LMFDB - Embedded newform 475.3.d.c.474.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,3,Mod(474,475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("475.474");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 475 = 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	475.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:	$12.9428125571$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 95)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			474.3
Character	$\chi$	$=$	475.474
Dual form			475.3.d.c.474.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.84623 q^{2} +2.18419 q^{3} +4.10101 q^{4} -6.21669 q^{6} +9.15076i q^{7} -0.287487 q^{8} -4.22932 q^{9} +O(q^{10})$	\(q-2.84623 q^{2} +2.18419 q^{3} +4.10101 q^{4} -6.21669 q^{6} +9.15076i q^{7} -0.287487 q^{8} -4.22932 q^{9} +9.89207 q^{11} +8.95737 q^{12} +12.3251 q^{13} -26.0451i q^{14} -15.5858 q^{16} +11.1895i q^{17} +12.0376 q^{18} +(-18.7659 - 2.97345i) q^{19} +19.9870i q^{21} -28.1551 q^{22} +20.8666i q^{23} -0.627927 q^{24} -35.0800 q^{26} -28.8953 q^{27} +37.5273i q^{28} +10.5011i q^{29} -52.3075i q^{31} +45.5106 q^{32} +21.6061 q^{33} -31.8479i q^{34} -17.3445 q^{36} +10.5041 q^{37} +(53.4120 + 8.46312i) q^{38} +26.9203 q^{39} +42.6288i q^{41} -56.8874i q^{42} +45.9768i q^{43} +40.5674 q^{44} -59.3912i q^{46} -4.22163i q^{47} -34.0423 q^{48} -34.7363 q^{49} +24.4400i q^{51} +50.5453 q^{52} -95.8254 q^{53} +82.2427 q^{54} -2.63073i q^{56} +(-40.9882 - 6.49458i) q^{57} -29.8886i q^{58} +6.38544i q^{59} +14.8591 q^{61} +148.879i q^{62} -38.7015i q^{63} -67.1904 q^{64} -61.4960 q^{66} +38.0693 q^{67} +45.8882i q^{68} +45.5767i q^{69} +125.169i q^{71} +1.21588 q^{72} +18.5252i q^{73} -29.8971 q^{74} +(-76.9590 - 12.1941i) q^{76} +90.5199i q^{77} -76.6214 q^{78} +137.419i q^{79} -25.0489 q^{81} -121.331i q^{82} +127.881i q^{83} +81.9667i q^{84} -130.860i q^{86} +22.9364i q^{87} -2.84385 q^{88} +19.8059i q^{89} +112.784i q^{91} +85.5743i q^{92} -114.249i q^{93} +12.0157i q^{94} +99.4037 q^{96} +42.4473 q^{97} +98.8675 q^{98} -41.8368 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 24 q^{4} - 56 q^{6} + 96 q^{9}+O(q^{10}) $ 24 * q + 24 * q^4 - 56 * q^6 + 96 * q^9	$ 24 q + 24 q^{4} - 56 q^{6} + 96 q^{9} + 64 q^{11} - 88 q^{16} - 16 q^{19} - 200 q^{24} + 216 q^{26} - 160 q^{36} - 152 q^{39} + 512 q^{44} - 144 q^{49} + 152 q^{54} - 592 q^{61} - 376 q^{64} + 304 q^{66} - 272 q^{74} + 496 q^{76} - 744 q^{81} - 88 q^{96} + 624 q^{99}+O(q^{100}) $ 24 * q + 24 * q^4 - 56 * q^6 + 96 * q^9 + 64 * q^11 - 88 * q^16 - 16 * q^19 - 200 * q^24 + 216 * q^26 - 160 * q^36 - 152 * q^39 + 512 * q^44 - 144 * q^49 + 152 * q^54 - 592 * q^61 - 376 * q^64 + 304 * q^66 - 272 * q^74 + 496 * q^76 - 744 * q^81 - 88 * q^96 + 624 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$.

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

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 $

$2$	−2.84623			−1.42311			−0.711557	−	0.702629i	$-0.752010\pi$
$2$	−2.84623			−1.42311			−0.711557	+	0.702629i	$0.752010\pi$
$3$	2.18419			0.728063			0.364031	−	0.931387i	$-0.381400\pi$
$3$	2.18419			0.728063			0.364031	+	0.931387i	$0.381400\pi$
$4$	4.10101			1.02525
$5$		0			0
$5$		0			0
$6$	−6.21669			−1.03612
$7$			9.15076i			1.30725i	0.756818	+	0.653625i	$0.226753\pi$
$7$			9.15076i			1.30725i	−0.756818	+	0.653625i	$0.773247\pi$
$8$	−0.287487			−0.0359359
$9$	−4.22932			−0.469925
$10$		0			0
$11$	9.89207			0.899279			0.449640	−	0.893210i	$-0.351552\pi$
$11$	9.89207			0.899279			0.449640	+	0.893210i	$0.351552\pi$
$12$	8.95737			0.746447
$13$	12.3251			0.948085			0.474042	−	0.880502i	$-0.342794\pi$
$13$	12.3251			0.948085			0.474042	+	0.880502i	$0.342794\pi$
$14$		−	26.0451i		−	1.86037i
$15$		0			0
$16$	−15.5858			−0.974111
$17$			11.1895i			0.658206i	0.944294	+	0.329103i	$0.106746\pi$
$17$			11.1895i			0.658206i	−0.944294	+	0.329103i	$0.893254\pi$
$18$	12.0376			0.668756
$19$	−18.7659	−	2.97345i	−0.987678	−	0.156497i
$19$	−18.7659	−	2.97345i	−0.987678	−	0.156497i
$20$		0			0
$21$			19.9870i			0.951760i
$22$	−28.1551			−1.27978
$23$			20.8666i			0.907246i	0.891194	+	0.453623i	$0.149869\pi$
$23$			20.8666i			0.907246i	−0.891194	+	0.453623i	$0.850131\pi$
$24$	−0.627927			−0.0261636
$25$		0			0
$26$	−35.0800			−1.34923
$27$	−28.8953			−1.07020
$28$			37.5273i			1.34026i
$29$			10.5011i			0.362107i	0.983473	+	0.181054i	$0.0579508\pi$
$29$			10.5011i			0.362107i	−0.983473	+	0.181054i	$0.942049\pi$
$30$		0			0
$31$		−	52.3075i		−	1.68734i	−0.536864	−	0.843669i	$-0.680391\pi$
$31$		−	52.3075i		−	1.68734i	0.536864	−	0.843669i	$-0.319609\pi$
$32$	45.5106			1.42221
$33$	21.6061			0.654732
$34$		−	31.8479i		−	0.936702i
$35$		0			0
$36$	−17.3445			−0.481791
$37$	10.5041			0.283895			0.141948	−	0.989874i	$-0.454664\pi$
$37$	10.5041			0.283895			0.141948	+	0.989874i	$0.454664\pi$
$38$	53.4120	+	8.46312i	1.40558	+	0.222714i
$39$	26.9203			0.690265
$40$		0			0
$41$			42.6288i			1.03973i	0.854250	+	0.519863i	$0.174017\pi$
$41$			42.6288i			1.03973i	−0.854250	+	0.519863i	$0.825983\pi$
$42$		−	56.8874i		−	1.35446i
$43$			45.9768i			1.06923i	0.845097	+	0.534614i	$0.179543\pi$
$43$			45.9768i			1.06923i	−0.845097	+	0.534614i	$0.820457\pi$
$44$	40.5674			0.921987
$45$		0			0
$46$		−	59.3912i		−	1.29111i
$47$		−	4.22163i		−	0.0898220i	−0.998991	−	0.0449110i	$-0.985700\pi$
$47$		−	4.22163i		−	0.0898220i	0.998991	−	0.0449110i	$-0.0143004\pi$
$48$	−34.0423			−0.709214
$49$	−34.7363			−0.708905
$50$		0			0
$51$			24.4400i			0.479215i
$52$	50.5453			0.972025
$53$	−95.8254			−1.80803			−0.904013	−	0.427505i	$-0.859393\pi$
$53$	−95.8254			−1.80803			−0.904013	+	0.427505i	$0.859393\pi$
$54$	82.2427			1.52301
$55$		0			0
$56$		−	2.63073i		−	0.0469773i
$57$	−40.9882	−	6.49458i	−0.719092	−	0.113940i
$58$		−	29.8886i		−	0.515320i
$59$			6.38544i			0.108228i	0.998535	+	0.0541139i	$0.0172334\pi$
$59$			6.38544i			0.108228i	−0.998535	+	0.0541139i	$0.982767\pi$
$60$		0			0
$61$	14.8591			0.243593			0.121796	−	0.992555i	$-0.461135\pi$
$61$	14.8591			0.243593			0.121796	+	0.992555i	$0.461135\pi$
$62$			148.879i			2.40127i
$63$		−	38.7015i		−	0.614310i
$64$	−67.1904			−1.04985
$65$		0			0
$66$	−61.4960			−0.931757
$67$	38.0693			0.568199			0.284099	−	0.958795i	$-0.408305\pi$
$67$	38.0693			0.568199			0.284099	+	0.958795i	$0.408305\pi$
$68$			45.8882i			0.674827i
$69$			45.5767i			0.660532i
$70$		0			0
$71$			125.169i			1.76295i	0.472230	+	0.881475i	$0.343449\pi$
$71$			125.169i			1.76295i	−0.472230	+	0.881475i	$0.656551\pi$
$72$	1.21588			0.0168872
$73$			18.5252i			0.253769i	0.991917	+	0.126885i	$0.0404978\pi$
$73$			18.5252i			0.253769i	−0.991917	+	0.126885i	$0.959502\pi$
$74$	−29.8971			−0.404015
$75$		0			0
$76$	−76.9590	−	12.1941i	−1.01262	−	0.160449i
$77$			90.5199i			1.17558i
$78$	−76.6214			−0.982325
$79$			137.419i			1.73948i	0.493510	+	0.869740i	$0.335714\pi$
$79$			137.419i			1.73948i	−0.493510	+	0.869740i	$0.664286\pi$
$80$		0			0
$81$	−25.0489			−0.309246
$82$		−	121.331i		−	1.47965i
$83$			127.881i			1.54073i	0.637603	+	0.770365i	$0.279926\pi$
$83$			127.881i			1.54073i	−0.637603	+	0.770365i	$0.720074\pi$
$84$			81.9667i			0.975794i
$85$		0			0
$86$		−	130.860i		−	1.52163i
$87$			22.9364i			0.263637i
$88$	−2.84385			−0.0323164
$89$			19.8059i			0.222538i	0.993790	+	0.111269i	$0.0354916\pi$
$89$			19.8059i			0.222538i	−0.993790	+	0.111269i	$0.964508\pi$
$90$		0			0
$91$			112.784i			1.23938i
$92$			85.5743i			0.930155i
$93$		−	114.249i		−	1.22849i
$94$			12.0157i			0.127827i
$95$		0			0
$96$	99.4037			1.03545
$97$	42.4473			0.437601			0.218800	−	0.975770i	$-0.429786\pi$
$97$	42.4473			0.437601			0.218800	+	0.975770i	$0.429786\pi$
$98$	98.8675			1.00885
$99$	−41.8368			−0.422594
$100$		0			0
$101$	−111.435			−1.10332			−0.551660	−	0.834069i	$-0.686005\pi$
$101$	−111.435			−1.10332			−0.551660	+	0.834069i	$0.686005\pi$
$102$		−	69.5617i		−	0.681978i
$103$	5.71536			0.0554889			0.0277445	−	0.999615i	$-0.491168\pi$
$103$	5.71536			0.0554889			0.0277445	+	0.999615i	$0.491168\pi$
$104$	−3.54331			−0.0340703
$105$		0			0
$106$	272.741			2.57303
$107$	−164.563			−1.53797			−0.768987	−	0.639264i	$-0.779239\pi$
$107$	−164.563			−1.53797			−0.768987	+	0.639264i	$0.779239\pi$
$108$	−118.500			−1.09722
$109$		−	102.267i		−	0.938227i	−0.883138	−	0.469113i	$-0.844574\pi$
$109$		−	102.267i		−	0.938227i	0.883138	−	0.469113i	$-0.155426\pi$
$110$		0			0
$111$	22.9430			0.206693
$112$		−	142.622i		−	1.27341i
$113$	−24.5050			−0.216858			−0.108429	−	0.994104i	$-0.534582\pi$
$113$	−24.5050			−0.216858			−0.108429	+	0.994104i	$0.534582\pi$
$114$	116.662	+	18.4850i	1.02335	+	0.162149i
$115$		0			0
$116$			43.0651i			0.371251i
$117$	−52.1268			−0.445528
$118$		−	18.1744i		−	0.154021i
$119$	−102.392			−0.860440
$120$		0			0
$121$	−23.1469			−0.191297
$122$	−42.2925			−0.346660
$123$			93.1093i			0.756986i
$124$		−	214.513i		−	1.72995i
$125$		0			0
$126$			110.153i			0.874232i
$127$	58.5427			0.460966			0.230483	−	0.973076i	$-0.425969\pi$
$127$	58.5427			0.460966			0.230483	+	0.973076i	$0.425969\pi$
$128$	9.19667			0.0718489
$129$			100.422i			0.778464i
$130$		0			0
$131$	223.883			1.70903			0.854516	−	0.519425i	$-0.173854\pi$
$131$	223.883			1.70903			0.854516	+	0.519425i	$0.173854\pi$
$132$	88.6069			0.671265
$133$	27.2093	−	171.722i	0.204581	−	1.29114i
$134$	−108.354			−0.808611
$135$		0			0
$136$		−	3.21684i		−	0.0236532i
$137$		−	6.24728i		−	0.0456006i	−0.999740	−	0.0228003i	$-0.992742\pi$
$137$		−	6.24728i		−	0.0456006i	0.999740	−	0.0228003i	$-0.00725818\pi$
$138$		−	129.722i		−	0.940011i
$139$	204.969			1.47460			0.737299	−	0.675567i	$-0.236101\pi$
$139$	204.969			1.47460			0.737299	+	0.675567i	$0.236101\pi$
$140$		0			0
$141$		−	9.22084i		−	0.0653960i
$142$		−	356.261i		−	2.50888i
$143$	121.921			0.852593
$144$	65.9173			0.457759
$145$		0			0
$146$		−	52.7268i		−	0.361143i
$147$	−75.8707			−0.516127
$148$	43.0775			0.291064
$149$	83.5502			0.560740			0.280370	−	0.959892i	$-0.409543\pi$
$149$	83.5502			0.560740			0.280370	+	0.959892i	$0.409543\pi$
$150$		0			0
$151$		−	189.386i		−	1.25421i	−0.778934	−	0.627106i	$-0.784239\pi$
$151$		−	189.386i		−	1.25421i	0.778934	−	0.627106i	$-0.215761\pi$
$152$	5.39496	+	0.854830i	0.0354931	+	0.00562388i
$153$		−	47.3240i		−	0.309307i
$154$		−	257.640i		−	1.67299i
$155$		0			0
$156$	110.400			0.707695
$157$			195.060i			1.24242i	0.783643	+	0.621212i	$0.213359\pi$
$157$			195.060i			1.24242i	−0.783643	+	0.621212i	$0.786641\pi$
$158$		−	391.126i		−	2.47548i
$159$	−209.301			−1.31636
$160$		0			0
$161$	−190.946			−1.18600
$162$	71.2949			0.440092
$163$		−	171.677i		−	1.05323i	−0.850103	−	0.526616i	$-0.823461\pi$
$163$		−	171.677i		−	1.05323i	0.850103	−	0.526616i	$-0.176539\pi$
$164$			174.821i			1.06598i
$165$		0			0
$166$		−	363.977i		−	2.19263i
$167$	−40.7628			−0.244088			−0.122044	−	0.992525i	$-0.538945\pi$
$167$	−40.7628			−0.244088			−0.122044	+	0.992525i	$0.538945\pi$
$168$		−	5.74600i		−	0.0342024i
$169$	−17.0919			−0.101136
$170$		0			0
$171$	79.3670	+	12.5757i	0.464135	+	0.0735420i
$172$			188.551i			1.09623i
$173$	279.193			1.61384			0.806918	−	0.590664i	$-0.201134\pi$
$173$	279.193			1.61384			0.806918	+	0.590664i	$0.201134\pi$
$174$		−	65.2822i		−	0.375185i
$175$		0			0
$176$	−154.176			−0.875997
$177$			13.9470i			0.0787967i
$178$		−	56.3721i		−	0.316697i
$179$		−	308.777i		−	1.72501i	−0.506048	−	0.862505i	$-0.668894\pi$
$179$		−	308.777i		−	1.72501i	0.506048	−	0.862505i	$-0.331106\pi$
$180$		0			0
$181$		−	252.084i		−	1.39273i	−0.717688	−	0.696365i	$-0.754799\pi$
$181$		−	252.084i		−	1.39273i	0.717688	−	0.696365i	$-0.245201\pi$
$182$		−	321.009i		−	1.76378i
$183$	32.4552			0.177351
$184$		−	5.99890i		−	0.0326027i
$185$		0			0
$186$			325.180i			1.74828i
$187$			110.687i			0.591911i
$188$		−	17.3129i		−	0.0920901i
$189$		−	264.414i		−	1.39902i
$190$		0			0
$191$	210.693			1.10311			0.551553	−	0.834140i	$-0.314035\pi$
$191$	210.693			1.10311			0.551553	+	0.834140i	$0.314035\pi$
$192$	−146.756			−0.764356
$193$	−113.346			−0.587285			−0.293643	−	0.955915i	$-0.594868\pi$
$193$	−113.346			−0.587285			−0.293643	+	0.955915i	$0.594868\pi$
$194$	−120.815			−0.622755
$195$		0			0
$196$	−142.454			−0.726806
$197$			206.358i			1.04750i	0.851871	+	0.523752i	$0.175468\pi$
$197$			206.358i			1.04750i	−0.851871	+	0.523752i	$0.824532\pi$
$198$	119.077			0.601399
$199$	−181.976			−0.914450			−0.457225	−	0.889351i	$-0.651157\pi$
$199$	−181.976			−0.914450			−0.457225	+	0.889351i	$0.651157\pi$
$200$		0			0
$201$	83.1505			0.413684
$202$	317.170			1.57015
$203$	−96.0931			−0.473365
$204$			100.229i			0.491316i
$205$		0			0
$206$	−16.2672			−0.0789670
$207$		−	88.2518i		−	0.426337i
$208$	−192.096			−0.923539
$209$	−185.633	−	29.4136i	−0.888199	−	0.140735i
$210$		0			0
$211$			106.729i			0.505824i	0.967489	+	0.252912i	$0.0813883\pi$
$211$			106.729i			0.505824i	−0.967489	+	0.252912i	$0.918612\pi$
$212$	−392.981			−1.85368
$213$			273.394i			1.28354i
$214$	468.384			2.18871
$215$		0			0
$216$	8.30704			0.0384585
$217$	478.653			2.20577
$218$			291.074i			1.33520i
$219$			40.4624i			0.184760i
$220$		0			0
$221$			137.912i			0.624035i
$222$	−65.3009			−0.294148
$223$	−209.351			−0.938793			−0.469397	−	0.882987i	$-0.655529\pi$
$223$	−209.351			−0.938793			−0.469397	+	0.882987i	$0.655529\pi$
$224$			416.456i			1.85918i
$225$		0			0
$226$	69.7467			0.308614
$227$	244.295			1.07619			0.538094	−	0.842885i	$-0.319144\pi$
$227$	244.295			1.07619			0.538094	+	0.842885i	$0.319144\pi$
$228$	−168.093	−	26.6343i	−0.737250	−	0.116817i
$229$	−254.372			−1.11079			−0.555397	−	0.831586i	$-0.687434\pi$
$229$	−254.372			−1.11079			−0.555397	+	0.831586i	$0.687434\pi$
$230$		0			0
$231$			197.713i			0.855898i
$232$		−	3.01894i		−	0.0130127i
$233$			347.037i			1.48943i	0.667384	+	0.744714i	$0.267414\pi$
$233$			347.037i			1.48943i	−0.667384	+	0.744714i	$0.732586\pi$
$234$	148.365			0.634038
$235$		0			0
$236$			26.1867i			0.110961i
$237$			300.149i			1.26645i
$238$	291.432			1.22450
$239$	−0.0478766			−0.000200320			−0.000100160	−	1.00000i	$-0.500032\pi$
$239$	−0.0478766			−0.000200320			−0.000100160		1.00000i	$0.500032\pi$
$240$		0			0
$241$		−	422.817i		−	1.75443i	−0.480100	−	0.877214i	$-0.659399\pi$
$241$		−	422.817i		−	1.75443i	0.480100	−	0.877214i	$-0.340601\pi$
$242$	65.8815			0.272237
$243$	205.346			0.845047
$244$	60.9375			0.249744
$245$		0			0
$246$		−	265.010i		−	1.07728i
$247$	−231.291	−	36.6481i	−0.936403	−	0.148373i
$248$			15.0377i			0.0606361i
$249$			279.315i			1.12175i
$250$		0			0
$251$	142.321			0.567015			0.283507	−	0.958970i	$-0.408502\pi$
$251$	142.321			0.567015			0.283507	+	0.958970i	$0.408502\pi$
$252$		−	158.715i		−	0.629822i
$253$			206.414i			0.815867i
$254$	−166.626			−0.656007
$255$		0			0
$256$	242.586			0.947600
$257$	198.179			0.771123			0.385562	−	0.922682i	$-0.374008\pi$
$257$	198.179			0.771123			0.385562	+	0.922682i	$0.374008\pi$
$258$		−	285.823i		−	1.10784i
$259$			96.1206i			0.371122i
$260$		0			0
$261$		−	44.4126i		−	0.170163i
$262$	−637.223			−2.43215
$263$		−	149.067i		−	0.566794i	−0.959003	−	0.283397i	$-0.908539\pi$
$263$		−	149.067i		−	0.566794i	0.959003	−	0.283397i	$-0.0914614\pi$
$264$	−6.21149			−0.0235284
$265$		0			0
$266$	−77.4439	+	488.760i	−0.291143	+	1.83744i
$267$			43.2599i			0.162022i
$268$	156.123			0.582547
$269$		−	204.113i		−	0.758785i	−0.925236	−	0.379392i	$-0.876133\pi$
$269$		−	204.113i		−	0.758785i	0.925236	−	0.379392i	$-0.123867\pi$
$270$		0			0
$271$	−240.731			−0.888305			−0.444152	−	0.895951i	$-0.646495\pi$
$271$	−240.731			−0.888305			−0.444152	+	0.895951i	$0.646495\pi$
$272$		−	174.397i		−	0.641166i
$273$			246.341i			0.902349i
$274$			17.7812i			0.0648948i
$275$		0			0
$276$			186.910i			0.677211i
$277$		−	458.745i		−	1.65612i	−0.560641	−	0.828059i	$-0.689445\pi$
$277$		−	458.745i		−	1.65612i	0.560641	−	0.828059i	$-0.310555\pi$
$278$	−583.388			−2.09852
$279$			221.225i			0.792922i
$280$		0			0
$281$			249.476i			0.887816i	0.896072	+	0.443908i	$0.146408\pi$
$281$			249.476i			0.887816i	−0.896072	+	0.443908i	$0.853592\pi$
$282$			26.2446i			0.0930659i
$283$		−	365.612i		−	1.29192i	−0.763373	−	0.645958i	$-0.776458\pi$
$283$		−	365.612i		−	1.29192i	0.763373	−	0.645958i	$-0.223542\pi$
$284$			513.321i			1.80747i
$285$		0			0
$286$	−347.014			−1.21334
$287$	−390.086			−1.35918
$288$	−192.479			−0.668330
$289$	163.795			0.566765
$290$		0			0
$291$	92.7128			0.318601
$292$			75.9718i			0.260177i
$293$	−173.088			−0.590745			−0.295373	−	0.955382i	$-0.595444\pi$
$293$	−173.088			−0.590745			−0.295373	+	0.955382i	$0.595444\pi$
$294$	215.945			0.734507
$295$		0			0
$296$	−3.01980			−0.0102020
$297$	−285.835			−0.962406
$298$	−237.803			−0.797996
$299$			257.184i			0.860146i
$300$		0			0
$301$	−420.722			−1.39775
$302$			539.036i			1.78489i
$303$	−243.396			−0.803286
$304$	292.481	+	46.3435i	0.962108	+	0.152446i
$305$		0			0
$306$			134.695i			0.440179i
$307$	452.211			1.47300			0.736500	−	0.676438i	$-0.236477\pi$
$307$	452.211			1.47300			0.736500	+	0.676438i	$0.236477\pi$
$308$			371.223i			1.20527i
$309$	12.4834			0.0403994
$310$		0			0
$311$	179.346			0.576675			0.288337	−	0.957529i	$-0.406898\pi$
$311$	179.346			0.576675			0.288337	+	0.957529i	$0.406898\pi$
$312$	−7.73926			−0.0248053
$313$			137.714i			0.439981i	0.975502	+	0.219990i	$0.0706026\pi$
$313$			137.714i			0.439981i	−0.975502	+	0.219990i	$0.929397\pi$
$314$		−	555.186i		−	1.76811i
$315$		0			0
$316$			563.556i			1.78341i
$317$	602.099			1.89936			0.949682	−	0.313215i	$-0.101406\pi$
$317$	602.099			1.89936			0.949682	+	0.313215i	$0.101406\pi$
$318$	595.717			1.87332
$319$			103.878i			0.325636i
$320$		0			0
$321$	−359.437			−1.11974
$322$	543.474			1.68781
$323$	33.2714	−	209.981i	0.103008	−	0.650096i
$324$	−102.726			−0.317055
$325$		0			0
$326$			488.631i			1.49887i
$327$		−	223.370i		−	0.683088i
$328$		−	12.2552i		−	0.0373635i
$329$	38.6311			0.117420
$330$		0			0
$331$		−	190.514i		−	0.575571i	−0.957695	−	0.287785i	$-0.907081\pi$
$331$		−	190.514i		−	0.575571i	0.957695	−	0.287785i	$-0.0929189\pi$
$332$			524.439i			1.57964i
$333$	−44.4253			−0.133409
$334$	116.020			0.347366
$335$		0			0
$336$		−	311.512i		−	0.927120i
$337$	−394.039			−1.16926			−0.584628	−	0.811301i	$-0.698760\pi$
$337$	−394.039			−1.16926			−0.584628	+	0.811301i	$0.698760\pi$
$338$	48.6475			0.143927
$339$	−53.5235			−0.157886
$340$		0			0
$341$		−	517.429i		−	1.51739i
$342$	−225.896	−	35.7933i	−0.660516	−	0.104659i
$343$			130.523i			0.380535i
$344$		−	13.2177i		−	0.0384237i
$345$		0			0
$346$	−794.648			−2.29667
$347$		−	205.061i		−	0.590953i	−0.955350	−	0.295476i	$-0.904522\pi$
$347$		−	205.061i		−	0.590953i	0.955350	−	0.295476i	$-0.0954784\pi$
$348$			94.0624i			0.270294i
$349$	488.832			1.40066			0.700332	−	0.713817i	$-0.253035\pi$
$349$	488.832			1.40066			0.700332	+	0.713817i	$0.253035\pi$
$350$		0			0
$351$	−356.138			−1.01464
$352$	450.194			1.27896
$353$			466.761i			1.32227i	0.750268	+	0.661134i	$0.229925\pi$
$353$			466.761i			1.32227i	−0.750268	+	0.661134i	$0.770075\pi$
$354$		−	39.6964i		−	0.112137i
$355$		0			0
$356$			81.2242i			0.228158i
$357$	−223.644			−0.626455
$358$			878.849i			2.45489i
$359$	−487.730			−1.35858			−0.679289	−	0.733871i	$-0.737712\pi$
$359$	−487.730			−1.35858			−0.679289	+	0.733871i	$0.737712\pi$
$360$		0			0
$361$	343.317	+	111.599i	0.951017	+	0.309138i
$362$			717.489i			1.98201i
$363$	−50.5573			−0.139276
$364$			462.528i			1.27068i
$365$		0			0
$366$	−92.3748			−0.252390
$367$			63.0494i			0.171797i	0.996304	+	0.0858983i	$0.0273760\pi$
$367$			63.0494i			0.171797i	−0.996304	+	0.0858983i	$0.972624\pi$
$368$		−	325.223i		−	0.883758i
$369$		−	180.291i		−	0.488593i
$370$		0			0
$371$		−	876.875i		−	2.36354i
$372$		−	468.537i		−	1.25951i
$373$	137.833			0.369526			0.184763	−	0.982783i	$-0.440848\pi$
$373$	137.833			0.369526			0.184763	+	0.982783i	$0.440848\pi$
$374$		−	315.041i		−	0.842356i
$375$		0			0
$376$			1.21367i			0.00322784i
$377$			129.427i			0.343308i
$378$			752.582i			1.99096i
$379$		−	316.605i		−	0.835370i	−0.908592	−	0.417685i	$-0.862842\pi$
$379$		−	316.605i		−	0.835370i	0.908592	−	0.417685i	$-0.137158\pi$
$380$		0			0
$381$	127.868			0.335612
$382$	−599.681			−1.56985
$383$	197.083			0.514577			0.257288	−	0.966335i	$-0.417171\pi$
$383$	197.083			0.514577			0.257288	+	0.966335i	$0.417171\pi$
$384$	20.0872			0.0523105
$385$		0			0
$386$	322.609			0.835774
$387$		−	194.451i		−	0.502456i
$388$	174.076			0.448651
$389$	−27.8203			−0.0715175			−0.0357587	−	0.999360i	$-0.511385\pi$
$389$	−27.8203			−0.0715175			−0.0357587	+	0.999360i	$0.511385\pi$
$390$		0			0
$391$	−233.487			−0.597155
$392$	9.98626			0.0254751
$393$	489.003			1.24428
$394$		−	587.343i		−	1.49072i
$395$		0			0
$396$	−171.573			−0.433265
$397$			702.745i			1.77014i	0.465459	+	0.885069i	$0.345889\pi$
$397$			702.745i			1.77014i	−0.465459	+	0.885069i	$0.654111\pi$
$398$	517.944			1.30137
$399$	59.4303	−	375.073i	0.148948	−	0.940033i
$400$		0			0
$401$		−	295.297i		−	0.736401i	−0.929746	−	0.368200i	$-0.879974\pi$
$401$		−	295.297i		−	0.736401i	0.929746	−	0.368200i	$-0.120026\pi$
$402$	−236.665			−0.588720
$403$		−	644.695i		−	1.59974i
$404$	−456.997			−1.13118
$405$		0			0
$406$	273.503			0.673652
$407$	103.907			0.255301
$408$		−	7.02619i		−	0.0172210i
$409$		−	371.735i		−	0.908887i	−0.890775	−	0.454444i	$-0.849838\pi$
$409$		−	371.735i		−	0.908887i	0.890775	−	0.454444i	$-0.150162\pi$
$410$		0			0
$411$		−	13.6452i		−	0.0332001i
$412$	23.4387			0.0568901
$413$	−58.4316			−0.141481
$414$			251.185i			0.606726i
$415$		0			0
$416$	560.923			1.34837
$417$	447.691			1.07360
$418$	528.355	+	83.7177i	1.26401	+	0.200282i
$419$	109.242			0.260721			0.130360	−	0.991467i	$-0.458387\pi$
$419$	109.242			0.260721			0.130360	+	0.991467i	$0.458387\pi$
$420$		0			0
$421$			762.528i			1.81123i	0.424100	+	0.905615i	$0.360591\pi$
$421$			762.528i			1.81123i	−0.424100	+	0.905615i	$0.639409\pi$
$422$		−	303.774i		−	0.719845i
$423$			17.8546i			0.0422096i
$424$	27.5486			0.0649731
$425$		0			0
$426$		−	778.140i		−	1.82662i
$427$			135.972i			0.318437i
$428$	−674.875			−1.57681
$429$	266.298			0.620741
$430$		0			0
$431$			131.893i			0.306016i	0.988225	+	0.153008i	$0.0488960\pi$
$431$			131.893i			0.306016i	−0.988225	+	0.153008i	$0.951104\pi$
$432$	450.356			1.04249
$433$	−401.234			−0.926638			−0.463319	−	0.886191i	$-0.653342\pi$
$433$	−401.234			−0.926638			−0.463319	+	0.886191i	$0.653342\pi$
$434$	−1362.35			−3.13907
$435$		0			0
$436$		−	419.397i		−	0.961919i
$437$	62.0460	−	391.581i	0.141982	−	0.896067i
$438$		−	115.165i		−	0.262934i
$439$			264.338i			0.602136i	0.953603	+	0.301068i	$0.0973432\pi$
$439$			264.338i			0.602136i	−0.953603	+	0.301068i	$0.902657\pi$
$440$		0			0
$441$	146.911			0.333132
$442$		−	392.528i		−	0.888073i
$443$		−	451.274i		−	1.01868i	−0.860566	−	0.509338i	$-0.829890\pi$
$443$		−	451.274i		−	1.01868i	0.860566	−	0.509338i	$-0.170110\pi$
$444$	94.0893			0.211913
$445$		0			0
$446$	595.860			1.33601
$447$	182.489			0.408254
$448$		−	614.843i		−	1.37242i
$449$			156.631i			0.348844i	0.984671	+	0.174422i	$0.0558057\pi$
$449$			156.631i			0.348844i	−0.984671	+	0.174422i	$0.944194\pi$
$450$		0			0
$451$			421.687i			0.935004i
$452$	−100.495			−0.222334
$453$		−	413.655i		−	0.913145i
$454$	−695.318			−1.53154
$455$		0			0
$456$	11.7836	+	1.86711i	0.0258412	+	0.00409454i
$457$		−	906.098i		−	1.98271i	−0.131209	−	0.991355i	$-0.541886\pi$
$457$		−	906.098i		−	1.98271i	0.131209	−	0.991355i	$-0.458114\pi$
$458$	724.000			1.58078
$459$		−	323.324i		−	0.704410i
$460$		0			0
$461$	−169.733			−0.368184			−0.184092	−	0.982909i	$-0.558934\pi$
$461$	−169.733			−0.368184			−0.184092	+	0.982909i	$0.558934\pi$
$462$		−	562.735i		−	1.21804i
$463$			207.991i			0.449225i	0.974448	+	0.224613i	$0.0721117\pi$
$463$			207.991i			0.449225i	−0.974448	+	0.224613i	$0.927888\pi$
$464$		−	163.668i		−	0.352733i
$465$		0			0
$466$		−	987.745i		−	2.11962i
$467$			479.390i			1.02653i	0.858230	+	0.513266i	$0.171565\pi$
$467$			479.390i			1.02653i	−0.858230	+	0.513266i	$0.828435\pi$
$468$	−213.772			−0.456779
$469$			348.363i			0.742778i
$470$		0			0
$471$			426.049i			0.904562i
$472$		−	1.83573i		−	0.00388927i
$473$			454.805i			0.961534i
$474$		−	854.292i		−	1.80230i
$475$		0			0
$476$	−419.912			−0.882168
$477$	405.277			0.849636
$478$	0.136268			0.000285079
$479$	712.482			1.48744			0.743718	−	0.668493i	$-0.233060\pi$
$479$	712.482			1.48744			0.743718	+	0.668493i	$0.233060\pi$
$480$		0			0
$481$	129.464			0.269157
$482$			1203.43i			2.49675i
$483$	−417.061			−0.863480
$484$	−94.9258			−0.196128
$485$		0			0
$486$	−584.462			−1.20260
$487$	−34.4191			−0.0706757			−0.0353379	−	0.999375i	$-0.511251\pi$
$487$	−34.4191			−0.0706757			−0.0353379	+	0.999375i	$0.511251\pi$
$488$	−4.27182			−0.00875373
$489$		−	374.975i		−	0.766819i
$490$		0			0
$491$	361.608			0.736472			0.368236	−	0.929732i	$-0.379962\pi$
$491$	361.608			0.736472			0.368236	+	0.929732i	$0.379962\pi$
$492$			381.842i			0.776101i
$493$	−117.502			−0.238341
$494$	658.308	+	104.309i	1.33261	+	0.211151i
$495$		0			0
$496$			815.252i			1.64365i
$497$	−1145.40			−2.30462
$498$		−	794.994i		−	1.59637i
$499$	189.977			0.380715			0.190358	−	0.981715i	$-0.439035\pi$
$499$	189.977			0.380715			0.190358	+	0.981715i	$0.439035\pi$
$500$		0			0
$501$	−89.0336			−0.177712
$502$	−405.077			−0.806926
$503$		−	687.468i		−	1.36674i	−0.730074	−	0.683368i	$-0.760514\pi$
$503$		−	687.468i		−	1.36674i	0.730074	−	0.683368i	$-0.239486\pi$
$504$			11.1262i			0.0220758i
$505$		0			0
$506$		−	587.502i		−	1.16107i
$507$	−37.3319			−0.0736330
$508$	240.084			0.472607
$509$			348.233i			0.684151i	0.939672	+	0.342076i	$0.111130\pi$
$509$			348.233i			0.684151i	−0.939672	+	0.342076i	$0.888870\pi$
$510$		0			0
$511$	−169.519			−0.331740
$512$	−727.241			−1.42039
$513$	542.247	+	85.9189i	1.05701	+	0.167483i
$514$	−564.061			−1.09740
$515$		0			0
$516$			411.831i			0.798122i
$517$		−	41.7607i		−	0.0807750i
$518$		−	273.581i		−	0.528149i
$519$	609.811			1.17497
$520$		0			0
$521$			222.778i			0.427597i	0.976878	+	0.213798i	$0.0685835\pi$
$521$			222.778i			0.427597i	−0.976878	+	0.213798i	$0.931416\pi$
$522$			126.408i			0.242162i
$523$	45.6247			0.0872365			0.0436182	−	0.999048i	$-0.486111\pi$
$523$	45.6247			0.0872365			0.0436182	+	0.999048i	$0.486111\pi$
$524$	918.147			1.75219
$525$		0			0
$526$			424.278i			0.806612i
$527$	585.295			1.11062
$528$	−336.748			−0.637781
$529$	93.5830			0.176905
$530$		0			0
$531$		−	27.0061i		−	0.0508590i
$532$	111.586	−	704.233i	0.209747	−	1.32375i
$533$			525.404i			0.985749i
$534$		−	123.127i		−	0.230576i
$535$		0			0
$536$	−10.9444			−0.0204187
$537$		−	674.427i		−	1.25592i
$538$			580.952i			1.07984i
$539$	−343.614			−0.637503
$540$		0			0
$541$	671.919			1.24199			0.620997	−	0.783813i	$-0.286728\pi$
$541$	671.919			1.24199			0.620997	+	0.783813i	$0.286728\pi$
$542$	685.174			1.26416
$543$		−	550.599i		−	1.01399i
$544$			509.241i			0.936105i
$545$		0			0
$546$		−	701.143i		−	1.28415i
$547$	−910.710			−1.66492			−0.832459	−	0.554087i	$-0.813068\pi$
$547$	−910.710			−1.66492			−0.832459	+	0.554087i	$0.813068\pi$
$548$		−	25.6201i		−	0.0467520i
$549$	−62.8441			−0.114470
$550$		0			0
$551$	31.2246	−	197.063i	0.0566689	−	0.357646i
$552$		−	13.1027i		−	0.0237368i
$553$	−1257.49			−2.27394
$554$			1305.69i			2.35684i
$555$		0			0
$556$	840.579			1.51183
$557$			246.347i			0.442275i	0.975243	+	0.221138i	$0.0709770\pi$
$557$			246.347i			0.442275i	−0.975243	+	0.221138i	$0.929023\pi$
$558$		−	629.657i		−	1.12842i
$559$			566.668i			1.01372i
$560$		0			0
$561$			241.762i			0.430948i
$562$		−	710.066i		−	1.26346i
$563$	772.769			1.37259			0.686296	−	0.727322i	$-0.259235\pi$
$563$	772.769			1.37259			0.686296	+	0.727322i	$0.259235\pi$
$564$		−	37.8147i		−	0.0670474i
$565$		0			0
$566$			1040.61i			1.83854i
$567$		−	229.217i		−	0.404262i
$568$		−	35.9847i		−	0.0633533i
$569$			123.962i			0.217860i	0.994049	+	0.108930i	$0.0347424\pi$
$569$			123.962i			0.217860i	−0.994049	+	0.108930i	$0.965258\pi$
$570$		0			0
$571$	535.555			0.937925			0.468963	−	0.883218i	$-0.344628\pi$
$571$	535.555			0.937925			0.468963	+	0.883218i	$0.344628\pi$
$572$	499.998			0.874122
$573$	460.194			0.803131
$574$	1110.27			1.93427
$575$		0			0
$576$	284.170			0.493350
$577$			210.295i			0.364463i	0.983256	+	0.182231i	$0.0583320\pi$
$577$			210.295i			0.364463i	−0.983256	+	0.182231i	$0.941668\pi$
$578$	−466.198			−0.806570
$579$	−247.569			−0.427580
$580$		0			0
$581$	−1170.20			−2.01412
$582$	−263.882			−0.453405
$583$	−947.912			−1.62592
$584$		−	5.32575i		−	0.00911944i
$585$		0			0
$586$	492.649			0.840697
$587$			508.069i			0.865534i	0.901506	+	0.432767i	$0.142463\pi$
$587$			508.069i			0.865534i	−0.901506	+	0.432767i	$0.857537\pi$
$588$	−311.146			−0.529160
$589$	−155.534	+	981.596i	−0.264064	+	1.66655i
$590$		0			0
$591$			450.726i			0.762649i
$592$	−163.715			−0.276545
$593$		−	32.8735i		−	0.0554358i	−0.999616	−	0.0277179i	$-0.991176\pi$
$593$		−	32.8735i		−	0.0554358i	0.999616	−	0.0277179i	$-0.00882402\pi$
$594$	813.550			1.36961
$595$		0			0
$596$	342.640			0.574899
$597$	−397.469			−0.665777
$598$		−	732.003i		−	1.22408i
$599$			225.201i			0.375961i	0.982173	+	0.187981i	$0.0601942\pi$
$599$			225.201i			0.375961i	−0.982173	+	0.187981i	$0.939806\pi$
$600$		0			0
$601$		−	169.930i		−	0.282746i	−0.989956	−	0.141373i	$-0.954848\pi$
$601$		−	169.930i		−	0.282746i	0.989956	−	0.141373i	$-0.0451517\pi$
$602$	1197.47			1.98915
$603$	−161.007			−0.267011
$604$		−	776.674i		−	1.28588i
$605$		0			0
$606$	692.760			1.14317
$607$	522.217			0.860325			0.430162	−	0.902752i	$-0.358456\pi$
$607$	522.217			0.860325			0.430162	+	0.902752i	$0.358456\pi$
$608$	−854.047	−	135.324i	−1.40468	−	0.222572i
$609$	−209.885			−0.344640
$610$		0			0
$611$		−	52.0320i		−	0.0851588i
$612$		−	194.076i		−	0.317118i
$613$		−	639.666i		−	1.04350i	−0.853098	−	0.521750i	$-0.825279\pi$
$613$		−	639.666i		−	1.04350i	0.853098	−	0.521750i	$-0.174721\pi$
$614$	−1287.09			−2.09625
$615$		0			0
$616$		−	26.0233i		−	0.0422457i
$617$		−	200.342i		−	0.324704i	−0.986733	−	0.162352i	$-0.948092\pi$
$617$		−	200.342i		−	0.324704i	0.986733	−	0.162352i	$-0.0519079\pi$
$618$	−35.5306			−0.0574929
$619$	347.156			0.560833			0.280417	−	0.959878i	$-0.409527\pi$
$619$	347.156			0.560833			0.280417	+	0.959878i	$0.409527\pi$
$620$		0			0
$621$		−	602.949i		−	0.970932i
$622$	−510.459			−0.820674
$623$	−181.239			−0.290914
$624$	−419.574			−0.672395
$625$		0			0
$626$		−	391.965i		−	0.626143i
$627$	−405.458	−	64.2448i	−0.646664	−	0.102464i
$628$			799.944i			1.27380i
$629$			117.536i			0.186861i
$630$		0			0
$631$	204.105			0.323463			0.161732	−	0.986835i	$-0.448292\pi$
$631$	204.105			0.323463			0.161732	+	0.986835i	$0.448292\pi$
$632$		−	39.5062i		−	0.0625099i
$633$			233.116i			0.368271i
$634$	−1713.71			−2.70301
$635$		0			0
$636$	−858.343			−1.34960
$637$	−428.129			−0.672102
$638$		−	295.660i		−	0.463416i
$639$		−	529.382i		−	0.828454i
$640$		0			0
$641$			334.435i			0.521740i	0.965374	+	0.260870i	$0.0840094\pi$
$641$			334.435i			0.521740i	−0.965374	+	0.260870i	$0.915991\pi$
$642$	1023.04			1.59352
$643$		−	22.8316i		−	0.0355079i	−0.999842	−	0.0177540i	$-0.994348\pi$
$643$		−	22.8316i		−	0.0355079i	0.999842	−	0.0177540i	$-0.00565156\pi$
$644$	−783.069			−1.21595
$645$		0			0
$646$	−94.6981	+	597.653i	−0.146591	+	0.925160i
$647$		−	280.968i		−	0.434263i	−0.976142	−	0.217131i	$-0.930330\pi$
$647$		−	280.968i		−	0.434263i	0.976142	−	0.217131i	$-0.0696700\pi$
$648$	7.20125			0.0111130
$649$			63.1653i			0.0973271i
$650$		0			0
$651$	1045.47			1.60594
$652$		−	704.048i		−	1.07983i
$653$			891.423i			1.36512i	0.730830	+	0.682559i	$0.239133\pi$
$653$			891.423i			1.36512i	−0.730830	+	0.682559i	$0.760867\pi$
$654$			635.761i			0.972112i
$655$		0			0
$656$		−	664.402i		−	1.01281i
$657$		−	78.3489i		−	0.119253i
$658$	−109.953			−0.167102
$659$		−	734.584i		−	1.11470i	−0.830279	−	0.557348i	$-0.811819\pi$
$659$		−	734.584i		−	1.11470i	0.830279	−	0.557348i	$-0.188181\pi$
$660$		0			0
$661$			968.622i			1.46539i	0.680558	+	0.732694i	$0.261737\pi$
$661$			968.622i			1.46539i	−0.680558	+	0.732694i	$0.738263\pi$
$662$			542.246i			0.819102i
$663$			301.225i			0.454337i
$664$		−	36.7641i		−	0.0553676i
$665$		0			0
$666$	126.445			0.189857
$667$	−219.123			−0.328520
$668$	−167.168			−0.250252
$669$	−457.262			−0.683500
$670$		0			0
$671$	146.988			0.219058
$672$			909.619i			1.35360i
$673$	948.399			1.40921			0.704606	−	0.709599i	$-0.251124\pi$
$673$	948.399			1.40921			0.704606	+	0.709599i	$0.251124\pi$
$674$	1121.53			1.66398
$675$		0			0
$676$	−70.0940			−0.103689
$677$	−82.3399			−0.121625			−0.0608123	−	0.998149i	$-0.519369\pi$
$677$	−82.3399			−0.121625			−0.0608123	+	0.998149i	$0.519369\pi$
$678$	152.340			0.224690
$679$			388.425i			0.572054i
$680$		0			0
$681$	533.586			0.783533
$682$			1472.72i			2.15941i
$683$	−10.8842			−0.0159359			−0.00796796	−	0.999968i	$-0.502536\pi$
$683$	−10.8842			−0.0159359			−0.00796796	+	0.999968i	$0.502536\pi$
$684$	325.485	+	51.5730i	0.475855	+	0.0753991i
$685$		0			0
$686$		−	371.499i		−	0.541544i
$687$	−555.596			−0.808727
$688$		−	716.583i		−	1.04155i
$689$	−1181.06			−1.71416
$690$		0			0
$691$	188.047			0.272138			0.136069	−	0.990699i	$-0.456553\pi$
$691$	188.047			0.272138			0.136069	+	0.990699i	$0.456553\pi$
$692$	1144.97			1.65459
$693$		−	382.838i		−	0.552436i
$694$			583.649i			0.840993i
$695$		0			0
$696$		−	6.59393i		−	0.00947404i
$697$	−476.995			−0.684354
$698$	−1391.33			−1.99330
$699$			757.993i			1.08440i
$700$		0			0
$701$	−1184.52			−1.68976			−0.844879	−	0.534957i	$-0.820328\pi$
$701$	−1184.52			−1.68976			−0.844879	+	0.534957i	$0.820328\pi$
$702$	1013.65			1.44394
$703$	−197.119	−	31.2335i	−0.280397	−	0.0444289i
$704$	−664.652			−0.944108
$705$		0			0
$706$		−	1328.51i		−	1.88174i
$707$		−	1019.72i		−	1.44232i
$708$			57.1968i			0.0807864i
$709$	541.289			0.763455			0.381727	−	0.924275i	$-0.375329\pi$
$709$	541.289			0.763455			0.381727	+	0.924275i	$0.375329\pi$
$710$		0			0
$711$		−	581.189i		−	0.817425i
$712$		−	5.69395i		−	0.00799713i
$713$	1091.48			1.53083
$714$	636.542			0.891516
$715$		0			0
$716$		−	1266.30i		−	1.76857i
$717$	−0.104571			−0.000145846
$718$	1388.19			1.93341
$719$	388.854			0.540826			0.270413	−	0.962744i	$-0.412840\pi$
$719$	388.854			0.540826			0.270413	+	0.962744i	$0.412840\pi$
$720$		0			0
$721$			52.2998i			0.0725379i
$722$	−977.159	−	317.636i	−1.35341	−	0.439939i
$723$		−	923.512i		−	1.27733i
$724$		−	1033.80i		−	1.42790i
$725$		0			0
$726$	143.897			0.198206
$727$		−	1450.46i		−	1.99513i	−0.0697458	−	0.997565i	$-0.522219\pi$
$727$		−	1450.46i		−	1.99513i	0.0697458	−	0.997565i	$-0.477781\pi$
$728$		−	32.4240i		−	0.0445384i
$729$	673.955			0.924493
$730$		0			0
$731$	−514.457			−0.703772
$732$	133.099			0.181829
$733$			873.057i			1.19107i	0.803328	+	0.595537i	$0.203061\pi$
$733$			873.057i			1.19107i	−0.803328	+	0.595537i	$0.796939\pi$
$734$		−	179.453i		−	0.244486i
$735$		0			0
$736$			949.653i			1.29029i
$737$	376.584			0.510969
$738$			513.149i			0.695323i
$739$	456.992			0.618393			0.309196	−	0.950998i	$-0.399940\pi$
$739$	456.992			0.618393			0.309196	+	0.950998i	$0.399940\pi$
$740$		0			0
$741$	−505.184	−	80.0463i	−0.681760	−	0.108025i
$742$			2495.78i			3.36359i
$743$	−60.6398			−0.0816148			−0.0408074	−	0.999167i	$-0.512993\pi$
$743$	−60.6398			−0.0816148			−0.0408074	+	0.999167i	$0.512993\pi$
$744$			32.8453i			0.0441468i
$745$		0			0
$746$	−392.304			−0.525877
$747$		−	540.848i		−	0.724027i
$748$			453.930i			0.606858i
$749$		−	1505.88i		−	2.01052i
$750$		0			0
$751$			1177.72i			1.56821i	0.620629	+	0.784104i	$0.286877\pi$
$751$			1177.72i			1.56821i	−0.620629	+	0.784104i	$0.713123\pi$
$752$			65.7974i			0.0874965i
$753$	310.855			0.412822
$754$		−	368.379i		−	0.488567i
$755$		0			0
$756$		−	1084.36i		−	1.43434i
$757$			112.664i			0.148830i	0.997227	+	0.0744151i	$0.0237090\pi$
$757$			112.664i			0.148830i	−0.997227	+	0.0744151i	$0.976291\pi$
$758$			901.130i			1.18883i
$759$			450.848i			0.594002i
$760$		0			0
$761$	−857.698			−1.12707			−0.563534	−	0.826093i	$-0.690558\pi$
$761$	−857.698			−1.12707			−0.563534	+	0.826093i	$0.690558\pi$
$762$	−363.942			−0.477615
$763$	935.818			1.22650
$764$	864.055			1.13096
$765$		0			0
$766$	−560.943			−0.732301
$767$			78.7012i			0.102609i
$768$	529.853			0.689912
$769$	146.990			0.191144			0.0955719	−	0.995423i	$-0.469532\pi$
$769$	146.990			0.191144			0.0955719	+	0.995423i	$0.469532\pi$
$770$		0			0
$771$	432.859			0.561426
$772$	−464.833			−0.602115
$773$	−624.217			−0.807525			−0.403762	−	0.914864i	$-0.632298\pi$
$773$	−624.217			−0.807525			−0.403762	+	0.914864i	$0.632298\pi$
$774$			553.451i			0.715052i
$775$		0			0
$776$	−12.2031			−0.0157256
$777$			209.946i			0.270200i
$778$	79.1829			0.101778
$779$	126.755	−	799.967i	0.162715	−	1.02692i
$780$		0			0
$781$			1238.19i			1.58538i
$782$	664.558			0.849819
$783$		−	303.433i		−	0.387526i
$784$	541.392			0.690552
$785$		0			0
$786$	−1391.81			−1.77076
$787$	551.370			0.700597			0.350298	−	0.936638i	$-0.386080\pi$
$787$	551.370			0.700597			0.350298	+	0.936638i	$0.386080\pi$
$788$			846.277i			1.07396i
$789$		−	325.590i		−	0.412661i
$790$		0			0
$791$		−	224.239i		−	0.283488i
$792$	12.0275			0.0151863
$793$	183.140			0.230946
$794$		−	2000.17i		−	2.51911i
$795$		0			0
$796$	−746.283			−0.937541
$797$	202.295			0.253821			0.126911	−	0.991914i	$-0.459494\pi$
$797$	202.295			0.253821			0.126911	+	0.991914i	$0.459494\pi$
$798$	−169.152	+	1067.54i	−0.211970	+	1.33777i
$799$	47.2380			0.0591214
$800$		0			0
$801$		−	83.7656i		−	0.104576i
$802$			840.481i			1.04798i
$803$			183.252i			0.228209i
$804$	341.001			0.424130
$805$		0			0
$806$			1834.95i			2.27661i
$807$		−	445.821i		−	0.552443i
$808$	32.0363			0.0396489
$809$	1164.20			1.43906			0.719531	−	0.694461i	$-0.244357\pi$
$809$	1164.20			1.43906			0.719531	+	0.694461i	$0.244357\pi$
$810$		0			0
$811$		−	617.970i		−	0.761985i	−0.924578	−	0.380993i	$-0.875582\pi$
$811$		−	617.970i		−	0.761985i	0.924578	−	0.380993i	$-0.124418\pi$
$812$	−394.079			−0.485318
$813$	−525.801			−0.646741
$814$	−295.744			−0.363322
$815$		0			0
$816$		−	380.916i		−	0.466809i
$817$	136.710	−	862.795i	0.167331	−	1.05605i
$818$			1058.04i			1.29345i
$819$		−	477.000i		−	0.582417i
$820$		0			0
$821$	255.141			0.310769			0.155384	−	0.987854i	$-0.450338\pi$
$821$	255.141			0.310769			0.155384	+	0.987854i	$0.450338\pi$
$822$			38.8374i			0.0472474i
$823$		−	658.923i		−	0.800636i	−0.916376	−	0.400318i	$-0.868900\pi$
$823$		−	658.923i		−	0.800636i	0.916376	−	0.400318i	$-0.131100\pi$
$824$	−1.64309			−0.00199405
$825$		0			0
$826$	166.310			0.201343
$827$	−633.264			−0.765736			−0.382868	−	0.923803i	$-0.625064\pi$
$827$	−633.264			−0.765736			−0.382868	+	0.923803i	$0.625064\pi$
$828$		−	361.921i		−	0.437103i
$829$		−	459.961i		−	0.554838i	−0.960749	−	0.277419i	$-0.910521\pi$
$829$		−	459.961i		−	0.554838i	0.960749	−	0.277419i	$-0.0894790\pi$
$830$		0			0
$831$		−	1001.98i		−	1.20576i
$832$	−828.128			−0.995346
$833$		−	388.682i		−	0.466605i
$834$	−1274.23			−1.52785
$835$		0			0
$836$	−761.284	−	120.625i	−0.910627	−	0.144289i
$837$			1511.44i			1.80578i
$838$	−310.927			−0.371035
$839$		−	1033.89i		−	1.23228i	−0.787635	−	0.616142i	$-0.788695\pi$
$839$		−	1033.89i		−	1.23228i	0.787635	−	0.616142i	$-0.211305\pi$
$840$		0			0
$841$	730.727			0.868878
$842$		−	2170.33i		−	2.57759i
$843$			544.903i			0.646386i
$844$			437.696i			0.518597i
$845$		0			0
$846$		−	50.8184i		−	0.0600690i
$847$		−	211.812i		−	0.250073i
$848$	1493.51			1.76122
$849$		−	798.565i		−	0.940595i
$850$		0			0
$851$			219.186i			0.257563i
$852$			1121.19i			1.31595i
$853$		−	333.243i		−	0.390671i	−0.980736	−	0.195336i	$-0.937420\pi$
$853$		−	333.243i		−	0.390671i	0.980736	−	0.195336i	$-0.0625796\pi$
$854$		−	387.008i		−	0.453171i
$855$		0			0
$856$	47.3099			0.0552685
$857$	−55.2946			−0.0645211			−0.0322605	−	0.999479i	$-0.510271\pi$
$857$	−55.2946			−0.0645211			−0.0322605	+	0.999479i	$0.510271\pi$
$858$	−757.944			−0.883385
$859$	−417.893			−0.486488			−0.243244	−	0.969965i	$-0.578212\pi$
$859$	−417.893			−0.486488			−0.243244	+	0.969965i	$0.578212\pi$
$860$		0			0
$861$	−852.020			−0.989570
$862$		−	375.396i		−	0.435495i
$863$	1042.76			1.20830			0.604150	−	0.796871i	$-0.293513\pi$
$863$	1042.76			1.20830			0.604150	+	0.796871i	$0.293513\pi$
$864$	−1315.04			−1.52204
$865$		0			0
$866$	1142.00			1.31871
$867$	357.759			0.412640
$868$	1962.96			2.26147
$869$			1359.36i			1.56428i
$870$		0			0
$871$	469.208			0.538700
$872$			29.4004i			0.0337161i
$873$	−179.523			−0.205639
$874$	−176.597	+	1114.53i	−0.202056	+	1.27520i
$875$		0			0
$876$			165.937i			0.189425i
$877$	−143.302			−0.163400			−0.0817002	−	0.996657i	$-0.526035\pi$
$877$	−143.302			−0.163400			−0.0817002	+	0.996657i	$0.526035\pi$
$878$		−	752.366i		−	0.856908i
$879$	−378.057			−0.430099
$880$		0			0
$881$	−944.916			−1.07255			−0.536275	−	0.844043i	$-0.680169\pi$
$881$	−944.916			−1.07255			−0.536275	+	0.844043i	$0.680169\pi$
$882$	−418.142			−0.474084
$883$		−	1406.88i		−	1.59329i	−0.604446	−	0.796646i	$-0.706605\pi$
$883$		−	1406.88i		−	1.59329i	0.604446	−	0.796646i	$-0.293395\pi$
$884$			565.577i			0.639793i
$885$		0			0
$886$			1284.43i			1.44969i
$887$	−437.734			−0.493500			−0.246750	−	0.969079i	$-0.579363\pi$
$887$	−437.734			−0.493500			−0.246750	+	0.969079i	$0.579363\pi$
$888$	−6.59582			−0.00742772
$889$			535.710i			0.602599i
$890$		0			0
$891$	−247.786			−0.278098
$892$	−858.549			−0.962499
$893$	−12.5528	+	79.2227i	−0.0140569	+	0.0887152i
$894$	−519.406			−0.580991
$895$		0			0
$896$			84.1564i			0.0939246i
$897$			561.737i			0.626240i
$898$		−	445.807i		−	0.496445i
$899$	549.287			0.610998
$900$		0			0
$901$		−	1072.24i		−	1.19005i
$902$		−	1200.22i		−	1.33062i
$903$	−918.936			−1.01765
$904$	7.04488			0.00779300
$905$		0			0
$906$			1177.36i			1.29951i
$907$	174.898			0.192832			0.0964158	−	0.995341i	$-0.469262\pi$
$907$	174.898			0.192832			0.0964158	+	0.995341i	$0.469262\pi$
$908$	1001.85			1.10336
$909$	471.296			0.518478
$910$		0			0
$911$		−	475.064i		−	0.521476i	−0.965410	−	0.260738i	$-0.916034\pi$
$911$		−	475.064i		−	0.521476i	0.965410	−	0.260738i	$-0.0839658\pi$
$912$	638.833	+	101.223i	0.700475	+	0.110990i
$913$			1265.00i			1.38555i
$914$			2578.96i			2.82162i
$915$		0			0
$916$	−1043.18			−1.13884
$917$			2048.70i			2.23413i
$918$			920.254i			1.00246i
$919$	−456.862			−0.497130			−0.248565	−	0.968615i	$-0.579959\pi$
$919$	−456.862			−0.497130			−0.248565	+	0.968615i	$0.579959\pi$
$920$		0			0
$921$	987.714			1.07244
$922$	483.098			0.523967
$923$			1542.73i			1.67143i
$924$			810.820i			0.877511i
$925$		0			0
$926$		−	591.991i		−	0.639299i
$927$	−24.1721			−0.0260756
$928$			477.912i			0.514991i
$929$	−714.272			−0.768861			−0.384430	−	0.923154i	$-0.625602\pi$
$929$	−714.272			−0.768861			−0.384430	+	0.923154i	$0.625602\pi$
$930$		0			0
$931$	651.858	+	103.287i	0.700170	+	0.110942i
$932$			1423.20i			1.52704i
$933$	391.725			0.419855
$934$		−	1364.45i		−	1.46087i
$935$		0			0
$936$	14.9858			0.0160105
$937$		−	711.794i		−	0.759652i	−0.925058	−	0.379826i	$-0.875984\pi$
$937$		−	711.794i		−	0.759652i	0.925058	−	0.379826i	$-0.124016\pi$
$938$		−	991.520i		−	1.05706i
$939$			300.793i			0.320334i
$940$		0			0
$941$			1055.43i			1.12161i	0.827950	+	0.560803i	$0.189507\pi$
$941$			1055.43i			1.12161i	−0.827950	+	0.560803i	$0.810493\pi$
$942$		−	1212.63i		−	1.28729i
$943$	−889.520			−0.943287
$944$		−	99.5221i		−	0.105426i
$945$		0			0
$946$		−	1294.48i		−	1.36837i
$947$			611.188i			0.645394i	0.946502	+	0.322697i	$0.104589\pi$
$947$			611.188i			0.645394i	−0.946502	+	0.322697i	$0.895411\pi$
$948$			1230.91i			1.29843i
$949$			228.324i			0.240595i
$950$		0			0
$951$	1315.10			1.38286
$952$	29.4365			0.0309207
$953$	−847.554			−0.889354			−0.444677	−	0.895691i	$-0.646682\pi$
$953$	−847.554			−0.889354			−0.444677	+	0.895691i	$0.646682\pi$
$954$	−1153.51			−1.20913
$955$		0			0
$956$	−0.196342			−0.000205379
$957$			226.889i			0.237083i
$958$	−2027.89			−2.11679
$959$	57.1673			0.0596114
$960$		0			0
$961$	−1775.07			−1.84711
$962$	−368.485			−0.383040
$963$	695.991			0.722732
$964$		−	1733.98i		−	1.79873i
$965$		0			0
$966$	1187.05			1.22883
$967$			293.882i			0.303911i	0.988387	+	0.151955i	$0.0485570\pi$
$967$			293.882i			0.303911i	−0.988387	+	0.151955i	$0.951443\pi$
$968$	6.65446			0.00687444
$969$	72.6711	−	458.638i	0.0749960	−	0.473311i
$970$		0			0
$971$		−	1289.96i		−	1.32849i	−0.747517	−	0.664243i	$-0.768754\pi$
$971$		−	1289.96i		−	1.32849i	0.747517	−	0.664243i	$-0.231246\pi$
$972$	842.127			0.866386
$973$			1875.62i			1.92767i
$974$	97.9645			0.100580
$975$		0			0
$976$	−231.591			−0.237286
$977$	−362.429			−0.370961			−0.185481	−	0.982648i	$-0.559384\pi$
$977$	−362.429			−0.370961			−0.185481	+	0.982648i	$0.559384\pi$
$978$			1067.26i			1.09127i
$979$			195.922i			0.200124i
$980$		0			0
$981$			432.519i			0.440896i
$982$	−1029.22			−1.04808
$983$	−236.553			−0.240644			−0.120322	−	0.992735i	$-0.538393\pi$
$983$	−236.553			−0.240644			−0.120322	+	0.992735i	$0.538393\pi$
$984$		−	26.7677i		−	0.0272030i
$985$		0			0
$986$	334.438			0.339187
$987$	84.3776			0.0854890
$988$	−948.528	−	150.294i	−0.960048	−	0.152119i
$989$	−959.381			−0.970052
$990$		0			0
$991$		−	328.650i		−	0.331635i	−0.986156	−	0.165817i	$-0.946974\pi$
$991$		−	328.650i		−	0.331635i	0.986156	−	0.165817i	$-0.0530262\pi$
$992$		−	2380.54i		−	2.39974i
$993$		−	416.118i		−	0.419051i
$994$	3260.06			3.27973
$995$		0			0
$996$			1145.47i			1.15007i
$997$			537.051i			0.538667i	0.963047	+	0.269333i	$0.0868033\pi$
$997$			537.051i			0.538667i	−0.963047	+	0.269333i	$0.913197\pi$
$998$	−540.718			−0.541801
$999$	−303.520			−0.303824

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.3.d.c.474.3		24
5.2	odd	4		95.3.c.a.56.2	✓	12
5.3	odd	4		475.3.c.g.151.11		12
5.4	even	2	inner	475.3.d.c.474.22		24
15.2	even	4		855.3.e.a.721.11		12
19.18	odd	2	inner	475.3.d.c.474.21		24
20.7	even	4		1520.3.h.a.721.8		12
95.18	even	4		475.3.c.g.151.2		12
95.37	even	4		95.3.c.a.56.11	yes	12
95.94	odd	2	inner	475.3.d.c.474.4		24
285.227	odd	4		855.3.e.a.721.2		12
380.227	odd	4		1520.3.h.a.721.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.3.c.a.56.2	✓	12	5.2	odd	4
95.3.c.a.56.11	yes	12	95.37	even	4
475.3.c.g.151.2		12	95.18	even	4
475.3.c.g.151.11		12	5.3	odd	4
475.3.d.c.474.3		24	1.1	even	1	trivial
475.3.d.c.474.4		24	95.94	odd	2	inner
475.3.d.c.474.21		24	19.18	odd	2	inner
475.3.d.c.474.22		24	5.4	even	2	inner
855.3.e.a.721.2		12	285.227	odd	4
855.3.e.a.721.11		12	15.2	even	4
1520.3.h.a.721.5		12	380.227	odd	4
1520.3.h.a.721.8		12	20.7	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:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	475.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.84623 q^{2} +2.18419 q^{3} +4.10101 q^{4} -6.21669 q^{6} +9.15076i q^{7} -0.287487 q^{8} -4.22932 q^{9} +O(q^{10})\)	\(q-2.84623 q^{2} +2.18419 q^{3} +4.10101 q^{4} -6.21669 q^{6} +9.15076i q^{7} -0.287487 q^{8} -4.22932 q^{9} +9.89207 q^{11} +8.95737 q^{12} +12.3251 q^{13} -26.0451i q^{14} -15.5858 q^{16} +11.1895i q^{17} +12.0376 q^{18} +(-18.7659 - 2.97345i) q^{19} +19.9870i q^{21} -28.1551 q^{22} +20.8666i q^{23} -0.627927 q^{24} -35.0800 q^{26} -28.8953 q^{27} +37.5273i q^{28} +10.5011i q^{29} -52.3075i q^{31} +45.5106 q^{32} +21.6061 q^{33} -31.8479i q^{34} -17.3445 q^{36} +10.5041 q^{37} +(53.4120 + 8.46312i) q^{38} +26.9203 q^{39} +42.6288i q^{41} -56.8874i q^{42} +45.9768i q^{43} +40.5674 q^{44} -59.3912i q^{46} -4.22163i q^{47} -34.0423 q^{48} -34.7363 q^{49} +24.4400i q^{51} +50.5453 q^{52} -95.8254 q^{53} +82.2427 q^{54} -2.63073i q^{56} +(-40.9882 - 6.49458i) q^{57} -29.8886i q^{58} +6.38544i q^{59} +14.8591 q^{61} +148.879i q^{62} -38.7015i q^{63} -67.1904 q^{64} -61.4960 q^{66} +38.0693 q^{67} +45.8882i q^{68} +45.5767i q^{69} +125.169i q^{71} +1.21588 q^{72} +18.5252i q^{73} -29.8971 q^{74} +(-76.9590 - 12.1941i) q^{76} +90.5199i q^{77} -76.6214 q^{78} +137.419i q^{79} -25.0489 q^{81} -121.331i q^{82} +127.881i q^{83} +81.9667i q^{84} -130.860i q^{86} +22.9364i q^{87} -2.84385 q^{88} +19.8059i q^{89} +112.784i q^{91} +85.5743i q^{92} -114.249i q^{93} +12.0157i q^{94} +99.4037 q^{96} +42.4473 q^{97} +98.8675 q^{98} -41.8368 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 24 q^{4} - 56 q^{6} + 96 q^{9}+O(q^{10}) \) 24 * q + 24 * q^4 - 56 * q^6 + 96 * q^9	\( 24 q + 24 q^{4} - 56 q^{6} + 96 q^{9} + 64 q^{11} - 88 q^{16} - 16 q^{19} - 200 q^{24} + 216 q^{26} - 160 q^{36} - 152 q^{39} + 512 q^{44} - 144 q^{49} + 152 q^{54} - 592 q^{61} - 376 q^{64} + 304 q^{66} - 272 q^{74} + 496 q^{76} - 744 q^{81} - 88 q^{96} + 624 q^{99}+O(q^{100}) \) 24 * q + 24 * q^4 - 56 * q^6 + 96 * q^9 + 64 * q^11 - 88 * q^16 - 16 * q^19 - 200 * q^24 + 216 * q^26 - 160 * q^36 - 152 * q^39 + 512 * q^44 - 144 * q^49 + 152 * q^54 - 592 * q^61 - 376 * q^64 + 304 * q^66 - 272 * q^74 + 496 * q^76 - 744 * q^81 - 88 * q^96 + 624 * q^99

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