LMFDB - Embedded newform 608.3.e.b.417.18

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [608,3,Mod(417,608)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("608.417"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(608, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 608 = 2^{5} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	608.e (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [20,0,0,0,0,0,0,0,-52,0,0,0,0,0,0,0,56] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$16.5668000731$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 8 x^{19} - 196 x^{18} + 1676 x^{17} + 16346 x^{16} - 161824 x^{15} - 667200 x^{14} + \cdots + 1135285065792 $ x^20 - 8x^19 - 196x^18 + 1676x^17 + 16346x^16 - 161824x^15 - 667200x^14 + 8850712x^13 + 9918758x^12 - 288078272x^11 + 224256138x^10 + 5383493580x^9 - 11116604703x^8 - 53755955160x^7 + 166723956672x^6 + 262105513072x^5 - 1053812084224x^4 - 855143026944x^3 + 3318150008320x^2 + 3262428344448*x + 1135285065792
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{27} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			417.18
Root			$5.68070 - 2.14790i$ of defining polynomial
Character	$\chi$	$=$	608.417
Dual form			608.3.e.b.417.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+4.29580i q^{3} +2.32882 q^{5} +2.95740 q^{7} -9.45390 q^{9} -19.4859 q^{11} -14.0235i q^{13} +10.0041i q^{15} -22.2731 q^{17} +(10.5756 + 15.7847i) q^{19} +12.7044i q^{21} -41.7289 q^{23} -19.5766 q^{25} -1.94987i q^{27} -22.9343i q^{29} +30.9806i q^{31} -83.7075i q^{33} +6.88725 q^{35} +62.1601i q^{37} +60.2423 q^{39} -20.2283i q^{41} +53.2356 q^{43} -22.0164 q^{45} -21.3514 q^{47} -40.2538 q^{49} -95.6807i q^{51} +12.4978i q^{53} -45.3791 q^{55} +(-67.8080 + 45.4305i) q^{57} -19.2741i q^{59} +4.96429 q^{61} -27.9590 q^{63} -32.6583i q^{65} -58.9495i q^{67} -179.259i q^{69} -61.6822i q^{71} +14.5807 q^{73} -84.0972i q^{75} -57.6276 q^{77} +113.346i q^{79} -76.7089 q^{81} +48.9861 q^{83} -51.8699 q^{85} +98.5213 q^{87} +27.8155i q^{89} -41.4733i q^{91} -133.087 q^{93} +(24.6285 + 36.7597i) q^{95} -144.982i q^{97} +184.218 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 52 q^{9} + 56 q^{17} + 36 q^{25} + 64 q^{45} + 332 q^{49} + 88 q^{57} - 32 q^{61} - 152 q^{73} + 360 q^{77} - 476 q^{81} - 552 q^{85} + 336 q^{93}+O(q^{100}) $ 20 * q - 52 * q^9 + 56 * q^17 + 36 * q^25 + 64 * q^45 + 332 * q^49 + 88 * q^57 - 32 * q^61 - 152 * q^73 + 360 * q^77 - 476 * q^81 - 552 * q^85 + 336 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/608\mathbb{Z}\right)^\times$.

$n$	$97$	$191$	$229$
$\chi(n)$	$-1$	$1$	$1$

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
$3$			4.29580i			1.43193i	0.698134	+	0.715967i	$0.254014\pi$
$3$			4.29580i			1.43193i	−0.698134	+	0.715967i	$0.745986\pi$
$4$		0			0
$5$	2.32882			0.465763			0.232882	−	0.972505i	$-0.425184\pi$
$5$	2.32882			0.465763			0.232882	+	0.972505i	$0.425184\pi$
$6$		0			0
$7$	2.95740			0.422486			0.211243	−	0.977434i	$-0.432249\pi$
$7$	2.95740			0.422486			0.211243	+	0.977434i	$0.432249\pi$
$8$		0			0
$9$	−9.45390			−1.05043
$10$		0			0
$11$	−19.4859			−1.77144			−0.885722	−	0.464216i	$-0.846336\pi$
$11$	−19.4859			−1.77144			−0.885722	+	0.464216i	$0.846336\pi$
$12$		0			0
$13$		−	14.0235i		−	1.07873i	−0.842071	−	0.539367i	$-0.818664\pi$
$13$		−	14.0235i		−	1.07873i	0.842071	−	0.539367i	$-0.181336\pi$
$14$		0			0
$15$			10.0041i			0.666942i
$16$		0			0
$17$	−22.2731			−1.31018			−0.655090	−	0.755551i	$-0.727369\pi$
$17$	−22.2731			−1.31018			−0.655090	+	0.755551i	$0.727369\pi$
$18$		0			0
$19$	10.5756	+	15.7847i	0.556608	+	0.830775i
$19$	10.5756	+	15.7847i	0.556608	+	0.830775i
$20$		0			0
$21$			12.7044i			0.604972i
$22$		0			0
$23$	−41.7289			−1.81430			−0.907150	−	0.420808i	$-0.861747\pi$
$23$	−41.7289			−1.81430			−0.907150	+	0.420808i	$0.861747\pi$
$24$		0			0
$25$	−19.5766			−0.783064
$26$		0			0
$27$		−	1.94987i		−	0.0722174i
$28$		0			0
$29$		−	22.9343i		−	0.790839i	−0.918501	−	0.395420i	$-0.870599\pi$
$29$		−	22.9343i		−	0.790839i	0.918501	−	0.395420i	$-0.129401\pi$
$30$		0			0
$31$			30.9806i			0.999375i	0.866206	+	0.499688i	$0.166552\pi$
$31$			30.9806i			0.999375i	−0.866206	+	0.499688i	$0.833448\pi$
$32$		0			0
$33$		−	83.7075i		−	2.53659i
$34$		0			0
$35$	6.88725			0.196779
$36$		0			0
$37$			62.1601i			1.68000i	0.542584	+	0.840001i	$0.317446\pi$
$37$			62.1601i			1.68000i	−0.542584	+	0.840001i	$0.682554\pi$
$38$		0			0
$39$	60.2423			1.54468
$40$		0			0
$41$		−	20.2283i		−	0.493373i	−0.969095	−	0.246686i	$-0.920658\pi$
$41$		−	20.2283i		−	0.493373i	0.969095	−	0.246686i	$-0.0793417\pi$
$42$		0			0
$43$	53.2356			1.23804			0.619019	−	0.785376i	$-0.287530\pi$
$43$	53.2356			1.23804			0.619019	+	0.785376i	$0.287530\pi$
$44$		0			0
$45$	−22.0164			−0.489254
$46$		0			0
$47$	−21.3514			−0.454285			−0.227143	−	0.973861i	$-0.572938\pi$
$47$	−21.3514			−0.454285			−0.227143	+	0.973861i	$0.572938\pi$
$48$		0			0
$49$	−40.2538			−0.821505
$50$		0			0
$51$		−	95.6807i		−	1.87609i
$52$		0			0
$53$			12.4978i			0.235808i	0.993025	+	0.117904i	$0.0376174\pi$
$53$			12.4978i			0.235808i	−0.993025	+	0.117904i	$0.962383\pi$
$54$		0			0
$55$	−45.3791			−0.825074
$56$		0			0
$57$	−67.8080	+	45.4305i	−1.18961	+	0.797026i
$58$		0			0
$59$		−	19.2741i		−	0.326680i	−0.986570	−	0.163340i	$-0.947773\pi$
$59$		−	19.2741i		−	0.326680i	0.986570	−	0.163340i	$-0.0522267\pi$
$60$		0			0
$61$	4.96429			0.0813817			0.0406909	−	0.999172i	$-0.487044\pi$
$61$	4.96429			0.0813817			0.0406909	+	0.999172i	$0.487044\pi$
$62$		0			0
$63$	−27.9590			−0.443794
$64$		0			0
$65$		−	32.6583i		−	0.502435i
$66$		0			0
$67$		−	58.9495i		−	0.879844i	−0.898036	−	0.439922i	$-0.855006\pi$
$67$		−	58.9495i		−	0.879844i	0.898036	−	0.439922i	$-0.144994\pi$
$68$		0			0
$69$		−	179.259i		−	2.59796i
$70$		0			0
$71$		−	61.6822i		−	0.868763i	−0.900729	−	0.434382i	$-0.856967\pi$
$71$		−	61.6822i		−	0.868763i	0.900729	−	0.434382i	$-0.143033\pi$
$72$		0			0
$73$	14.5807			0.199736			0.0998680	−	0.995001i	$-0.468158\pi$
$73$	14.5807			0.199736			0.0998680	+	0.995001i	$0.468158\pi$
$74$		0			0
$75$		−	84.0972i		−	1.12130i
$76$		0			0
$77$	−57.6276			−0.748411
$78$		0			0
$79$			113.346i			1.43475i	0.696685	+	0.717377i	$0.254658\pi$
$79$			113.346i			1.43475i	−0.696685	+	0.717377i	$0.745342\pi$
$80$		0			0
$81$	−76.7089			−0.947023
$82$		0			0
$83$	48.9861			0.590194			0.295097	−	0.955467i	$-0.404648\pi$
$83$	48.9861			0.590194			0.295097	+	0.955467i	$0.404648\pi$
$84$		0			0
$85$	−51.8699			−0.610234
$86$		0			0
$87$	98.5213			1.13243
$88$		0			0
$89$			27.8155i			0.312534i	0.987715	+	0.156267i	$0.0499460\pi$
$89$			27.8155i			0.312534i	−0.987715	+	0.156267i	$0.950054\pi$
$90$		0			0
$91$		−	41.4733i		−	0.455750i
$92$		0			0
$93$	−133.087			−1.43104
$94$		0			0
$95$	24.6285	+	36.7597i	0.259248	+	0.386945i
$96$		0			0
$97$		−	144.982i		−	1.49466i	−0.664452	−	0.747331i	$-0.731335\pi$
$97$		−	144.982i		−	1.49466i	0.664452	−	0.747331i	$-0.268665\pi$
$98$		0			0
$99$	184.218			1.86078
$100$		0			0
$101$	150.334			1.48845			0.744226	−	0.667927i	$-0.232818\pi$
$101$	150.334			1.48845			0.744226	+	0.667927i	$0.232818\pi$
$102$		0			0
$103$			74.2695i			0.721063i	0.932747	+	0.360532i	$0.117405\pi$
$103$			74.2695i			0.721063i	−0.932747	+	0.360532i	$0.882595\pi$
$104$		0			0
$105$			29.5863i			0.281774i
$106$		0			0
$107$			139.704i			1.30564i	0.757511	+	0.652822i	$0.226415\pi$
$107$			139.704i			1.30564i	−0.757511	+	0.652822i	$0.773585\pi$
$108$		0			0
$109$		−	42.5178i		−	0.390072i	−0.980796	−	0.195036i	$-0.937518\pi$
$109$		−	42.5178i		−	0.390072i	0.980796	−	0.195036i	$-0.0624823\pi$
$110$		0			0
$111$	−267.027			−2.40565
$112$		0			0
$113$			109.368i			0.967862i	0.875106	+	0.483931i	$0.160791\pi$
$113$			109.368i			0.967862i	−0.875106	+	0.483931i	$0.839209\pi$
$114$		0			0
$115$	−97.1790			−0.845034
$116$		0			0
$117$			132.577i			1.13314i
$118$		0			0
$119$	−65.8705			−0.553533
$120$		0			0
$121$	258.700			2.13802
$122$		0			0
$123$	86.8966			0.706477
$124$		0			0
$125$	−103.811			−0.830486
$126$		0			0
$127$		−	40.2362i		−	0.316821i	−0.987373	−	0.158410i	$-0.949363\pi$
$127$		−	40.2362i		−	0.316821i	0.987373	−	0.158410i	$-0.0506369\pi$
$128$		0			0
$129$			228.690i			1.77279i
$130$		0			0
$131$	−65.6183			−0.500903			−0.250452	−	0.968129i	$-0.580579\pi$
$131$	−65.6183			−0.500903			−0.250452	+	0.968129i	$0.580579\pi$
$132$		0			0
$133$	31.2762	+	46.6818i	0.235159	+	0.350991i
$134$		0			0
$135$		−	4.54089i		−	0.0336362i
$136$		0			0
$137$	168.934			1.23310			0.616549	−	0.787316i	$-0.288530\pi$
$137$	168.934			1.23310			0.616549	+	0.787316i	$0.288530\pi$
$138$		0			0
$139$	−137.539			−0.989487			−0.494743	−	0.869039i	$-0.664738\pi$
$139$	−137.539			−0.989487			−0.494743	+	0.869039i	$0.664738\pi$
$140$		0			0
$141$		−	91.7214i		−	0.650506i
$142$		0			0
$143$			273.261i			1.91092i
$144$		0			0
$145$		−	53.4099i		−	0.368344i
$146$		0			0
$147$		−	172.922i		−	1.17634i
$148$		0			0
$149$	−50.0197			−0.335703			−0.167851	−	0.985812i	$-0.553683\pi$
$149$	−50.0197			−0.335703			−0.167851	+	0.985812i	$0.553683\pi$
$150$		0			0
$151$			204.762i			1.35604i	0.735045	+	0.678019i	$0.237161\pi$
$151$			204.762i			1.35604i	−0.735045	+	0.678019i	$0.762839\pi$
$152$		0			0
$153$	210.567			1.37626
$154$		0			0
$155$			72.1482i			0.465472i
$156$		0			0
$157$	57.6193			0.367002			0.183501	−	0.983020i	$-0.441257\pi$
$157$	57.6193			0.367002			0.183501	+	0.983020i	$0.441257\pi$
$158$		0			0
$159$	−53.6881			−0.337661
$160$		0			0
$161$	−123.409			−0.766517
$162$		0			0
$163$	−97.9480			−0.600908			−0.300454	−	0.953796i	$-0.597138\pi$
$163$	−97.9480			−0.600908			−0.300454	+	0.953796i	$0.597138\pi$
$164$		0			0
$165$		−	194.939i		−	1.18145i
$166$		0			0
$167$		−	84.7060i		−	0.507222i	−0.967306	−	0.253611i	$-0.918382\pi$
$167$		−	84.7060i		−	0.507222i	0.967306	−	0.253611i	$-0.0816182\pi$
$168$		0			0
$169$	−27.6597			−0.163667
$170$		0			0
$171$	−99.9803	−	149.227i	−0.584680	−	0.872674i
$172$		0			0
$173$		−	22.3618i		−	0.129259i	−0.997909	−	0.0646294i	$-0.979413\pi$
$173$		−	22.3618i		−	0.129259i	0.997909	−	0.0646294i	$-0.0205865\pi$
$174$		0			0
$175$	−57.8959			−0.330834
$176$		0			0
$177$	82.7977			0.467784
$178$		0			0
$179$			72.7922i			0.406661i	0.979110	+	0.203330i	$0.0651765\pi$
$179$			72.7922i			0.406661i	−0.979110	+	0.203330i	$0.934823\pi$
$180$		0			0
$181$			350.286i			1.93528i	0.252326	+	0.967642i	$0.418804\pi$
$181$			350.286i			1.93528i	−0.252326	+	0.967642i	$0.581196\pi$
$182$		0			0
$183$			21.3256i			0.116533i
$184$		0			0
$185$			144.760i			0.782484i
$186$		0			0
$187$	434.011			2.32091
$188$		0			0
$189$		−	5.76656i		−	0.0305109i
$190$		0			0
$191$	107.952			0.565194			0.282597	−	0.959239i	$-0.408804\pi$
$191$	107.952			0.565194			0.282597	+	0.959239i	$0.408804\pi$
$192$		0			0
$193$			267.636i			1.38672i	0.720593	+	0.693359i	$0.243870\pi$
$193$			267.636i			1.38672i	−0.720593	+	0.693359i	$0.756130\pi$
$194$		0			0
$195$	140.293			0.719453
$196$		0			0
$197$	290.715			1.47571			0.737854	−	0.674960i	$-0.235839\pi$
$197$	290.715			1.47571			0.737854	+	0.674960i	$0.235839\pi$
$198$		0			0
$199$	−302.296			−1.51907			−0.759537	−	0.650464i	$-0.774574\pi$
$199$	−302.296			−1.51907			−0.759537	+	0.650464i	$0.774574\pi$
$200$		0			0
$201$	253.235			1.25988
$202$		0			0
$203$		−	67.8261i		−	0.334119i
$204$		0			0
$205$		−	47.1080i		−	0.229795i
$206$		0			0
$207$	394.501			1.90580
$208$		0			0
$209$	−206.074	−	307.579i	−0.986001	−	1.47167i
$210$		0			0
$211$			73.5756i			0.348700i	0.984684	+	0.174350i	$0.0557824\pi$
$211$			73.5756i			0.348700i	−0.984684	+	0.174350i	$0.944218\pi$
$212$		0			0
$213$	264.974			1.24401
$214$		0			0
$215$	123.976			0.576633
$216$		0			0
$217$			91.6222i			0.422222i
$218$		0			0
$219$			62.6359i			0.286009i
$220$		0			0
$221$			312.347i			1.41334i
$222$		0			0
$223$		−	225.342i		−	1.01050i	−0.862973	−	0.505251i	$-0.831400\pi$
$223$		−	225.342i		−	1.01050i	0.862973	−	0.505251i	$-0.168600\pi$
$224$		0			0
$225$	185.075			0.822557
$226$		0			0
$227$		−	19.4056i		−	0.0854872i	−0.999086	−	0.0427436i	$-0.986390\pi$
$227$		−	19.4056i		−	0.0854872i	0.999086	−	0.0427436i	$-0.0136099\pi$
$228$		0			0
$229$	285.505			1.24675			0.623374	−	0.781924i	$-0.285762\pi$
$229$	285.505			1.24675			0.623374	+	0.781924i	$0.285762\pi$
$230$		0			0
$231$		−	247.557i		−	1.07167i
$232$		0			0
$233$	−359.060			−1.54103			−0.770515	−	0.637422i	$-0.780001\pi$
$233$	−359.060			−1.54103			−0.770515	+	0.637422i	$0.780001\pi$
$234$		0			0
$235$	−49.7235			−0.211590
$236$		0			0
$237$	−486.910			−2.05447
$238$		0			0
$239$	−182.395			−0.763160			−0.381580	−	0.924336i	$-0.624620\pi$
$239$	−182.395			−0.763160			−0.381580	+	0.924336i	$0.624620\pi$
$240$		0			0
$241$			119.880i			0.497429i	0.968577	+	0.248714i	$0.0800081\pi$
$241$			119.880i			0.497429i	−0.968577	+	0.248714i	$0.919992\pi$
$242$		0			0
$243$		−	347.075i		−	1.42829i
$244$		0			0
$245$	−93.7437			−0.382627
$246$		0			0
$247$	221.358	−	148.307i	0.896185	−	0.600432i
$248$		0			0
$249$			210.434i			0.845118i
$250$		0			0
$251$	−449.306			−1.79007			−0.895033	−	0.446000i	$-0.852848\pi$
$251$	−449.306			−1.79007			−0.895033	+	0.446000i	$0.852848\pi$
$252$		0			0
$253$	813.125			3.21393
$254$		0			0
$255$		−	222.823i		−	0.873815i
$256$		0			0
$257$		−	469.173i		−	1.82558i	−0.408432	−	0.912789i	$-0.633924\pi$
$257$		−	469.173i		−	1.82558i	0.408432	−	0.912789i	$-0.366076\pi$
$258$		0			0
$259$			183.833i			0.709778i
$260$		0			0
$261$			216.819i			0.830724i
$262$		0			0
$263$	265.109			1.00802			0.504010	−	0.863698i	$-0.331858\pi$
$263$	265.109			1.00802			0.504010	+	0.863698i	$0.331858\pi$
$264$		0			0
$265$			29.1051i			0.109831i
$266$		0			0
$267$	−119.490			−0.447528
$268$		0			0
$269$		−	303.491i		−	1.12822i	−0.825700	−	0.564109i	$-0.809220\pi$
$269$		−	303.491i		−	1.12822i	0.825700	−	0.564109i	$-0.190780\pi$
$270$		0			0
$271$	−200.301			−0.739117			−0.369559	−	0.929207i	$-0.620491\pi$
$271$	−200.301			−0.739117			−0.369559	+	0.929207i	$0.620491\pi$
$272$		0			0
$273$	178.161			0.652604
$274$		0			0
$275$	381.468			1.38716
$276$		0			0
$277$	−160.105			−0.577998			−0.288999	−	0.957329i	$-0.593322\pi$
$277$	−160.105			−0.577998			−0.288999	+	0.957329i	$0.593322\pi$
$278$		0			0
$279$		−	292.888i		−	1.04978i
$280$		0			0
$281$		−	383.009i		−	1.36302i	−0.731808	−	0.681511i	$-0.761323\pi$
$281$		−	383.009i		−	1.36302i	0.731808	−	0.681511i	$-0.238677\pi$
$282$		0			0
$283$	−226.723			−0.801140			−0.400570	−	0.916266i	$-0.631188\pi$
$283$	−226.723			−0.801140			−0.400570	+	0.916266i	$0.631188\pi$
$284$		0			0
$285$	−157.913	+	105.799i	−0.554079	+	0.371226i
$286$		0			0
$287$		−	59.8232i		−	0.208443i
$288$		0			0
$289$	207.090			0.716573
$290$		0			0
$291$	622.815			2.14026
$292$		0			0
$293$			429.962i			1.46745i	0.679447	+	0.733724i	$0.262220\pi$
$293$			429.962i			1.46745i	−0.679447	+	0.733724i	$0.737780\pi$
$294$		0			0
$295$		−	44.8859i		−	0.152155i
$296$		0			0
$297$			37.9950i			0.127929i
$298$		0			0
$299$			585.187i			1.95715i
$300$		0			0
$301$	157.439			0.523054
$302$		0			0
$303$			645.804i			2.13137i
$304$		0			0
$305$	11.5609			0.0379046
$306$		0			0
$307$		−	208.273i		−	0.678414i	−0.940712	−	0.339207i	$-0.889841\pi$
$307$		−	208.273i		−	0.678414i	0.940712	−	0.339207i	$-0.110159\pi$
$308$		0			0
$309$	−319.047			−1.03251
$310$		0			0
$311$	135.179			0.434659			0.217329	−	0.976098i	$-0.430265\pi$
$311$	135.179			0.434659			0.217329	+	0.976098i	$0.430265\pi$
$312$		0			0
$313$	−412.048			−1.31645			−0.658224	−	0.752823i	$-0.728692\pi$
$313$	−412.048			−1.31645			−0.658224	+	0.752823i	$0.728692\pi$
$314$		0			0
$315$	−65.1114			−0.206703
$316$		0			0
$317$			386.723i			1.21995i	0.792422	+	0.609973i	$0.208820\pi$
$317$			386.723i			1.21995i	−0.792422	+	0.609973i	$0.791180\pi$
$318$		0			0
$319$			446.896i			1.40093i
$320$		0			0
$321$	−600.140			−1.86960
$322$		0			0
$323$	−235.550	−	351.574i	−0.729257	−	1.08847i
$324$		0			0
$325$			274.533i			0.844718i
$326$		0			0
$327$	182.648			0.558557
$328$		0			0
$329$	−63.1447			−0.191929
$330$		0			0
$331$			143.670i			0.434047i	0.976166	+	0.217024i	$0.0696349\pi$
$331$			143.670i			0.434047i	−0.976166	+	0.217024i	$0.930365\pi$
$332$		0			0
$333$		−	587.655i		−	1.76473i
$334$		0			0
$335$		−	137.283i		−	0.409799i
$336$		0			0
$337$			162.296i			0.481591i	0.970576	+	0.240795i	$0.0774082\pi$
$337$			162.296i			0.481591i	−0.970576	+	0.240795i	$0.922592\pi$
$338$		0			0
$339$	−469.825			−1.38591
$340$		0			0
$341$		−	603.685i		−	1.77034i
$342$		0			0
$343$	−263.959			−0.769561
$344$		0			0
$345$		−	417.461i		−	1.21003i
$346$		0			0
$347$	−429.273			−1.23710			−0.618549	−	0.785746i	$-0.712279\pi$
$347$	−429.273			−1.23710			−0.618549	+	0.785746i	$0.712279\pi$
$348$		0			0
$349$	463.869			1.32914			0.664568	−	0.747227i	$-0.268615\pi$
$349$	463.869			1.32914			0.664568	+	0.747227i	$0.268615\pi$
$350$		0			0
$351$	−27.3441			−0.0779034
$352$		0			0
$353$	78.3436			0.221937			0.110968	−	0.993824i	$-0.464605\pi$
$353$	78.3436			0.221937			0.110968	+	0.993824i	$0.464605\pi$
$354$		0			0
$355$		−	143.647i		−	0.404638i
$356$		0			0
$357$		−	282.966i		−	0.792623i
$358$		0			0
$359$	−243.233			−0.677528			−0.338764	−	0.940871i	$-0.610009\pi$
$359$	−243.233			−0.677528			−0.338764	+	0.940871i	$0.610009\pi$
$360$		0			0
$361$	−137.315	+	333.865i	−0.380374	+	0.924833i
$362$		0			0
$363$			1111.32i			3.06150i
$364$		0			0
$365$	33.9558			0.0930297
$366$		0			0
$367$	−323.083			−0.880335			−0.440167	−	0.897916i	$-0.645081\pi$
$367$	−323.083			−0.880335			−0.440167	+	0.897916i	$0.645081\pi$
$368$		0			0
$369$			191.236i			0.518255i
$370$		0			0
$371$			36.9610i			0.0996255i
$372$		0			0
$373$		−	86.8197i		−	0.232761i	−0.993205	−	0.116380i	$-0.962871\pi$
$373$		−	86.8197i		−	0.232761i	0.993205	−	0.116380i	$-0.0371291\pi$
$374$		0			0
$375$		−	445.950i		−	1.18920i
$376$		0			0
$377$	−321.621			−0.853105
$378$		0			0
$379$			20.8856i			0.0551071i	0.999620	+	0.0275535i	$0.00877168\pi$
$379$			20.8856i			0.0551071i	−0.999620	+	0.0275535i	$0.991228\pi$
$380$		0			0
$381$	172.847			0.453666
$382$		0			0
$383$			310.036i			0.809492i	0.914429	+	0.404746i	$0.132640\pi$
$383$			310.036i			0.809492i	−0.914429	+	0.404746i	$0.867360\pi$
$384$		0			0
$385$	−134.204			−0.348582
$386$		0			0
$387$	−503.285			−1.30048
$388$		0			0
$389$	−483.760			−1.24360			−0.621799	−	0.783177i	$-0.713598\pi$
$389$	−483.760			−1.24360			−0.621799	+	0.783177i	$0.713598\pi$
$390$		0			0
$391$	929.430			2.37706
$392$		0			0
$393$		−	281.883i		−	0.717260i
$394$		0			0
$395$			263.961i			0.668256i
$396$		0			0
$397$	−114.551			−0.288543			−0.144271	−	0.989538i	$-0.546084\pi$
$397$	−114.551			−0.288543			−0.144271	+	0.989538i	$0.546084\pi$
$398$		0			0
$399$	−200.536	+	134.356i	−0.502596	+	0.336733i
$400$		0			0
$401$		−	20.2091i		−	0.0503968i	−0.999682	−	0.0251984i	$-0.991978\pi$
$401$		−	20.2091i		−	0.0503968i	0.999682	−	0.0251984i	$-0.00802175\pi$
$402$		0			0
$403$	434.458			1.07806
$404$		0			0
$405$	−178.641			−0.441089
$406$		0			0
$407$		−	1211.24i		−	2.97603i
$408$		0			0
$409$			137.213i			0.335485i	0.985831	+	0.167742i	$0.0536477\pi$
$409$			137.213i			0.335485i	−0.985831	+	0.167742i	$0.946352\pi$
$410$		0			0
$411$			725.709i			1.76571i
$412$		0			0
$413$		−	57.0013i		−	0.138018i
$414$		0			0
$415$	114.080			0.274891
$416$		0			0
$417$		−	590.839i		−	1.41688i
$418$		0			0
$419$	−69.2314			−0.165230			−0.0826151	−	0.996582i	$-0.526327\pi$
$419$	−69.2314			−0.165230			−0.0826151	+	0.996582i	$0.526327\pi$
$420$		0			0
$421$		−	96.9103i		−	0.230191i	−0.993354	−	0.115095i	$-0.963283\pi$
$421$		−	96.9103i		−	0.230191i	0.993354	−	0.115095i	$-0.0367174\pi$
$422$		0			0
$423$	201.854			0.477197
$424$		0			0
$425$	436.031			1.02596
$426$		0			0
$427$	14.6814			0.0343827
$428$		0			0
$429$	−1173.88			−2.73631
$430$		0			0
$431$		−	473.629i		−	1.09891i	−0.835524	−	0.549453i	$-0.814836\pi$
$431$		−	473.629i		−	1.09891i	0.835524	−	0.549453i	$-0.185164\pi$
$432$		0			0
$433$		−	371.336i		−	0.857588i	−0.903402	−	0.428794i	$-0.858939\pi$
$433$		−	371.336i		−	0.857588i	0.903402	−	0.428794i	$-0.141061\pi$
$434$		0			0
$435$	229.438			0.527444
$436$		0			0
$437$	−441.306	−	658.679i	−1.00985	−	1.50727i
$438$		0			0
$439$			576.675i			1.31361i	0.754060	+	0.656806i	$0.228093\pi$
$439$			576.675i			1.31361i	−0.754060	+	0.656806i	$0.771907\pi$
$440$		0			0
$441$	380.555			0.862937
$442$		0			0
$443$	−436.152			−0.984542			−0.492271	−	0.870442i	$-0.663833\pi$
$443$	−436.152			−0.984542			−0.492271	+	0.870442i	$0.663833\pi$
$444$		0			0
$445$			64.7773i			0.145567i
$446$		0			0
$447$		−	214.875i		−	0.480704i
$448$		0			0
$449$			875.653i			1.95023i	0.221699	+	0.975115i	$0.428840\pi$
$449$			875.653i			1.95023i	−0.221699	+	0.975115i	$0.571160\pi$
$450$		0			0
$451$			394.166i			0.873982i
$452$		0			0
$453$	−879.615			−1.94176
$454$		0			0
$455$		−	96.5837i		−	0.212272i
$456$		0			0
$457$	−135.889			−0.297350			−0.148675	−	0.988886i	$-0.547501\pi$
$457$	−135.889			−0.297350			−0.148675	+	0.988886i	$0.547501\pi$
$458$		0			0
$459$			43.4296i			0.0946179i
$460$		0			0
$461$	−258.134			−0.559943			−0.279972	−	0.960008i	$-0.590325\pi$
$461$	−258.134			−0.559943			−0.279972	+	0.960008i	$0.590325\pi$
$462$		0			0
$463$	83.7030			0.180784			0.0903920	−	0.995906i	$-0.471188\pi$
$463$	83.7030			0.180784			0.0903920	+	0.995906i	$0.471188\pi$
$464$		0			0
$465$	−309.934			−0.666526
$466$		0			0
$467$	210.142			0.449984			0.224992	−	0.974361i	$-0.427764\pi$
$467$	210.142			0.449984			0.224992	+	0.974361i	$0.427764\pi$
$468$		0			0
$469$		−	174.338i		−	0.371722i
$470$		0			0
$471$			247.521i			0.525522i
$472$		0			0
$473$	−1037.34			−2.19312
$474$		0			0
$475$	−207.034	−	309.011i	−0.435860	−	0.650550i
$476$		0			0
$477$		−	118.153i		−	0.247700i
$478$		0			0
$479$	226.280			0.472400			0.236200	−	0.971704i	$-0.424098\pi$
$479$	226.280			0.472400			0.236200	+	0.971704i	$0.424098\pi$
$480$		0			0
$481$	871.705			1.81228
$482$		0			0
$483$		−	530.141i		−	1.09760i
$484$		0			0
$485$		−	337.637i		−	0.696159i
$486$		0			0
$487$		−	899.572i		−	1.84717i	−0.383394	−	0.923585i	$-0.625245\pi$
$487$		−	899.572i		−	1.84717i	0.383394	−	0.923585i	$-0.374755\pi$
$488$		0			0
$489$		−	420.765i		−	0.860460i
$490$		0			0
$491$	−405.337			−0.825533			−0.412767	−	0.910837i	$-0.635438\pi$
$491$	−405.337			−0.825533			−0.412767	+	0.910837i	$0.635438\pi$
$492$		0			0
$493$			510.818i			1.03614i
$494$		0			0
$495$	429.009			0.866686
$496$		0			0
$497$		−	182.419i		−	0.367041i
$498$		0			0
$499$	194.024			0.388825			0.194413	−	0.980920i	$-0.437720\pi$
$499$	194.024			0.388825			0.194413	+	0.980920i	$0.437720\pi$
$500$		0			0
$501$	363.880			0.726308
$502$		0			0
$503$	623.065			1.23870			0.619348	−	0.785116i	$-0.287397\pi$
$503$	623.065			1.23870			0.619348	+	0.785116i	$0.287397\pi$
$504$		0			0
$505$	350.100			0.693267
$506$		0			0
$507$		−	118.821i		−	0.234360i
$508$		0			0
$509$			321.176i			0.630995i	0.948926	+	0.315497i	$0.102171\pi$
$509$			321.176i			0.630995i	−0.948926	+	0.315497i	$0.897829\pi$
$510$		0			0
$511$	43.1211			0.0843857
$512$		0			0
$513$	30.7782	−	20.6210i	0.0599964	−	0.0401968i
$514$		0			0
$515$			172.960i			0.335845i
$516$		0			0
$517$	416.051			0.804741
$518$		0			0
$519$	96.0618			0.185090
$520$		0			0
$521$		−	535.331i		−	1.02751i	−0.857938	−	0.513753i	$-0.828255\pi$
$521$		−	535.331i		−	1.02751i	0.857938	−	0.513753i	$-0.171745\pi$
$522$		0			0
$523$		−	873.472i		−	1.67012i	−0.550160	−	0.835059i	$-0.685433\pi$
$523$		−	873.472i		−	1.67012i	0.550160	−	0.835059i	$-0.314567\pi$
$524$		0			0
$525$		−	248.709i		−	0.473732i
$526$		0			0
$527$		−	690.034i		−	1.30936i
$528$		0			0
$529$	1212.30			2.29168
$530$		0			0
$531$			182.215i			0.343155i
$532$		0			0
$533$	−283.672			−0.532218
$534$		0			0
$535$			325.345i			0.608122i
$536$		0			0
$537$	−312.701			−0.582311
$538$		0			0
$539$	784.380			1.45525
$540$		0			0
$541$	302.002			0.558229			0.279114	−	0.960258i	$-0.409959\pi$
$541$	302.002			0.558229			0.279114	+	0.960258i	$0.409959\pi$
$542$		0			0
$543$	−1504.76			−2.77120
$544$		0			0
$545$		−	99.0162i		−	0.181681i
$546$		0			0
$547$		−	1004.10i		−	1.83564i	−0.396994	−	0.917821i	$-0.629946\pi$
$547$		−	1004.10i		−	1.83564i	0.396994	−	0.917821i	$-0.370054\pi$
$548$		0			0
$549$	−46.9319			−0.0854861
$550$		0			0
$551$	362.012	−	242.543i	0.657009	−	0.440188i
$552$		0			0
$553$			335.209i			0.606164i
$554$		0			0
$555$	−621.858			−1.12046
$556$		0			0
$557$	449.650			0.807271			0.403635	−	0.914920i	$-0.367746\pi$
$557$	449.650			0.807271			0.403635	+	0.914920i	$0.367746\pi$
$558$		0			0
$559$		−	746.552i		−	1.33551i
$560$		0			0
$561$			1864.42i			3.32339i
$562$		0			0
$563$			829.613i			1.47356i	0.676133	+	0.736779i	$0.263654\pi$
$563$			829.613i			1.47356i	−0.676133	+	0.736779i	$0.736346\pi$
$564$		0			0
$565$			254.699i			0.450795i
$566$		0			0
$567$	−226.859			−0.400104
$568$		0			0
$569$			46.4574i			0.0816475i	0.999166	+	0.0408238i	$0.0129982\pi$
$569$			46.4574i			0.0816475i	−0.999166	+	0.0408238i	$0.987002\pi$
$570$		0			0
$571$	788.963			1.38172			0.690860	−	0.722988i	$-0.257232\pi$
$571$	788.963			1.38172			0.690860	+	0.722988i	$0.257232\pi$
$572$		0			0
$573$			463.740i			0.809320i
$574$		0			0
$575$	816.910			1.42071
$576$		0			0
$577$	−25.1604			−0.0436055			−0.0218028	−	0.999762i	$-0.506941\pi$
$577$	−25.1604			−0.0436055			−0.0218028	+	0.999762i	$0.506941\pi$
$578$		0			0
$579$	−1149.71			−1.98569
$580$		0			0
$581$	144.872			0.249349
$582$		0			0
$583$		−	243.531i		−	0.417720i
$584$		0			0
$585$			308.748i			0.527774i
$586$		0			0
$587$	779.591			1.32809			0.664047	−	0.747691i	$-0.268838\pi$
$587$	779.591			1.32809			0.664047	+	0.747691i	$0.268838\pi$
$588$		0			0
$589$	−489.021	+	327.638i	−0.830256	+	0.556261i
$590$		0			0
$591$			1248.85i			2.11312i
$592$		0			0
$593$	−657.878			−1.10941			−0.554703	−	0.832048i	$-0.687168\pi$
$593$	−657.878			−1.10941			−0.554703	+	0.832048i	$0.687168\pi$
$594$		0			0
$595$	−153.400			−0.257816
$596$		0			0
$597$		−	1298.60i		−	2.17521i
$598$		0			0
$599$			761.121i			1.27065i	0.772244	+	0.635326i	$0.219134\pi$
$599$			761.121i			1.27065i	−0.772244	+	0.635326i	$0.780866\pi$
$600$		0			0
$601$		−	798.347i		−	1.32837i	−0.747570	−	0.664183i	$-0.768780\pi$
$601$		−	798.347i		−	1.32837i	0.747570	−	0.664183i	$-0.231220\pi$
$602$		0			0
$603$			557.303i			0.924218i
$604$		0			0
$605$	602.465			0.995810
$606$		0			0
$607$			752.935i			1.24042i	0.784436	+	0.620210i	$0.212953\pi$
$607$			752.935i			1.24042i	−0.784436	+	0.620210i	$0.787047\pi$
$608$		0			0
$609$	291.367			0.478436
$610$		0			0
$611$			299.422i			0.490053i
$612$		0			0
$613$	536.480			0.875171			0.437586	−	0.899177i	$-0.355834\pi$
$613$	536.480			0.875171			0.437586	+	0.899177i	$0.355834\pi$
$614$		0			0
$615$	202.366			0.329051
$616$		0			0
$617$	−317.064			−0.513879			−0.256940	−	0.966427i	$-0.582714\pi$
$617$	−317.064			−0.513879			−0.256940	+	0.966427i	$0.582714\pi$
$618$		0			0
$619$	−351.867			−0.568445			−0.284222	−	0.958758i	$-0.591735\pi$
$619$	−351.867			−0.568445			−0.284222	+	0.958758i	$0.591735\pi$
$620$		0			0
$621$			81.3660i			0.131024i
$622$		0			0
$623$			82.2617i			0.132041i
$624$		0			0
$625$	247.659			0.396254
$626$		0			0
$627$	1321.30	−	885.254i	2.10734	−	1.41189i
$628$		0			0
$629$		−	1384.50i		−	2.20111i
$630$		0			0
$631$	−371.411			−0.588606			−0.294303	−	0.955712i	$-0.595088\pi$
$631$	−371.411			−0.588606			−0.294303	+	0.955712i	$0.595088\pi$
$632$		0			0
$633$	−316.066			−0.499315
$634$		0			0
$635$		−	93.7029i		−	0.147564i
$636$		0			0
$637$			564.500i			0.886186i
$638$		0			0
$639$			583.137i			0.912578i
$640$		0			0
$641$			256.605i			0.400319i	0.979763	+	0.200160i	$0.0641461\pi$
$641$			256.605i			0.400319i	−0.979763	+	0.200160i	$0.935854\pi$
$642$		0			0
$643$	759.280			1.18084			0.590420	−	0.807096i	$-0.298962\pi$
$643$	759.280			1.18084			0.590420	+	0.807096i	$0.298962\pi$
$644$		0			0
$645$			532.577i			0.825700i
$646$		0			0
$647$	−398.410			−0.615780			−0.307890	−	0.951422i	$-0.599623\pi$
$647$	−398.410			−0.615780			−0.307890	+	0.951422i	$0.599623\pi$
$648$		0			0
$649$			375.573i			0.578695i
$650$		0			0
$651$	−393.591			−0.604594
$652$		0			0
$653$	−453.692			−0.694781			−0.347391	−	0.937721i	$-0.612932\pi$
$653$	−453.692			−0.694781			−0.347391	+	0.937721i	$0.612932\pi$
$654$		0			0
$655$	−152.813			−0.233302
$656$		0			0
$657$	−137.845			−0.209809
$658$		0			0
$659$		−	193.340i		−	0.293383i	−0.989182	−	0.146692i	$-0.953138\pi$
$659$		−	193.340i		−	0.293383i	0.989182	−	0.146692i	$-0.0468625\pi$
$660$		0			0
$661$			902.276i			1.36502i	0.730878	+	0.682508i	$0.239111\pi$
$661$			902.276i			1.36502i	−0.730878	+	0.682508i	$0.760889\pi$
$662$		0			0
$663$	−1341.78			−2.02380
$664$		0			0
$665$	72.8366	+	108.713i	0.109529	+	0.163479i
$666$		0			0
$667$			957.024i			1.43482i
$668$		0			0
$669$	968.023			1.44697
$670$		0			0
$671$	−96.7335			−0.144163
$672$		0			0
$673$		−	486.217i		−	0.722463i	−0.932476	−	0.361231i	$-0.882356\pi$
$673$		−	486.217i		−	0.722463i	0.932476	−	0.361231i	$-0.117644\pi$
$674$		0			0
$675$			38.1719i			0.0565509i
$676$		0			0
$677$			1034.98i			1.52877i	0.644759	+	0.764386i	$0.276958\pi$
$677$			1034.98i			1.52877i	−0.644759	+	0.764386i	$0.723042\pi$
$678$		0			0
$679$		−	428.771i		−	0.631474i
$680$		0			0
$681$	83.3626			0.122412
$682$		0			0
$683$			100.984i			0.147853i	0.997264	+	0.0739267i	$0.0235531\pi$
$683$			100.984i			0.147853i	−0.997264	+	0.0739267i	$0.976447\pi$
$684$		0			0
$685$	393.417			0.574332
$686$		0			0
$687$			1226.47i			1.78526i
$688$		0			0
$689$	175.263			0.254374
$690$		0			0
$691$	−774.181			−1.12038			−0.560189	−	0.828365i	$-0.689272\pi$
$691$	−774.181			−1.12038			−0.560189	+	0.828365i	$0.689272\pi$
$692$		0			0
$693$	544.806			0.786156
$694$		0			0
$695$	−320.302			−0.460867
$696$		0			0
$697$			450.546i			0.646407i
$698$		0			0
$699$		−	1542.45i		−	2.20665i
$700$		0			0
$701$	1010.71			1.44181			0.720906	−	0.693033i	$-0.243726\pi$
$701$	1010.71			1.44181			0.720906	+	0.693033i	$0.243726\pi$
$702$		0			0
$703$	−981.180	+	657.378i	−1.39570	+	0.935104i
$704$		0			0
$705$		−	213.602i		−	0.302982i
$706$		0			0
$707$	444.598			0.628851
$708$		0			0
$709$	−542.239			−0.764794			−0.382397	−	0.923998i	$-0.624901\pi$
$709$	−542.239			−0.764794			−0.382397	+	0.923998i	$0.624901\pi$
$710$		0			0
$711$		−	1071.56i		−	1.50711i
$712$		0			0
$713$		−	1292.79i		−	1.81317i
$714$		0			0
$715$			636.375i			0.890035i
$716$		0			0
$717$		−	783.534i		−	1.09280i
$718$		0			0
$719$	105.476			0.146699			0.0733493	−	0.997306i	$-0.476631\pi$
$719$	105.476			0.146699			0.0733493	+	0.997306i	$0.476631\pi$
$720$		0			0
$721$			219.645i			0.304639i
$722$		0			0
$723$	−514.982			−0.712285
$724$		0			0
$725$			448.977i			0.619278i
$726$		0			0
$727$	859.614			1.18241			0.591206	−	0.806520i	$-0.298652\pi$
$727$	859.614			1.18241			0.591206	+	0.806520i	$0.298652\pi$
$728$		0			0
$729$	800.584			1.09820
$730$		0			0
$731$	−1185.72			−1.62205
$732$		0			0
$733$	1170.01			1.59619			0.798097	−	0.602528i	$-0.205840\pi$
$733$	1170.01			1.59619			0.798097	+	0.602528i	$0.205840\pi$
$734$		0			0
$735$		−	402.704i		−	0.547897i
$736$		0			0
$737$			1148.68i			1.55859i
$738$		0			0
$739$	−742.350			−1.00453			−0.502267	−	0.864713i	$-0.667500\pi$
$739$	−742.350			−1.00453			−0.502267	+	0.864713i	$0.667500\pi$
$740$		0			0
$741$	637.096	+	950.909i	0.859779	+	1.28328i
$742$		0			0
$743$		−	1306.08i		−	1.75784i	−0.476967	−	0.878921i	$-0.658264\pi$
$743$		−	1306.08i		−	1.75784i	0.476967	−	0.878921i	$-0.341736\pi$
$744$		0			0
$745$	−116.487			−0.156358
$746$		0			0
$747$	−463.109			−0.619959
$748$		0			0
$749$			413.161i			0.551617i
$750$		0			0
$751$			324.212i			0.431707i	0.976426	+	0.215853i	$0.0692533\pi$
$751$			324.212i			0.431707i	−0.976426	+	0.215853i	$0.930747\pi$
$752$		0			0
$753$		−	1930.13i		−	2.56325i
$754$		0			0
$755$			476.853i			0.631593i
$756$		0			0
$757$	−1118.35			−1.47734			−0.738672	−	0.674065i	$-0.764547\pi$
$757$	−1118.35			−1.47734			−0.738672	+	0.674065i	$0.764547\pi$
$758$		0			0
$759$			3493.02i			4.60214i
$760$		0			0
$761$	696.142			0.914772			0.457386	−	0.889268i	$-0.348786\pi$
$761$	696.142			0.914772			0.457386	+	0.889268i	$0.348786\pi$
$762$		0			0
$763$		−	125.742i		−	0.164800i
$764$		0			0
$765$	490.373			0.641010
$766$		0			0
$767$	−270.291			−0.352400
$768$		0			0
$769$	55.8346			0.0726067			0.0363034	−	0.999341i	$-0.488442\pi$
$769$	55.8346			0.0726067			0.0363034	+	0.999341i	$0.488442\pi$
$770$		0			0
$771$	2015.48			2.61411
$772$		0			0
$773$			1113.47i			1.44045i	0.693740	+	0.720226i	$0.255962\pi$
$773$			1113.47i			1.44045i	−0.693740	+	0.720226i	$0.744038\pi$
$774$		0			0
$775$		−	606.496i		−	0.782575i
$776$		0			0
$777$	−789.708			−1.01635
$778$		0			0
$779$	319.298	−	213.925i	0.409882	−	0.274615i
$780$		0			0
$781$			1201.93i			1.53897i
$782$		0			0
$783$	−44.7190			−0.0571124
$784$		0			0
$785$	134.185			0.170936
$786$		0			0
$787$		−	104.121i		−	0.132301i	−0.997810	−	0.0661504i	$-0.978928\pi$
$787$		−	104.121i		−	0.132301i	0.997810	−	0.0661504i	$-0.0210717\pi$
$788$		0			0
$789$			1138.86i			1.44342i
$790$		0			0
$791$			323.447i			0.408908i
$792$		0			0
$793$		−	69.6169i		−	0.0877892i
$794$		0			0
$795$	−125.030			−0.157270
$796$		0			0
$797$			292.129i			0.366536i	0.983063	+	0.183268i	$0.0586677\pi$
$797$			292.129i			0.366536i	−0.983063	+	0.183268i	$0.941332\pi$
$798$		0			0
$799$	475.561			0.595196
$800$		0			0
$801$		−	262.965i		−	0.328296i
$802$		0			0
$803$	−284.118			−0.353821
$804$		0			0
$805$	−287.397			−0.357015
$806$		0			0
$807$	1303.74			1.61553
$808$		0			0
$809$	−304.463			−0.376344			−0.188172	−	0.982136i	$-0.560256\pi$
$809$	−304.463			−0.376344			−0.188172	+	0.982136i	$0.560256\pi$
$810$		0			0
$811$		−	816.016i		−	1.00618i	−0.864233	−	0.503092i	$-0.832196\pi$
$811$		−	816.016i		−	1.00618i	0.864233	−	0.503092i	$-0.167804\pi$
$812$		0			0
$813$		−	860.452i		−	1.05837i
$814$		0			0
$815$	−228.103			−0.279881
$816$		0			0
$817$	562.997	+	840.310i	0.689102	+	1.02853i
$818$		0			0
$819$			392.084i			0.478735i
$820$		0			0
$821$	721.591			0.878917			0.439458	−	0.898263i	$-0.355170\pi$
$821$	721.591			0.878917			0.439458	+	0.898263i	$0.355170\pi$
$822$		0			0
$823$	414.065			0.503117			0.251558	−	0.967842i	$-0.419057\pi$
$823$	414.065			0.503117			0.251558	+	0.967842i	$0.419057\pi$
$824$		0			0
$825$			1638.71i			1.98631i
$826$		0			0
$827$			100.878i			0.121981i	0.998138	+	0.0609903i	$0.0194259\pi$
$827$			100.878i			0.121981i	−0.998138	+	0.0609903i	$0.980574\pi$
$828$		0			0
$829$		−	621.755i		−	0.750006i	−0.927024	−	0.375003i	$-0.877642\pi$
$829$		−	621.755i		−	0.750006i	0.927024	−	0.375003i	$-0.122358\pi$
$830$		0			0
$831$		−	687.781i		−	0.827655i
$832$		0			0
$833$	896.575			1.07632
$834$		0			0
$835$		−	197.265i		−	0.236245i
$836$		0			0
$837$	60.4082			0.0721723
$838$		0			0
$839$		−	115.138i		−	0.137232i	−0.997643	−	0.0686160i	$-0.978142\pi$
$839$		−	115.138i		−	0.137232i	0.997643	−	0.0686160i	$-0.0218583\pi$
$840$		0			0
$841$	315.016			0.374573
$842$		0			0
$843$	1645.33			1.95176
$844$		0			0
$845$	−64.4144			−0.0762300
$846$		0			0
$847$	765.080			0.903282
$848$		0			0
$849$		−	973.956i		−	1.14718i
$850$		0			0
$851$		−	2593.87i		−	3.04803i
$852$		0			0
$853$	762.387			0.893772			0.446886	−	0.894591i	$-0.352533\pi$
$853$	762.387			0.893772			0.446886	+	0.894591i	$0.352533\pi$
$854$		0			0
$855$	−232.836	−	347.523i	−0.272323	−	0.406460i
$856$		0			0
$857$		−	537.227i		−	0.626869i	−0.949610	−	0.313435i	$-0.898520\pi$
$857$		−	537.227i		−	0.626869i	0.949610	−	0.313435i	$-0.101480\pi$
$858$		0			0
$859$	645.182			0.751084			0.375542	−	0.926805i	$-0.377456\pi$
$859$	645.182			0.751084			0.375542	+	0.926805i	$0.377456\pi$
$860$		0			0
$861$	256.988			0.298477
$862$		0			0
$863$		−	892.080i		−	1.03370i	−0.856077	−	0.516848i	$-0.827105\pi$
$863$		−	892.080i		−	1.03370i	0.856077	−	0.516848i	$-0.172895\pi$
$864$		0			0
$865$		−	52.0765i		−	0.0602041i
$866$		0			0
$867$			889.615i			1.02608i
$868$		0			0
$869$		−	2208.64i		−	2.54159i
$870$		0			0
$871$	−826.681			−0.949118
$872$		0			0
$873$			1370.65i			1.57004i
$874$		0			0
$875$	−307.010			−0.350869
$876$		0			0
$877$		−	873.903i		−	0.996469i	−0.867042	−	0.498235i	$-0.833982\pi$
$877$		−	873.903i		−	0.996469i	0.867042	−	0.498235i	$-0.166018\pi$
$878$		0			0
$879$	−1847.03			−2.10129
$880$		0			0
$881$	−687.622			−0.780502			−0.390251	−	0.920709i	$-0.627612\pi$
$881$	−687.622			−0.780502			−0.390251	+	0.920709i	$0.627612\pi$
$882$		0			0
$883$	1305.56			1.47855			0.739276	−	0.673403i	$-0.235168\pi$
$883$	1305.56			1.47855			0.739276	+	0.673403i	$0.235168\pi$
$884$		0			0
$885$	192.821			0.217876
$886$		0			0
$887$		−	57.7531i		−	0.0651106i	−0.999470	−	0.0325553i	$-0.989635\pi$
$887$		−	57.7531i		−	0.0651106i	0.999470	−	0.0325553i	$-0.0103645\pi$
$888$		0			0
$889$		−	118.995i		−	0.133852i
$890$		0			0
$891$	1494.74			1.67760
$892$		0			0
$893$	−225.803	−	337.026i	−0.252859	−	0.377409i
$894$		0			0
$895$			169.520i			0.189408i
$896$		0			0
$897$	−2513.85			−2.80250
$898$		0			0
$899$	710.520			0.790345
$900$		0			0
$901$		−	278.364i		−	0.308951i
$902$		0			0
$903$			676.328i			0.748979i
$904$		0			0
$905$			815.753i			0.901385i
$906$		0			0
$907$		−	837.489i		−	0.923362i	−0.887046	−	0.461681i	$-0.847247\pi$
$907$		−	837.489i		−	0.923362i	0.887046	−	0.461681i	$-0.152753\pi$
$908$		0			0
$909$	−1421.24			−1.56352
$910$		0			0
$911$			662.892i			0.727653i	0.931467	+	0.363826i	$0.118530\pi$
$911$			662.892i			0.727653i	−0.931467	+	0.363826i	$0.881470\pi$
$912$		0			0
$913$	−954.537			−1.04550
$914$		0			0
$915$			49.6634i			0.0542769i
$916$		0			0
$917$	−194.060			−0.211625
$918$		0			0
$919$	−1000.18			−1.08834			−0.544169	−	0.838976i	$-0.683155\pi$
$919$	−1000.18			−1.08834			−0.544169	+	0.838976i	$0.683155\pi$
$920$		0			0
$921$	894.699			0.971443
$922$		0			0
$923$	−865.003			−0.937164
$924$		0			0
$925$		−	1216.88i		−	1.31555i
$926$		0			0
$927$		−	702.137i		−	0.757429i
$928$		0			0
$929$	1504.37			1.61934			0.809671	−	0.586885i	$-0.199646\pi$
$929$	1504.37			1.61934			0.809671	+	0.586885i	$0.199646\pi$
$930$		0			0
$931$	−425.706	−	635.395i	−0.457257	−	0.682486i
$932$		0			0
$933$			580.701i			0.622402i
$934$		0			0
$935$	1010.73			1.08100
$936$		0			0
$937$	−961.503			−1.02615			−0.513075	−	0.858344i	$-0.671494\pi$
$937$	−961.503			−1.02615			−0.513075	+	0.858344i	$0.671494\pi$
$938$		0			0
$939$		−	1770.08i		−	1.88506i
$940$		0			0
$941$		−	681.880i		−	0.724633i	−0.932055	−	0.362317i	$-0.881986\pi$
$941$		−	681.880i		−	0.724633i	0.932055	−	0.362317i	$-0.118014\pi$
$942$		0			0
$943$			844.103i			0.895126i
$944$		0			0
$945$		−	13.4293i		−	0.0142109i
$946$		0			0
$947$	−1441.83			−1.52253			−0.761263	−	0.648443i	$-0.775420\pi$
$947$	−1441.83			−1.52253			−0.761263	+	0.648443i	$0.775420\pi$
$948$		0			0
$949$		−	204.473i		−	0.215462i
$950$		0			0
$951$	−1661.29			−1.74688
$952$		0			0
$953$			495.317i			0.519745i	0.965643	+	0.259872i	$0.0836805\pi$
$953$			495.317i			0.519745i	−0.965643	+	0.259872i	$0.916320\pi$
$954$		0			0
$955$	251.400			0.263247
$956$		0			0
$957$	−1919.78			−2.00604
$958$		0			0
$959$	499.607			0.520967
$960$		0			0
$961$	1.20041			0.00124912
$962$		0			0
$963$		−	1320.75i		−	1.37149i
$964$		0			0
$965$			623.276i			0.645882i
$966$		0			0
$967$	1020.51			1.05534			0.527670	−	0.849449i	$-0.323066\pi$
$967$	1020.51			1.05534			0.527670	+	0.849449i	$0.323066\pi$
$968$		0			0
$969$	1510.29	−	1011.88i	1.55861	−	1.04425i
$970$		0			0
$971$			563.750i			0.580587i	0.956938	+	0.290293i	$0.0937529\pi$
$971$			563.750i			0.580587i	−0.956938	+	0.290293i	$0.906247\pi$
$972$		0			0
$973$	−406.757			−0.418045
$974$		0			0
$975$	−1179.34			−1.20958
$976$		0			0
$977$		−	786.557i		−	0.805073i	−0.915404	−	0.402537i	$-0.868129\pi$
$977$		−	786.557i		−	0.805073i	0.915404	−	0.402537i	$-0.131871\pi$
$978$		0			0
$979$		−	542.010i		−	0.553636i
$980$		0			0
$981$			401.959i			0.409744i
$982$		0			0
$983$		−	337.555i		−	0.343393i	−0.985150	−	0.171696i	$-0.945075\pi$
$983$		−	337.555i		−	0.343393i	0.985150	−	0.171696i	$-0.0549248\pi$
$984$		0			0
$985$	677.021			0.687331
$986$		0			0
$987$		−	271.257i		−	0.274830i
$988$		0			0
$989$	−2221.46			−2.24617
$990$		0			0
$991$			1048.22i			1.05774i	0.848703	+	0.528870i	$0.177384\pi$
$991$			1048.22i			1.05774i	−0.848703	+	0.528870i	$0.822616\pi$
$992$		0			0
$993$	−617.176			−0.621527
$994$		0			0
$995$	−703.991			−0.707529
$996$		0			0
$997$	1041.54			1.04467			0.522335	−	0.852741i	$-0.325061\pi$
$997$	1041.54			1.04467			0.522335	+	0.852741i	$0.325061\pi$
$998$		0			0
$999$	121.204			0.121325

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	608.3.e.b.417.18	yes	20
4.3	odd	2	inner	608.3.e.b.417.3	✓	20
8.3	odd	2		1216.3.e.p.1025.17		20
8.5	even	2		1216.3.e.p.1025.4		20
19.18	odd	2	inner	608.3.e.b.417.4	yes	20
76.75	even	2	inner	608.3.e.b.417.17	yes	20
152.37	odd	2		1216.3.e.p.1025.18		20
152.75	even	2		1216.3.e.p.1025.3		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
608.3.e.b.417.3	✓	20	4.3	odd	2	inner
608.3.e.b.417.4	yes	20	19.18	odd	2	inner
608.3.e.b.417.17	yes	20	76.75	even	2	inner
608.3.e.b.417.18	yes	20	1.1	even	1	trivial
1216.3.e.p.1025.3		20	152.75	even	2
1216.3.e.p.1025.4		20	8.5	even	2
1216.3.e.p.1025.17		20	8.3	odd	2
1216.3.e.p.1025.18		20	152.37	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 608 = 2^{5} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	608.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.29580i q^{3} +2.32882 q^{5} +2.95740 q^{7} -9.45390 q^{9} -19.4859 q^{11} -14.0235i q^{13} +10.0041i q^{15} -22.2731 q^{17} +(10.5756 + 15.7847i) q^{19} +12.7044i q^{21} -41.7289 q^{23} -19.5766 q^{25} -1.94987i q^{27} -22.9343i q^{29} +30.9806i q^{31} -83.7075i q^{33} +6.88725 q^{35} +62.1601i q^{37} +60.2423 q^{39} -20.2283i q^{41} +53.2356 q^{43} -22.0164 q^{45} -21.3514 q^{47} -40.2538 q^{49} -95.6807i q^{51} +12.4978i q^{53} -45.3791 q^{55} +(-67.8080 + 45.4305i) q^{57} -19.2741i q^{59} +4.96429 q^{61} -27.9590 q^{63} -32.6583i q^{65} -58.9495i q^{67} -179.259i q^{69} -61.6822i q^{71} +14.5807 q^{73} -84.0972i q^{75} -57.6276 q^{77} +113.346i q^{79} -76.7089 q^{81} +48.9861 q^{83} -51.8699 q^{85} +98.5213 q^{87} +27.8155i q^{89} -41.4733i q^{91} -133.087 q^{93} +(24.6285 + 36.7597i) q^{95} -144.982i q^{97} +184.218 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 52 q^{9} + 56 q^{17} + 36 q^{25} + 64 q^{45} + 332 q^{49} + 88 q^{57} - 32 q^{61} - 152 q^{73} + 360 q^{77} - 476 q^{81} - 552 q^{85} + 336 q^{93}+O(q^{100}) \) 20 * q - 52 * q^9 + 56 * q^17 + 36 * q^25 + 64 * q^45 + 332 * q^49 + 88 * q^57 - 32 * q^61 - 152 * q^73 + 360 * q^77 - 476 * q^81 - 552 * q^85 + 336 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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