
$124$		0			0
$125$		0			0
$126$		0			0
$127$			176.513i			1.38987i	0.719073	+	0.694934i	$0.244566\pi$
$127$			176.513i			1.38987i	−0.719073	+	0.694934i	$0.755434\pi$
$128$		0			0
$129$		−	43.1647i		−	0.334610i
$130$		0			0
$131$	256.391			1.95718			0.978591	−	0.205817i	$-0.0659851\pi$
$131$	256.391			1.95718			0.978591	+	0.205817i	$0.0659851\pi$
$132$		0			0
$133$			47.3912i			0.356324i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−112.682			−0.822499			−0.411250	−	0.911523i	$-0.634908\pi$
$137$	−112.682			−0.822499			−0.411250	+	0.911523i	$0.634908\pi$
$138$		0			0
$139$	−10.8850			−0.0783093			−0.0391546	−	0.999233i	$-0.512466\pi$
$139$	−10.8850			−0.0783093			−0.0391546	+	0.999233i	$0.512466\pi$
$140$		0			0
$141$	218.590			1.55029
$142$		0			0
$143$	178.320			1.24699
$144$		0			0
$145$		0			0
$146$		0			0
$147$		−	37.4738i		−	0.254924i
$148$		0			0
$149$		−	55.6405i		−	0.373426i	−0.982414	−	0.186713i	$-0.940216\pi$
$149$		−	55.6405i		−	0.373426i	0.982414	−	0.186713i	$-0.0597835\pi$
$150$		0			0
$151$	102.085			0.676058			0.338029	−	0.941136i	$-0.390240\pi$
$151$	102.085			0.676058			0.338029	+	0.941136i	$0.390240\pi$
$152$		0			0
$153$	−146.896			−0.960105
$154$		0			0
$155$		0			0
$156$		0			0
$157$	118.883			0.757215			0.378607	−	0.925557i	$-0.376403\pi$
$157$	118.883			0.757215			0.378607	+	0.925557i	$0.376403\pi$
$158$		0			0
$159$		−	359.938i		−	2.26376i
$160$		0			0
$161$	−115.437	−	86.6098i	−0.717000	−	0.537949i
$162$		0			0
$163$		−	212.056i		−	1.30096i	−0.759525	−	0.650479i	$-0.774568\pi$
$163$		−	212.056i		−	1.30096i	0.759525	−	0.650479i	$-0.225432\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	24.1061i		−	0.144348i	−0.997392	−	0.0721739i	$-0.977006\pi$
$167$		−	24.1061i		−	0.144348i	0.997392	−	0.0721739i	$-0.0229937\pi$
$168$		0			0
$169$	−85.9812			−0.508764
$170$		0			0
$171$			46.4054i			0.271376i
$172$		0			0
$173$			23.2761i			0.134544i	0.997735	+	0.0672720i	$0.0214295\pi$
$173$			23.2761i			0.134544i	−0.997735	+	0.0672720i	$0.978570\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			305.543i			1.72623i
$178$		0			0
$179$	−77.3667			−0.432216			−0.216108	−	0.976369i	$-0.569336\pi$
$179$	−77.3667			−0.432216			−0.216108	+	0.976369i	$0.569336\pi$
$180$		0			0
$181$			255.730i			1.41287i	0.707776	+	0.706437i	$0.249699\pi$
$181$			255.730i			1.41287i	−0.707776	+	0.706437i	$0.750301\pi$
$182$		0			0
$183$	−443.275			−2.42227
$184$		0			0
$185$		0			0
$186$		0			0
$187$			266.993i			1.42777i
$188$		0			0
$189$		−	69.7355i		−	0.368971i
$190$		0			0
$191$			162.830i			0.852512i	0.904602	+	0.426256i	$0.140168\pi$
$191$			162.830i			0.852512i	−0.904602	+	0.426256i	$0.859832\pi$
$192$		0			0
$193$		−	202.809i		−	1.05082i	−0.850849	−	0.525411i	$-0.823912\pi$
$193$		−	202.809i		−	1.05082i	0.850849	−	0.525411i	$-0.176088\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	209.095i		−	1.06140i	−0.847561	−	0.530699i	$-0.821930\pi$
$197$		−	209.095i		−	1.06140i	0.847561	−	0.530699i	$-0.178070\pi$
$198$		0			0
$199$			299.549i			1.50527i	0.658438	+	0.752635i	$0.271217\pi$
$199$			299.549i			1.50527i	−0.658438	+	0.752635i	$0.728783\pi$
$200$		0			0
$201$			217.346i			1.08132i
$202$		0			0
$203$	64.9158			0.319782
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−113.036	−	84.8083i	−0.546067	−	0.409702i
$208$		0			0
$209$	84.3447			0.403563
$210$		0			0
$211$	−108.872			−0.515982			−0.257991	−	0.966147i	$-0.583060\pi$
$211$	−108.872			−0.515982			−0.257991	+	0.966147i	$0.583060\pi$
$212$		0			0
$213$			231.470i			1.08672i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−333.062			−1.53485
$218$		0			0
$219$	−37.6395			−0.171870
$220$		0			0
$221$		−	381.775i		−	1.72749i
$222$		0			0
$223$			90.7595i			0.406993i	0.979076	+	0.203497i	$0.0652306\pi$
$223$			90.7595i			0.406993i	−0.979076	+	0.203497i	$0.934769\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−160.457			−0.706860			−0.353430	−	0.935461i	$-0.614985\pi$
$227$	−160.457			−0.706860			−0.353430	+	0.935461i	$0.614985\pi$
$228$		0			0
$229$		−	367.687i		−	1.60562i	−0.596235	−	0.802810i	$-0.703337\pi$
$229$		−	367.687i		−	1.60562i	0.596235	−	0.802810i	$-0.296663\pi$
$230$		0			0
$231$	272.679			1.18043
$232$		0			0
$233$			27.0239i			0.115983i	0.998317	+	0.0579913i	$0.0184696\pi$
$233$			27.0239i			0.115983i	−0.998317	+	0.0579913i	$0.981530\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−293.522			−1.23849
$238$		0			0
$239$	36.5799			0.153054			0.0765270	−	0.997068i	$-0.475617\pi$
$239$	36.5799			0.153054			0.0765270	+	0.997068i	$0.475617\pi$
$240$		0			0
$241$			75.2891i			0.312403i	0.987725	+	0.156202i	$0.0499249\pi$
$241$			75.2891i			0.312403i	−0.987725	+	0.156202i	$0.950075\pi$
$242$		0			0
$243$		−	283.474i		−	1.16656i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−120.605			−0.488280
$248$		0			0
$249$		−	550.819i		−	2.21213i
$250$		0			0
$251$			294.339i			1.17267i	0.810070	+	0.586334i	$0.199429\pi$
$251$			294.339i			1.17267i	−0.810070	+	0.586334i	$0.800571\pi$
$252$		0			0
$253$	−154.144	+	205.450i	−0.609267	+	0.812055i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			247.661i			0.963662i	0.876264	+	0.481831i	$0.160028\pi$
$257$			247.661i			0.963662i	−0.876264	+	0.481831i	$0.839972\pi$
$258$		0			0
$259$	−167.910			−0.648303
$260$		0			0
$261$	63.5655			0.243546
$262$		0			0
$263$	148.037			0.562877			0.281439	−	0.959579i	$-0.409188\pi$
$263$	148.037			0.562877			0.281439	+	0.959579i	$0.409188\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−84.5877			−0.316808
$268$		0			0
$269$	−306.902			−1.14090			−0.570449	−	0.821333i	$-0.693231\pi$
$269$	−306.902			−1.14090			−0.570449	+	0.821333i	$0.693231\pi$
$270$		0			0
$271$	1.96053			0.00723442			0.00361721	−	0.999993i	$-0.498849\pi$
$271$	1.96053			0.00723442			0.00361721	+	0.999993i	$0.498849\pi$
$272$		0			0
$273$	−389.907			−1.42823
$274$		0			0
$275$		0			0
$276$		0			0
$277$			315.295i			1.13825i	0.822251	+	0.569125i	$0.192718\pi$
$277$			315.295i			1.13825i	−0.822251	+	0.569125i	$0.807282\pi$
$278$		0			0
$279$	−326.134			−1.16894
$280$		0			0
$281$			102.967i			0.366431i	0.983073	+	0.183215i	$0.0586506\pi$
$281$			102.967i			0.366431i	−0.983073	+	0.183215i	$0.941349\pi$
$282$		0			0
$283$	349.071			1.23347			0.616734	−	0.787172i	$-0.288456\pi$
$283$	349.071			1.23347			0.616734	+	0.787172i	$0.288456\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−247.973			−0.864016
$288$		0			0
$289$	282.620			0.977925
$290$		0			0
$291$		−	76.4819i		−	0.262824i
$292$		0			0
$293$	180.647			0.616543			0.308271	−	0.951298i	$-0.400250\pi$
$293$	180.647			0.616543			0.308271	+	0.951298i	$0.400250\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−124.112			−0.417886
$298$		0			0
$299$	220.412	−	293.774i	0.737165	−	0.982523i
$300$		0			0
$301$	69.5973			0.231220
$302$		0			0
$303$			35.6982i			0.117816i
$304$		0			0
$305$		0			0
$306$		0			0
$307$			483.725i			1.57565i	0.615897	+	0.787826i	$0.288794\pi$
$307$			483.725i			1.57565i	−0.615897	+	0.787826i	$0.711206\pi$
$308$		0			0
$309$			570.122i			1.84505i
$310$		0			0
$311$	−244.285			−0.785483			−0.392741	−	0.919649i	$-0.628473\pi$
$311$	−244.285			−0.785483			−0.392741	+	0.919649i	$0.628473\pi$
$312$		0			0
$313$	−467.975			−1.49513			−0.747564	−	0.664190i	$-0.768777\pi$
$313$	−467.975			−1.49513			−0.747564	+	0.664190i	$0.768777\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			581.409i			1.83410i	0.398775	+	0.917049i	$0.369435\pi$
$317$			581.409i			1.83410i	−0.398775	+	0.917049i	$0.630565\pi$
$318$		0			0
$319$		−	115.534i		−	0.362177i
$320$		0			0
$321$		−	790.788i		−	2.46351i
$322$		0			0
$323$		−	180.578i		−	0.559066i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−118.521			−0.362449
$328$		0			0
$329$			352.448i			1.07127i
$330$		0			0
$331$	−400.525			−1.21004			−0.605022	−	0.796209i	$-0.706836\pi$
$331$	−400.525			−1.21004			−0.605022	+	0.796209i	$0.706836\pi$
$332$		0			0
$333$	−164.418			−0.493747
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−420.503			−1.24778			−0.623892	−	0.781510i	$-0.714450\pi$
$337$	−420.503			−1.24778			−0.623892	+	0.781510i	$0.714450\pi$
$338$		0			0
$339$		−	370.644i		−	1.09335i
$340$		0			0
$341$			592.770i			1.73833i
$342$		0			0
$343$	367.876			1.07253
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	210.984i		−	0.608024i	−0.952668	−	0.304012i	$-0.901674\pi$
$347$		−	210.984i		−	0.608024i	0.952668	−	0.304012i	$-0.0983263\pi$
$348$		0			0
$349$	57.7559			0.165490			0.0827449	−	0.996571i	$-0.473631\pi$
$349$	57.7559			0.165490			0.0827449	+	0.996571i	$0.473631\pi$
$350$		0			0
$351$	177.469			0.505610
$352$		0			0
$353$			300.742i			0.851959i	0.904733	+	0.425980i	$0.140070\pi$
$353$			300.742i			0.851959i	−0.904733	+	0.425980i	$0.859930\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		−	583.795i		−	1.63528i
$358$		0			0
$359$			478.851i			1.33385i	0.745126	+	0.666924i	$0.232389\pi$
$359$			478.851i			1.33385i	−0.745126	+	0.666924i	$0.767611\pi$
$360$		0			0
$361$	303.954			0.841978
$362$		0			0
$363$		−	14.4271i		−	0.0397442i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	629.925			1.71642			0.858209	−	0.513301i	$-0.171578\pi$
$367$	629.925			1.71642			0.858209	+	0.513301i	$0.171578\pi$
$368$		0			0
$369$	−242.815			−0.658034
$370$		0			0
$371$	580.353			1.56429
$372$		0			0
$373$	−54.4670			−0.146024			−0.0730121	−	0.997331i	$-0.523261\pi$
$373$	−54.4670			−0.146024			−0.0730121	+	0.997331i	$0.523261\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			165.204i			0.438206i
$378$		0			0
$379$		−	607.631i		−	1.60325i	−0.597829	−	0.801624i	$-0.703970\pi$
$379$		−	607.631i		−	1.60325i	0.597829	−	0.801624i	$-0.296030\pi$
$380$		0			0
$381$	−686.908			−1.80291
$382$		0			0
$383$	281.454			0.734868			0.367434	−	0.930050i	$-0.380236\pi$
$383$	281.454			0.734868			0.367434	+	0.930050i	$0.380236\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	68.1496			0.176097
$388$		0			0
$389$		−	371.606i		−	0.955286i	−0.878554	−	0.477643i	$-0.841491\pi$
$389$		−	371.606i		−	0.955286i	0.878554	−	0.477643i	$-0.158509\pi$
$390$		0			0
$391$	439.859	+	330.017i	1.12496	+	0.844032i
$392$		0			0
$393$			997.754i			2.53882i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	384.323i		−	0.968067i	−0.875050	−	0.484033i	$-0.839171\pi$
$397$		−	384.323i		−	0.968067i	0.875050	−	0.484033i	$-0.160829\pi$
$398$		0			0
$399$	−184.424			−0.462217
$400$		0			0
$401$		−	50.5408i		−	0.126037i	−0.998012	−	0.0630185i	$-0.979927\pi$
$401$		−	50.5408i		−	0.126037i	0.998012	−	0.0630185i	$-0.0200727\pi$
$402$		0			0
$403$		−	847.607i		−	2.10324i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			298.840i			0.734250i
$408$		0			0
$409$	−252.368			−0.617036			−0.308518	−	0.951219i	$-0.599833\pi$
$409$	−252.368			−0.617036			−0.308518	+	0.951219i	$0.599833\pi$
$410$		0			0
$411$		−	438.508i		−	1.06693i
$412$		0			0
$413$	−492.648			−1.19285
$414$		0			0
$415$		0			0
$416$		0			0
$417$		−	42.3594i		−	0.101581i
$418$		0			0
$419$			335.284i			0.800201i	0.916471	+	0.400100i	$0.131025\pi$
$419$			335.284i			0.800201i	−0.916471	+	0.400100i	$0.868975\pi$
$420$		0			0
$421$			255.121i			0.605987i	0.952993	+	0.302994i	$0.0979861\pi$
$421$			255.121i			0.605987i	−0.952993	+	0.302994i	$0.902014\pi$
$422$		0			0
$423$			345.117i			0.815879i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	714.722i		−	1.67382i
$428$		0			0
$429$			693.939i			1.61757i
$430$		0			0
$431$			461.701i			1.07123i	0.844462	+	0.535616i	$0.179921\pi$
$431$			461.701i			1.07123i	−0.844462	+	0.535616i	$0.820079\pi$
$432$		0			0
$433$	474.466			1.09576			0.547882	−	0.836555i	$-0.315434\pi$
$433$	474.466			1.09576			0.547882	+	0.836555i	$0.315434\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	104.254	−	138.954i	0.238568	−	0.317973i
$438$		0			0
$439$	115.013			0.261990			0.130995	−	0.991383i	$-0.458183\pi$
$439$	115.013			0.261990			0.130995	+	0.991383i	$0.458183\pi$
$440$		0			0
$441$	59.1646			0.134160
$442$		0			0
$443$		−	319.337i		−	0.720852i	−0.932788	−	0.360426i	$-0.882631\pi$
$443$		−	319.337i		−	0.720852i	0.932788	−	0.360426i	$-0.117369\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	216.527			0.484401
$448$		0			0
$449$	332.870			0.741358			0.370679	−	0.928761i	$-0.379125\pi$
$449$	332.870			0.741358			0.370679	+	0.928761i	$0.379125\pi$
$450$		0			0
$451$			441.331i			0.978561i
$452$		0			0
$453$			397.267i			0.876968i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	553.060			1.21020			0.605098	−	0.796151i	$-0.293134\pi$
$457$	553.060			1.21020			0.605098	+	0.796151i	$0.293134\pi$
$458$		0			0
$459$			265.719i			0.578908i
$460$		0			0
$461$	28.9872			0.0628790			0.0314395	−	0.999506i	$-0.489991\pi$
$461$	28.9872			0.0628790			0.0314395	+	0.999506i	$0.489991\pi$
$462$		0			0
$463$			784.874i			1.69519i	0.530642	+	0.847596i	$0.321951\pi$
$463$			784.874i			1.69519i	−0.530642	+	0.847596i	$0.678049\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−56.1876			−0.120316			−0.0601580	−	0.998189i	$-0.519160\pi$
$467$	−56.1876			−0.120316			−0.0601580	+	0.998189i	$0.519160\pi$
$468$		0			0
$469$	−350.442			−0.747210
$470$		0			0
$471$			462.637i			0.982243i
$472$		0			0
$473$		−	123.866i		−	0.261874i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	568.281			1.19136
$478$		0			0
$479$			415.801i			0.868061i	0.900898	+	0.434031i	$0.142909\pi$
$479$			415.801i			0.868061i	−0.900898	+	0.434031i	$0.857091\pi$
$480$		0			0
$481$		−	427.313i		−	0.888385i
$482$		0			0
$483$	337.045	−	449.228i	0.697817	−	0.930078i
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	776.789i		−	1.59505i	−0.603287	−	0.797524i	$-0.706142\pi$
$487$		−	776.789i		−	1.59505i	0.603287	−	0.797524i	$-0.293858\pi$
$488$		0			0
$489$	825.224			1.68757
$490$		0			0
$491$	−3.76293			−0.00766382			−0.00383191	−	0.999993i	$-0.501220\pi$
$491$	−3.76293			−0.00766382			−0.00383191	+	0.999993i	$0.501220\pi$
$492$		0			0
$493$	−247.354			−0.501733
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−373.215			−0.750936
$498$		0			0
$499$	−468.553			−0.938984			−0.469492	−	0.882937i	$-0.655563\pi$
$499$	−468.553			−0.938984			−0.469492	+	0.882937i	$0.655563\pi$
$500$		0			0
$501$	93.8097			0.187245
$502$		0			0
$503$	410.958			0.817014			0.408507	−	0.912755i	$-0.366050\pi$
$503$	410.958			0.817014			0.408507	+	0.912755i	$0.366050\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		−	334.599i		−	0.659959i
$508$		0			0
$509$	734.337			1.44271			0.721353	−	0.692567i	$-0.243521\pi$
$509$	734.337			1.44271			0.721353	+	0.692567i	$0.243521\pi$
$510$		0			0
$511$		−	60.6887i		−	0.118764i
$512$		0			0
$513$	83.9423			0.163630
$514$		0			0
$515$		0			0
$516$		0			0
$517$	627.272			1.21329
$518$		0			0
$519$	−90.5799			−0.174528
$520$		0			0
$521$		−	590.079i		−	1.13259i	−0.824203	−	0.566295i	$-0.808376\pi$
$521$		−	590.079i		−	1.13259i	0.824203	−	0.566295i	$-0.191624\pi$
$522$		0			0
$523$	363.326			0.694696			0.347348	−	0.937736i	$-0.387082\pi$
$523$	363.326			0.694696			0.347348	+	0.937736i	$0.387082\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1269.09			2.40815
$528$		0			0
$529$	147.940	+	507.893i	0.279659	+	0.960099i
$530$		0			0
$531$	−482.400			−0.908475
$532$		0			0
$533$		−	631.063i		−	1.18398i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		−	301.076i		−	0.560662i
$538$		0			0
$539$		−	107.536i		−	0.199509i
$540$		0			0
$541$	−280.552			−0.518580			−0.259290	−	0.965799i	$-0.583489\pi$
$541$	−280.552			−0.518580			−0.259290	+	0.965799i	$0.583489\pi$
$542$		0			0
$543$	−995.184			−1.83275
$544$		0			0
$545$		0			0
$546$		0			0
$547$			361.521i			0.660916i	0.943821	+	0.330458i	$0.107203\pi$
$547$			361.521i			0.660916i	−0.943821	+	0.330458i	$0.892797\pi$
$548$		0			0
$549$		−	699.855i		−	1.27478i
$550$		0			0
$551$			78.1407i			0.141816i
$552$		0			0
$553$		−	473.265i		−	0.855814i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	10.3628			0.0186046			0.00930229	−	0.999957i	$-0.497039\pi$
$557$	10.3628			0.0186046			0.00930229	+	0.999957i	$0.497039\pi$
$558$		0			0
$559$			177.117i			0.316847i
$560$		0			0
$561$	−1039.01			−1.85207
$562$		0			0
$563$	−191.929			−0.340903			−0.170452	−	0.985366i	$-0.554523\pi$
$563$	−191.929			−0.340903			−0.170452	+	0.985366i	$0.554523\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	618.342			1.09055
$568$		0			0
$569$			386.111i			0.678578i	0.940682	+	0.339289i	$0.110186\pi$
$569$			386.111i			0.678578i	−0.940682	+	0.339289i	$0.889814\pi$
$570$		0			0
$571$		−	497.294i		−	0.870917i	−0.900209	−	0.435459i	$-0.856586\pi$
$571$		−	497.294i		−	0.870917i	0.900209	−	0.435459i	$-0.143414\pi$
$572$		0			0
$573$	−633.659			−1.10586
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	1072.91i		−	1.85945i	−0.368249	−	0.929727i	$-0.620042\pi$
$577$		−	1072.91i		−	1.85945i	0.368249	−	0.929727i	$-0.379958\pi$
$578$		0			0
$579$	789.237			1.36310
$580$		0			0
$581$	888.123			1.52861
$582$		0			0
$583$		−	1032.89i		−	1.77168i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			514.836i			0.877064i	0.898716	+	0.438532i	$0.144501\pi$
$587$			514.836i			0.877064i	−0.898716	+	0.438532i	$0.855499\pi$
$588$		0			0
$589$		−	400.915i		−	0.680671i
$590$		0			0
$591$	813.702			1.37682
$592$		0			0
$593$			306.334i			0.516584i	0.966067	+	0.258292i	$0.0831597\pi$
$593$			306.334i			0.516584i	−0.966067	+	0.258292i	$0.916840\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−1165.70			−1.95260
$598$		0			0
$599$	674.082			1.12534			0.562672	−	0.826680i	$-0.309773\pi$
$599$	674.082			1.12534			0.562672	+	0.826680i	$0.309773\pi$
$600$		0			0
$601$	399.477			0.664687			0.332343	−	0.943158i	$-0.392161\pi$
$601$	399.477			0.664687			0.332343	+	0.943158i	$0.392161\pi$
$602$		0			0
$603$	−343.152			−0.569075
$604$		0			0
$605$		0			0
$606$		0			0
$607$			151.528i			0.249635i	0.992180	+	0.124817i	$0.0398345\pi$
$607$			151.528i			0.249635i	−0.992180	+	0.124817i	$0.960166\pi$
$608$		0			0
$609$			252.622i			0.414815i
$610$		0			0
$611$	−896.941			−1.46799
$612$		0			0
$613$	−361.724			−0.590088			−0.295044	−	0.955484i	$-0.595334\pi$
$613$	−361.724			−0.590088			−0.295044	+	0.955484i	$0.595334\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	235.561			0.381784			0.190892	−	0.981611i	$-0.438862\pi$
$617$	235.561			0.381784			0.190892	+	0.981611i	$0.438862\pi$
$618$		0			0
$619$		−	706.923i		−	1.14204i	−0.820936	−	0.571020i	$-0.806548\pi$
$619$		−	706.923i		−	1.14204i	0.820936	−	0.571020i	$-0.193452\pi$
$620$		0			0
$621$	−153.409	+	204.470i	−0.247035	+	0.329259i
$622$		0			0
$623$		−	136.386i		−	0.218919i
$624$		0			0
$625$		0			0
$626$		0			0
$627$			328.231i			0.523494i
$628$		0			0
$629$	639.803			1.01717
$630$		0			0
$631$			547.820i			0.868178i	0.900870	+	0.434089i	$0.142930\pi$
$631$			547.820i			0.868178i	−0.900870	+	0.434089i	$0.857070\pi$
$632$		0			0
$633$		−	423.680i		−	0.669321i
$634$		0			0
$635$		0			0
$636$		0			0
$637$			153.766i			0.241391i
$638$		0			0
$639$	−365.452			−0.571912
$640$		0			0
$641$		−	71.0439i		−	0.110833i	−0.998463	−	0.0554165i	$-0.982351\pi$
$641$		−	71.0439i		−	0.110833i	0.998463	−	0.0554165i	$-0.0176487\pi$
$642$		0			0
$643$	528.466			0.821875			0.410938	−	0.911663i	$-0.365201\pi$
$643$	528.466			0.821875			0.410938	+	0.911663i	$0.365201\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	402.365i		−	0.621894i	−0.950427	−	0.310947i	$-0.899354\pi$
$647$		−	402.365i		−	0.621894i	0.950427	−	0.310947i	$-0.100646\pi$
$648$		0			0
$649$			876.794i			1.35099i
$650$		0			0
$651$		−	1296.12i		−	1.99098i
$652$		0			0
$653$		−	814.505i		−	1.24733i	−0.781693	−	0.623664i	$-0.785644\pi$
$653$		−	814.505i		−	1.24733i	0.781693	−	0.623664i	$-0.214356\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		−	59.4263i		−	0.0904510i
$658$		0			0
$659$		−	195.734i		−	0.297017i	−0.988911	−	0.148508i	$-0.952553\pi$
$659$		−	195.734i		−	0.297017i	0.988911	−	0.148508i	$-0.0474472\pi$
$660$		0			0
$661$		−	971.202i		−	1.46929i	−0.678451	−	0.734646i	$-0.737348\pi$
$661$		−	971.202i		−	1.46929i	0.678451	−	0.734646i	$-0.262652\pi$
$662$		0			0
$663$	1485.69			2.24086
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−190.338	−	142.806i	−0.285364	−	0.214102i
$668$		0			0
$669$	−353.194			−0.527943
$670$		0			0
$671$	−1272.03			−1.89573
$672$		0			0
$673$			133.183i			0.197895i	0.995093	+	0.0989475i	$0.0315476\pi$
$673$			133.183i			0.197895i	−0.995093	+	0.0989475i	$0.968452\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−795.149			−1.17452			−0.587259	−	0.809399i	$-0.699793\pi$
$677$	−795.149			−1.17452			−0.587259	+	0.809399i	$0.699793\pi$
$678$		0			0
$679$	123.317			0.181615
$680$		0			0
$681$		−	624.425i		−	0.916924i
$682$		0			0
$683$			801.446i			1.17342i	0.809797	+	0.586710i	$0.199577\pi$
$683$			801.446i			1.17342i	−0.809797	+	0.586710i	$0.800423\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	1430.87			2.08278
$688$		0			0
$689$			1476.93i			2.14359i
$690$		0			0
$691$	1199.78			1.73629			0.868147	−	0.496307i	$-0.165311\pi$
$691$	1199.78			1.73629			0.868147	+	0.496307i	$0.165311\pi$
$692$		0			0
$693$			430.514i			0.621232i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	944.871			1.35563
$698$		0			0
$699$	−105.165			−0.150450
$700$		0			0
$701$		−	1045.49i		−	1.49143i	−0.666263	−	0.745717i	$-0.732107\pi$
$701$		−	1045.49i		−	1.49143i	0.666263	−	0.745717i	$-0.267893\pi$
$702$		0			0
$703$		−	202.118i		−	0.287508i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−57.5585			−0.0814124
$708$		0			0
$709$			1014.25i			1.43053i	0.698852	+	0.715266i	$0.253695\pi$
$709$			1014.25i			1.43053i	−0.698852	+	0.715266i	$0.746305\pi$
$710$		0			0
$711$		−	463.421i		−	0.651787i
$712$		0			0
$713$	976.563	+	732.693i	1.36965	+	1.02762i
$714$		0			0
$715$		0			0
$716$		0			0
$717$			142.352i			0.198538i
$718$		0			0
$719$	−172.880			−0.240445			−0.120222	−	0.992747i	$-0.538361\pi$
$719$	−172.880			−0.240445			−0.120222	+	0.992747i	$0.538361\pi$
$720$		0			0
$721$	−919.245			−1.27496
$722$		0			0
$723$	−292.991			−0.405243
$724$		0			0
$725$		0			0
$726$		0			0
$727$	586.143			0.806249			0.403125	−	0.915145i	$-0.367924\pi$
$727$	586.143			0.806249			0.403125	+	0.915145i	$0.367924\pi$
$728$		0			0
$729$	216.226			0.296607
$730$		0			0
$731$	−265.192			−0.362780
$732$		0			0
$733$	697.035			0.950934			0.475467	−	0.879734i	$-0.342279\pi$
$733$	697.035			0.950934			0.475467	+	0.879734i	$0.342279\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			623.701i			0.846270i
$738$		0			0
$739$	−1363.75			−1.84539			−0.922696	−	0.385527i	$-0.874019\pi$
$739$	−1363.75			−1.84539			−0.922696	+	0.385527i	$0.874019\pi$
$740$		0			0
$741$		−	469.340i		−	0.633387i
$742$		0			0
$743$	419.771			0.564967			0.282484	−	0.959272i	$-0.408842\pi$
$743$	419.771			0.564967			0.282484	+	0.959272i	$0.408842\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	869.649			1.16419
$748$		0			0
$749$	1275.04			1.70232
$750$		0			0
$751$			101.842i			0.135608i	0.997699	+	0.0678040i	$0.0215993\pi$
$751$			101.842i			0.135608i	−0.997699	+	0.0678040i	$0.978401\pi$
$752$		0			0
$753$	−1145.43			−1.52116
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−330.800			−0.436989			−0.218494	−	0.975838i	$-0.570114\pi$
$757$	−330.800			−0.436989			−0.218494	+	0.975838i	$0.570114\pi$
$758$		0			0
$759$	−799.516	−	599.859i	−1.05338	−	0.790328i
$760$		0			0
$761$	−673.019			−0.884388			−0.442194	−	0.896920i	$-0.645800\pi$
$761$	−673.019			−0.884388			−0.442194	+	0.896920i	$0.645800\pi$
$762$		0			0
$763$		−	191.099i		−	0.250457i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	1253.73i		−	1.63459i
$768$		0			0
$769$			511.255i			0.664831i	0.943133	+	0.332415i	$0.107864\pi$
$769$			511.255i			0.664831i	−0.943133	+	0.332415i	$0.892136\pi$
$770$		0			0
$771$	−963.783			−1.25004
$772$		0			0
$773$	1277.42			1.65254			0.826272	−	0.563272i	$-0.190458\pi$
$773$	1277.42			1.65254			0.826272	+	0.563272i	$0.190458\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		−	653.430i		−	0.840965i
$778$		0			0
$779$		−	298.491i		−	0.383172i
$780$		0			0
$781$			664.232i			0.850490i
$782$		0			0
$783$		−	114.983i		−	0.146849i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−1364.89			−1.73430			−0.867149	−	0.498049i	$-0.834050\pi$
$787$	−1364.89			−1.73430			−0.867149	+	0.498049i	$0.834050\pi$
$788$		0			0
$789$			576.090i			0.730152i
$790$		0			0
$791$	597.614			0.755517
$792$		0			0
$793$	1818.89			2.29368
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−164.706			−0.206657			−0.103329	−	0.994647i	$-0.532949\pi$
$797$	−164.706			−0.206657			−0.103329	+	0.994647i	$0.532949\pi$
$798$		0			0
$799$		−	1342.96i		−	1.68080i
$800$		0			0
$801$		−	133.549i		−	0.166728i
$802$		0			0
$803$	−108.011			−0.134509
$804$		0			0
$805$		0			0
$806$		0			0
$807$		−	1194.32i		−	1.47995i
$808$		0			0
$809$	−405.500			−0.501236			−0.250618	−	0.968086i	$-0.580634\pi$
$809$	−405.500			−0.501236			−0.250618	+	0.968086i	$0.580634\pi$
$810$		0			0
$811$	224.885			0.277294			0.138647	−	0.990342i	$-0.455725\pi$
$811$	224.885			0.277294			0.138647	+	0.990342i	$0.455725\pi$
$812$		0			0
$813$			7.62947i			0.00938434i
$814$		0			0
$815$		0			0
$816$		0			0
$817$			83.7760i			0.102541i
$818$		0			0
$819$		−	615.596i		−	0.751643i
$820$		0			0
$821$	589.271			0.717748			0.358874	−	0.933386i	$-0.383161\pi$
$821$	589.271			0.717748			0.358874	+	0.933386i	$0.383161\pi$
$822$		0			0
$823$			1009.61i			1.22674i	0.789796	+	0.613370i	$0.210186\pi$
$823$			1009.61i			1.22674i	−0.789796	+	0.613370i	$0.789814\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	359.328			0.434496			0.217248	−	0.976116i	$-0.430292\pi$
$827$	359.328			0.434496			0.217248	+	0.976116i	$0.430292\pi$
$828$		0			0
$829$	1320.55			1.59294			0.796469	−	0.604680i	$-0.206699\pi$
$829$	1320.55			1.59294			0.796469	+	0.604680i	$0.206699\pi$
$830$		0			0
$831$	−1226.98			−1.47652
$832$		0			0
$833$	−230.229			−0.276385
$834$		0			0
$835$		0			0
$836$		0			0
$837$			589.942i			0.704829i
$838$		0			0
$839$		−	771.287i		−	0.919294i	−0.888102	−	0.459647i	$-0.847976\pi$
$839$		−	771.287i		−	0.919294i	0.888102	−	0.459647i	$-0.152024\pi$
$840$		0			0
$841$	−733.964			−0.872727
$842$		0			0
$843$	−400.700			−0.475326
$844$		0			0
$845$		0			0
$846$		0			0
$847$	23.2618			0.0274638
$848$		0			0
$849$			1358.42i			1.60003i
$850$		0			0
$851$	492.326	+	369.381i	0.578526	+	0.434055i
$852$		0			0
$853$		−	1277.60i		−	1.49777i	−0.662697	−	0.748887i	$-0.730588\pi$
$853$		−	1277.60i		−	1.49777i	0.662697	−	0.748887i	$-0.269412\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			740.983i			0.864625i	0.901724	+	0.432312i	$0.142302\pi$
$857$			740.983i			0.864625i	−0.901724	+	0.432312i	$0.857698\pi$
$858$		0			0
$859$	−641.980			−0.747358			−0.373679	−	0.927558i	$-0.621904\pi$
$859$	−641.980			−0.747358			−0.373679	+	0.927558i	$0.621904\pi$
$860$		0			0
$861$		−	964.995i		−	1.12078i
$862$		0			0
$863$		−	752.491i		−	0.871948i	−0.899959	−	0.435974i	$-0.856404\pi$
$863$		−	752.491i		−	0.871948i	0.899959	−	0.435974i	$-0.143596\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			1099.83i			1.26854i
$868$		0			0
$869$	−842.297			−0.969271
$870$		0			0
$871$		−	891.835i		−	1.02392i
$872$		0			0
$873$	120.752			0.138318
$874$		0			0
$875$		0			0
$876$		0			0
$877$			1382.47i			1.57636i	0.615443	+	0.788182i	$0.288977\pi$
$877$			1382.47i			1.57636i	−0.615443	+	0.788182i	$0.711023\pi$
$878$		0			0
$879$			702.995i			0.799766i
$880$		0			0
$881$			1443.26i			1.63821i	0.573644	+	0.819104i	$0.305529\pi$
$881$			1443.26i			1.63821i	−0.573644	+	0.819104i	$0.694471\pi$
$882$		0			0
$883$			89.2549i			0.101081i	0.998722	+	0.0505407i	$0.0160945\pi$
$883$			89.2549i			0.101081i	−0.998722	+	0.0505407i	$0.983906\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	1753.34i		−	1.97671i	−0.152166	−	0.988355i	$-0.548625\pi$
$887$		−	1753.34i		−	1.97671i	0.152166	−	0.988355i	$-0.451375\pi$
$888$		0			0
$889$		−	1107.55i		−	1.24584i
$890$		0			0
$891$		−	1100.50i		−	1.23513i
$892$		0			0
$893$	−424.250			−0.475084
$894$		0			0
$895$		0			0
$896$		0			0
$897$	1143.23	+	857.743i	1.27451	+	0.956235i
$898$		0			0
$899$	−549.169			−0.610866
$900$		0			0
$901$	−2211.37			−2.45435
$902$		0			0
$903$			270.841i			0.299934i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−573.867			−0.632709			−0.316355	−	0.948641i	$-0.602459\pi$
$907$	−573.867			−0.632709			−0.316355	+	0.948641i	$0.602459\pi$
$908$		0			0
$909$	−56.3613			−0.0620036
$910$		0			0
$911$		−	786.535i		−	0.863375i	−0.902023	−	0.431687i	$-0.857918\pi$
$911$		−	786.535i		−	0.863375i	0.902023	−	0.431687i	$-0.142082\pi$
$912$		0			0
$913$		−	1580.64i		−	1.73126i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1608.75			−1.75436
$918$		0			0
$919$		−	282.876i		−	0.307808i	−0.988086	−	0.153904i	$-0.950815\pi$
$919$		−	282.876i		−	0.307808i	0.988086	−	0.153904i	$-0.0491847\pi$
$920$		0			0
$921$	−1882.44			−2.04390
$922$		0			0
$923$		−	949.791i		−	1.02903i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−900.124			−0.971008
$928$		0			0
$929$	−365.054			−0.392954			−0.196477	−	0.980508i	$-0.562950\pi$
$929$	−365.054			−0.392954			−0.196477	+	0.980508i	$0.562950\pi$
$930$		0			0
$931$			72.7307i			0.0781211i
$932$		0			0
$933$		−	950.645i		−	1.01891i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−883.636			−0.943048			−0.471524	−	0.881853i	$-0.656296\pi$
$937$	−883.636			−0.943048			−0.471524	+	0.881853i	$0.656296\pi$
$938$		0			0
$939$		−	1821.14i		−	1.93945i
$940$		0			0
$941$		−	424.118i		−	0.450710i	−0.974277	−	0.225355i	$-0.927646\pi$
$941$		−	424.118i		−	0.450710i	0.974277	−	0.225355i	$-0.0723542\pi$
$942$		0			0
$943$	727.074	+	545.507i	0.771022	+	0.578481i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	53.7644i		−	0.0567734i	−0.999597	−	0.0283867i	$-0.990963\pi$
$947$		−	53.7644i		−	0.0567734i	0.999597	−	0.0283867i	$-0.00903697\pi$
$948$		0			0
$949$	154.446			0.162746
$950$		0			0
$951$	−2262.57			−2.37915
$952$		0			0
$953$	768.678			0.806588			0.403294	−	0.915070i	$-0.367865\pi$
$953$	768.678			0.806588			0.403294	+	0.915070i	$0.367865\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	449.607			0.469808
$958$		0			0
$959$	707.036			0.737263
$960$		0			0
$961$	1856.61			1.93195
$962$		0			0
$963$	1248.52			1.29649
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	1687.63i		−	1.74522i	−0.488419	−	0.872609i	$-0.662426\pi$
$967$		−	1687.63i		−	1.74522i	0.488419	−	0.872609i	$-0.337574\pi$
$968$		0			0
$969$	702.728			0.725209
$970$		0			0
$971$		−	1409.22i		−	1.45131i	−0.688059	−	0.725655i	$-0.741537\pi$
$971$		−	1409.22i		−	1.45131i	0.688059	−	0.725655i	$-0.258463\pi$
$972$		0			0
$973$	68.2988			0.0701941
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−327.355			−0.335062			−0.167531	−	0.985867i	$-0.553579\pi$
$977$	−327.355			−0.335062			−0.167531	+	0.985867i	$0.553579\pi$
$978$		0			0
$979$	−242.735			−0.247941
$980$		0			0
$981$		−	187.124i		−	0.190748i
$982$		0			0
$983$	−725.006			−0.737544			−0.368772	−	0.929520i	$-0.620222\pi$
$983$	−725.006			−0.737544			−0.368772	+	0.929520i	$0.620222\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−1371.56			−1.38963
$988$		0			0
$989$	−204.064	−	153.105i	−0.206334	−	0.154808i
$990$		0			0
$991$	−17.7701			−0.0179315			−0.00896575	−	0.999960i	$-0.502854\pi$
$991$	−17.7701			−0.0179315			−0.00896575	+	0.999960i	$0.502854\pi$
$992$		0			0
$993$		−	1558.66i		−	1.56964i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	859.050i		−	0.861635i	−0.902439	−	0.430818i	$-0.858225\pi$
$997$		−	859.050i		−	0.861635i	0.902439	−	0.430818i	$-0.141775\pi$
$998$		0			0
$999$			297.414i			0.297712i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2300.3.d.c.1149.10		32
5.2	odd	4		2300.3.f.d.1701.13	yes	16
5.3	odd	4		2300.3.f.c.1701.4	yes	16
5.4	even	2	inner	2300.3.d.c.1149.23		32
23.22	odd	2	inner	2300.3.d.c.1149.24		32
115.22	even	4		2300.3.f.d.1701.14	yes	16
115.68	even	4		2300.3.f.c.1701.3	✓	16
115.114	odd	2	inner	2300.3.d.c.1149.9		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2300.3.d.c.1149.9		32	115.114	odd	2	inner
2300.3.d.c.1149.10		32	1.1	even	1	trivial
2300.3.d.c.1149.23		32	5.4	even	2	inner
2300.3.d.c.1149.24		32	23.22	odd	2	inner
2300.3.f.c.1701.3	✓	16	115.68	even	4
2300.3.f.c.1701.4	yes	16	5.3	odd	4
2300.3.f.d.1701.13	yes	16	5.2	odd	4
2300.3.f.d.1701.14	yes	16	115.22	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:	\( N \)	\(=\)	\( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2300.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.89154i q^{3} -6.27459 q^{7} -6.14407 q^{9} +O(q^{10})\)	\(q+3.89154i q^{3} -6.27459 q^{7} -6.14407 q^{9} +11.1672i q^{11} -15.9681i q^{13} +23.9086 q^{17} -7.55287i q^{19} -24.4178i q^{21} +(18.3975 + 13.8033i) q^{23} +11.1140i q^{27} -10.3458 q^{29} +53.0811 q^{31} -43.4578 q^{33} +26.7604 q^{37} +62.1406 q^{39} +39.5201 q^{41} -11.0919 q^{43} -56.1707i q^{47} -9.62955 q^{49} +93.0412i q^{51} -92.4926 q^{53} +29.3923 q^{57} +78.5148 q^{59} +113.907i q^{61} +38.5515 q^{63} +55.8509 q^{67} +(-53.7159 + 71.5948i) q^{69} +59.4804 q^{71} +9.67213i q^{73} -70.0698i q^{77} +75.4257i q^{79} -98.5470 q^{81} -141.543 q^{83} -40.2612i q^{87} +21.7363i q^{89} +100.193i q^{91} +206.567i q^{93} -19.6534 q^{97} -68.6123i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 128 q^{9}+O(q^{10}) \) 32 * q - 128 * q^9	\( 32 q - 128 q^{9} + 12 q^{29} + 56 q^{31} - 148 q^{39} - 180 q^{41} + 380 q^{49} + 348 q^{59} - 100 q^{69} + 232 q^{71} + 112 q^{81}+O(q^{100}) \) 32 * q - 128 * q^9 + 12 * q^29 + 56 * q^31 - 148 * q^39 - 180 * q^41 + 380 * q^49 + 348 * q^59 - 100 * q^69 + 232 * q^71 + 112 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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